Must know Formula on Physics' for AIPMT Parallelogram law of vector addition :
R = Let α be the angle, which the resultant makes with
Resolution of vectors
Scalar Product (Dot Product) :
If and are two vectors having angle θ between them, then their scalar (Dot) product written as and read as dot is defined as Properties of Scalar Product (i) It is always a scalar and it will be positive if angle between them is acute, negative if angle between them is obtuse and zero if angle between them is 90°.
(ii) It obeys commutative law
(iii) It obey distributive law
(iv) By definition
cos θ = ⇒ θ = where θ is the angle between two vectors. (v) Scalar (Dot) product of two mutually perpendicular vectors is zero i.e.,
(2) Must know Formula on Physics' for AIPMT
(vi) Scalar (Dot) product will be maximum when θ = 0° i.e.,
vectors are parallel to each other.
(vii) If and are unit vectors, then and
(viii) Dot product of unit vectors
(ix) Square of a vector (x) If the two vectors and , in terms of their rectangular components, are
and , then
= AxBx + AyBy + AzBz Vector Product (Cross Product)
If and are two vectors, then their vector product is written as is vector and is read as cross . It is defined as
...... (7)
where is unit vector along the direction of
Properties of Vector Product
(i) Cross product of two vectors is not commutative
;
(ii) Cross product is not associative
(iii) Cross product obey distributive law
(iv) If θ = 0 or π it means two vectors are collinear.
and conversely, if , then the vector are parallel provided are non-zero vectors.
(3) Must know Formula on Physics' for AIPMT This may be regarded as a test to decide whether the given two vectors are parallel or not.
(v) If θ = 90°, then
(vi) The vector product of any vector with itself is
(vii) If , then or or
(viii) If and are unit vectors, then
(ix) Cross product of unit vectors
(4) Must know Formula on Physics' for AIPMT
(5) Must know Formula on Physics' for AIPMT
Representation of errors Errors can be expressed in the following ways :
Absolute Propagation of errors in mathematical operations Rule I : If X = A + B or X = A – B and if ±∆A and ±∆B represent the absolute errors in A and B respectively, then maximum absolute error in X = ∆X = ∆A + ∆B and
Maximum percentage error = ....(6) The result will be written as X ± ∆X (in terms of absolute error)
or X %(in term of percentage error) Rule II : If X = A × B or X = A/B
then ....(7) Rule III :
If X = An then
If X = ApBqC r then
(6) Must know Formula on Physics' for AIPMT Motion in Straight Line Speed and Velocity Speed is related to distance and it is a scalar while velocity is related to displacement and it is a vector. For a moving body speed can’t have zero or negative values but velocity can have. Average Velocity
Let a particle is at a point A at time t1 and B at time t2. Position vectors of A and B are and . The displacement in this time interval is the vector = . The average velocity in this time interval is,
here = change in position vector. The direction of average velocity is in the direction of displacement .
Average Speed Average speed is defined for a time interval and is given by the total distance traversed in the given time interval divided by the time interval.
Average speed of a trip vav =
If a particle travels a distance S in time t1 to t2, the average speed is vav = Acceleration
The motion of an object is said to be accelerated if its velocity changes with time (when either
the speed changes or the direction of motion changes). The acceleration is defined as the rate of change of velocity vector w.r.t. time. Average Acceleration During an interval of time when the velocity of particle changes from to , the average acceleration is defined as the change in velocity divided by interval of time (say ∆t).
Average acceleration = ⇒
The magnitude of average acceleration The Acceleration-time Graph Acceleration time curves give information about the variation of acceleration with time. Area under the acceleration time curve gives the change in velocity of the particle in the given time interval.
(7) Must know Formula on Physics' for AIPMT Analysis of Uniformaly Accelerated Motion
Case I : For uniformaly accelerated motion with initial velocity u and initial position x0.
Velocity time graph
θ In every case, tan = a0 Position time graph
Initial position x of the body in every case is x0 (> 0)
Case II : For uniformly retarded motion with initial velocity u and initial position x0.
Velocity time graph
θ In every case tan = – a0 Position time graph
Initial position x of the body in every case is x0 (>0) Concepts
1. Slopes of v-t or s-t graphs can never be infinite at any point, because infinite slope ofv-t graph means infinite acceleration. Similarly, infinite slope s-tof graph means infinite velocity. Hence, the following graphs are not possible.
2. At one time, two values of velocity or displacement are not possible Hence, the following graphs are not acceptable.
(8) Must know Formula on Physics' for AIPMT
3. The displacement-time graph cannot take sharp turns because it gives two different velocities at that point.e.g.
4. The displacement-time graph cannot be symmetric about the time-axis because at an in- stant a particle cannot have two displacements. But the graph may be symmetric about the displacement-axis. e.g.
5. The distance-time graph is always an increasing curve for a moving body. 6. The displacement-time graph does not show the trajectory of the particle. If a body dropped from some height (initial velocity zero) (i) Equation of motion : Taking initial position as origin and direction of motion (i.e., downward direction) as a positive, here we have u = 0 [As body starts from rest] a = +g [As acceleration is in the direction of motion v = g t …(i)
…(ii) v2 = 2gh …(iii)
...(iv) (ii) Graph of distance, velocity and acceleration with respect to time :
(iii) As h = (1/2)gt2, i.e., h ∝ t2, distance covered in time t, 2t, 3t, etc., will be in the ratio of 21 : 22 : 32, i.e., square of integers.
(iv) The distance covered in the nth sec, So distance covered in I, II, III sec, etc., will be in the ratio of 1 : 3 : 5,i.e. , odd integers only. If a body is projected vertically downward with some initial velocity Equation of motion : v = u + g t
v2 = u2 + 2 gh
(9) Must know Formula on Physics' for AIPMT If a body is projected vertically upward (i) Equation of motion : Taking initial position as origin and direction of motion (i.e., vertically up) as positive a = – g [As acceleration is downwards while motion upwards] So, if the body is projected with velocity u and after timet it reaches up to height h then
v = u – g t ; ; v2 = u2 – 2gh ; (ii) For maximum height v = 0 So from above equation u = gt,
and u2 = 2gh
(iii) Graph of distance, displacement, velocity and acceleration with respect to time (for maximum height) :
It is clear that both quantities do not depend upon the mass of the body or we can say that in absence of air resistance, all bodies fall on the surface of the earth with the same rate. Applications of Relative Velocity Relative velocity concept is very useful in (1) Motion analysis between two bodies. (2) Rain / wind / aircraft based problems. (3) River (Swimmer ) based problem. Swimming into the river A man can swim with velocity , i.e., it is the velocity of man w.r.t. still water.
If water is also flowing with velocity then velocity of man relative to ground (i) If the swimming is in the direction of flow of water or along the downstream then
vm = v + vR (ii) If the swimming is in the direction opposite to the flow of water or along the upstream then
(10) Must know Formula on Physics' for AIPMT
vm = v – vR
(iii) If man is crossing the river i.e., and not collinear then use the vector algebra
For shortest path : If man wants to cross the river such that his “displacement should be minimum”. It means he wants to reach just opposite point across the river. Man should start swimming at an angle θ with the perpendicular to the flow of river towards upstream.
So that its resultant velocity . It is in the direction of displacement AB.
To reach at B v sin θ = vR ⇒ sin θ =
component of velocity of v along AB is = vcos θ. so time taken For minimum time To cross the river in minimum time the velocity alongAB (v cos θ) should be maximum. It is possible if θ = 0, i.e., swimming should start perpendicular to water current. Due to effect of river velocity, man will reach at point C along resultant velocity, i.e., his displacement will not be minimum but time taken to cross the river will be minimum.
∴ In timet min swimmer travels distance BC along the river with speed of river vR BC = tmin vR Distance
travelled along river flow = drift of man = tmin vR = Oblique Projectile Motion Equation of Trajectory :A projectile thrown with velocityu at an angle θ with the horizontal. The velocity u can be resolved into two rectangular components.
(11) Must know Formula on Physics' for AIPMT This equation shows that the trajectory of projectile is parabolic because it is similar to equation of parabola y = ax - bx2 Horizontal range : It is the horizontal distance travelled by a body during the time of flight. So by using second equation of motion
(a) Range of projectile can also be expressed as :
2u u R = x y g ∴ (where ux and uy are the horizontal and vertical component of initial velocity) Newton’s Second Law of Motion
or
if m = constant then Impulse Impulse = product of force with time.
Suppose a force acts for a short time dt then impulse =
For a finite interval of time from t1 to t2 then the impulse = If constant force act for an interval ∆t then Impulse = Impulse - Momentum theorem
Impulse of a force is equal to the change of momentum Conservation Of Linear Momentum (COLM)
If = 0, then
= 0 ⇒ = 0 ⇒ Motion of Bodies In contact Case I :
(i) When the force F acts on the body with mass m1 as shown in figure
F = (m1 + m2)a.
If the force exerted by m2 on m1 is f1 (force of contact)
then for body m1 : (F – f1) = m1 a
for body m2 : f1 = m2a
⇒ action of m1 on m2 : f1 = (12) Must know Formula on Physics' for AIPMT
(ii) When the force F acts on the body with mass m2 as shown in figure
F = (m1 + m2)a
for body with mass m2
F – f2 = m2 a
for body m1: f2 = m1a
⇒ action on m1: f2 = Case II :
f1 = (action on m1 alone) and f2 = (action on m1 and m2) Motion of Bodies Connected by Strings
Acceleration a =
T1 = m1a =
∴ T2 = F – Some Cases of Pulley
Case I : Case II :
a = and T =
acceleration = Reaction at the suspension of pulley or thurst or pressure force
(13) Must know Formula on Physics' for AIPMT
Tension = R = 2T = Case III :
acceleration a = ⇒ T = Pressure force or thurst on pulley P = √2 T
Case IV : (m1 > m2)
a = Case V :
acceleration a =
T = Case VI :
Acceleration a =
Tension T = Case VII :
For mass m1 : T1 – m1g = m1a
For mass m2 : m2g + T2 – T1 = m2a
For mass m3 : m3g – T2 = m3a
a = Coefficient of Friction
Static friction coefficients µ = Sliding friction coefficientk µ =
The values of µs and µk depend on the nature of both the surface in contact. Angle of Friction
(14) Must know Formula on Physics' for AIPMT
by λ tanλ =
Angle of Repose (θ) If a body is placed on an inclined plane and if its angle of inclination is gradually increased, then at some angle of inclinationθ the body will just on the point to slide down. The angle is called angle of repose (θ).
FS = mg sinθ and N = mg cosθ so Relation between angle of friction (λ) and angle of repose (θ) tan λ = µ and µ = tan θ, hence tan λ = tan θ ⇒ θ = λ Thus, angle of repose = angle of friction Acceleration of a body down a rough inclined plane
a = g [sinθ – µk cosθ] hence a < g acceleration of a body down a rough inclined plane is always less than ‘g’. By Banking of Roads only
Nsinθ = and Ncosθ = mg ⇒ tanθ = ⇒ v = Friction and banking of Road both
.
WORK POWER AND ENERGY W = F s cosθ = If and
(15) Must know Formula on Physics' for AIPMT
Then W = Dimension : M1L2T–2 UNIT SI : joule C.G.S. : erg 1 joule = 107 erg Work Done by a Variable Force
WAB = POWER
Instantaneous power P = 1 horsepower = 746 watt
Kinetic Energy If a body of mass m is moving with velocity v, its kinetic energy KE = mv2/2.
If linear momentum of body is p, the kinetic energy for translatory motion is KE = mv2. Work Energy Theorem
W = ∆KE or ⇒ ∆ W = K2 – K1 W = KE W = change in K.E. of body. Potential Energy
If force varies only with one dimension (along x-axis) then F = Potential Energy Curve and Equilibrium It is a curve which shows change in potential energy with position of a particle. Stable Equilibrium : When a particle is slightly displaced from equilibrium position and it tends to come back towards equilibrium
(16) Must know Formula on Physics' for AIPMT then it is said to be in stable equilibrium
At point C : slope is negative so F is positive
At point D : slope is positive so F is negative At point A : it is the point of stable equilibrium.
At point A : U = Umin , = 0 and = positive Unstable equilibrium : When a particle is slightly displaced from equilibrium and it tends to move away from equilibrium position then it is said to be in unstable equilibrium
At point E : slope is positive so F is negative.
At point G : slope is negative so F is positive At point B : it is the point of unstable equilibrium.
At point B : U = Umax, = 0 and = negative Neutral equilibrium : When a particle is slightly displaced from equilibrium position and no force acts on it then equilibrium is said to be neutral equilibrium
Point H is at neutral equilibrium ⇒ U = constant ; = 0, = 0 Law of Conservation of Mechanical Energy KE + U = constant COLLISION Types of collision according to the conservation law of kinetic energy :
(a) Elastic collision : kinetic energy is conserved. KEbefore collision = KEafter collision (b) Inelastic collision : kinetic energy is not conserved.
Some energy is lost in collision KEbefore collision > KEafter collision (c) Perfect inelastic collision : Two bodies stick together after the collision. Coefficient of restitutione ( ) The coefficient of restitution is defined as the ratio of the impulses of recovery and deformation of either body.
e = Value of e is 1 for elastic collision, 0 for perfectly inelastic collision, 0 e< < 1 for inelastic collision. Head on Inelastic Collision of two Particles
Let the coefficient of restitution for collision is e
(i) Momentum is conserved m1u1 + m2u2 = m1v1 + m2v2 ...(1) (ii) Kinetic energy is not conserved
(17) Must know Formula on Physics' for AIPMT
(iii) According to Newton’s law ...(2) By solving equation (1) and (2)
SYSTEM OF PARTICLES AND ROTATIONAL MOTION Co-ordinates of Centre of mass
Velocity and acceleration of Centre of Mass
velocity of CM
acceleration of CM MOMENT OF INERTIA
Moment of inertia of system of particle I =
For continuous body I = Radius of Gyration K( )
K =
Theorems of Moment of Inertia 1. Theorem of perpendicular axes (applicable only for two dimensional bodies or plane laminas)
Iz = Ix + Iy where
Ix = MI of the body about x-axis
Iy = MI of the body about y-axis
Iz = MI of the body about z-axis 2. Theorem of parallel axes (for all type of bodies)
(18) Must know Formula on Physics' for AIPMT
2 I = ICM + Md
TORQUE (OR MOMENt OF A FORCE)
where θ is smaller angle between and , is unit vector along . Power Associated with Torque P = τω ANGULAR MOMENTUM (MOMENT OF LINEAR MOMENTUM) L = mv × r sinθ ⇒ Angular Impulse
Angular impulse = τav∆t = = change in angular momentum Relation Between Angular Momentum and Torque
Rotational Kinetic Energy
KE = Work energy theorem in rotational motion
The work done by torque = Change in kinetic energy of rotation W = ROLLING MOTION
Total Energy
Rolling Motion on an Inclined Plane
Time of descend when body rolling
(19) Must know Formula on Physics' for AIPMT
Time of descend when body sliding so If different body are rolled down on an inclined plane then body which has
1. Least, will reach first
2. Maximum, will reach last
3. equal, will reach together θ θ θ 4. From figure 3 > 2 > 1
a3 > a2 > a1
t3 < t2 > t1
v1 = v2 = v3
In next figure if v2 > v1 Change in kinetic energy due to rolling
=
= Rolling/Sliding :
Pure Rolling If the velocity of point of contact with respect to the surface is zero then it is known as pure rolling. If a body is performing rolling then the velocity of any point of the body with respect to the surface is given by
Rolling Motion on an Inclined Plane Rolling Motion on an Inclined Plane
(20) Must know Formula on Physics' for AIPMT
for pure rolling µmin = NEWTON’S LAW OF GRAVITATION
GRAVITATIONAL FIELD
Gravitational Field Intensity I[ g or Eg]
Gravitational Field Intensity for Solid and Hollow Sphere
(21) Must know Formula on Physics' for AIPMT
GRAVITATIONAL POTENTIAL
Joule/kg. Gravitational Potential for Solid and Hollow Sphere
(22) Must know Formula on Physics' for AIPMT
ACCELERATION DUE TO GRAVITY
VARIATION IN ACCELERATION DUE TO GRAVITY
Due to Altitude (height)
If h << Re then Due to Depth
valid for 100% depth Due to Rotation of the Earth
(23) Must know Formula on Physics' for AIPMT
GRAVITATIONAL POTENTIAL ENERGY
Work done by Gravitational force in shifting a mass from one place to another place.
W = U = –
ESCAPE VELOCITY (ve)
1. (In form of mass) If M = constant 2. (In form of g) If g = constant
3. (In form of density) If ρ = constant 4. Escape velocity does not depend on mass of body, angle of projection or direction of projection.
5. Escape velocity at : Earth surface ve = 11.2 km/s Moon surface ve = 2.31 km/s 6. Atmosphere on Moon is absent because root mean square velocity of gas particle is greater then
escape velocity. (vrms > ve).
Orbital Velocity (v0)
∴ v0 = v0 = (Puting r = Re + h)
For near by satellite h << Re
2 v0 = = 8 km/sec (on puting GMe = gRe ) √ If orbital velocity of a near by satellite becomes 2 v0 (or increased by 41.4%, or K.E. is doubled) then the satellite escapes from gravitational field of earth.
Time Period of a Satellite
For Geostationary Satellites
T = 24 hr, h = 36,000 km = 6 Re (r = 7 Re), v0 = 3.1 km/s For Near by Satellite
v0 = = 8 km/s = 84 minute = 1 hour 24 minute = 1.4 hr = 5063 s
In terms of density Energies of a Satellite
Kinetic energy K.E. =
(24) Must know Formula on Physics' for AIPMT
Potential energy P.E. =
Total mechanical energy T.E. = P.E. + K.E. = Binding Energy
B.E. = – T.E. B.E. = Hence B.E. = K.E. = – T.E. = Work done in Changing the Orbit of Satellite
W = Change in mechanical energy of system but E = so W = E2 – E1 = Planet Moves Around Sun in an Elliptical Orbit A planet moves around sun in an elliptical orbit of semin major axis a and eccentricity e.
For this planet areal velocity = = constant.
⇒ MECHANICAL PROPERTIES OF SOLIDS AND FLUIDS
Stress = Breaking Stress : The stress required to cause actual fracture of a material is called the breaking stress or the ulti-
mate strength. Breaking stress =
Strain =
Longitudinal Strain =
(25) Must know Formula on Physics' for AIPMT
Volume strain = Shear strain(φ)
φ = Hooke's law Within elastic limit, stress is directly proportional to strain.
E = Elastic constant or modulus of elasticity. Types of Elasticity Coefficients
Young's Modulus Elongation of a wire due to its own weight Let, M = mass of rope, L = length of rope, A = cross-sectional area of the rope,ρ = density of rope, ∆L = increase in length of the rope due to its own weight.
Work done in stretching a wire (Potential energy of a stretched wire)
The total work done in stretching the wire from x = 0 to x = ∆ is, then
Energy density,
Bulk modulus of elasticity (K) : Volume stress and volume strain are in constant ratio which is called Bulk modulus.
(26) Must know Formula on Physics' for AIPMT
Isothermal bulk modulus Kisothermal = P γ Adiabatic bulk modulus Kadiabatic = P
Compressibility :
The reciprocal of bulk modulus of elasticity is defined as compressibility. Modulus of Rigidity or Shear modulus (η) : Shearing stress and shearing strain are in constant ratio which is called Shear modulus.
where angle of shear 'φ' is always taken in radian.
Poisson's Ratio (σ) : In elastic limit, the ratio of lateral strain and longitudinal strain is called poisson's ratio.
–1 < σ < 0.5 (theoritical limit) σ ≈ 0.2 to 0.4 (experiments) Relation between (Y, K, η & σ) :
(1) Y = 3K (1 – 2σ) (2) Y = 2η (1 + σ) (3) (4)
Relation between angle of twist and angle of shear. rθ = φ θ → angle of twist, φ → angle of shear
surface tension
Surface Tension T = 1. Work done (surface energy) in formation of a drop of radiusr = Work done against surface tension
(27) Must know Formula on Physics' for AIPMT W = Surface tension (T) × change in area (∆A) = T × 4πr2 = 4πr2T 2. Work done (surface energy) in formation of a soap bubble of radius r. W = T × ∆A or W = T × 2 × 4πr2 = 8πr2T [ soap bubble has two surfaces] n drops coalesce (combine) to form a big drop
Now surface energy of one small drop = 4 πr2T. π 2 The total initial surface energy of n drops is Einitial = n(4 r T) π 2 The final surface energy of big drop is Einitial = 4 R T. Therefore, the amount of surface energy released or surface energy loss is 2 2 ∆E = Ei – Ef = 4πnr T – 4πR T
∴ ∆E or Work (W) Excess Pressure Inside a Curved Liquid Surface (i) Excess pressure inside the drop
(ii) Excess pressure inside soap bubble
⇒ excess pressure = Pi – PO = (iii) Excess pressure inside the cavity or air bubble in liquid
Pexcess = Pin – Pout = hdg +
Pinside = Patm + hdg + characteristic properties of liquid Pressure If a uniform force is exerted normal to an area (A), then pressure (P) is defined as the normal force (F) per unit area i.e. P = F/A Types of Pressure : Pressure is of three types
(a) Atmospheric pressure(P0) (b) Gauge pressure (Pgauge) (c) Absolute pressure (Pabs.)
Absolute Pressure : Sum of atmospheric and Gauge pressure is called absolute pressure.
∴ Relative Density:
If a body is weighed in air (WA), in water (Ww) and in a liquid (WL), then
Specific gravity of liquid = Equation of Continuity
A1V1 = A2V2 Based on conservation of mass
(28) Must know Formula on Physics' for AIPMT Bernoulli's theorem : The sum of pressure energy, kinetic energy and potential energy per unit volume remains constant along a streamline in an ideal fluid flow i.e.,
= constant (Energy per unit volume)
or = constant (Energy per unit weight) Here P/ρg is called pressure head, v2/2g is called velocity/kinetic headand h is called gravitational/ potential head Torricelli's law of Efflux (Fluid Outflow)The : area of cross-section atA is very large as compared to that
at orifice B, speed at A i.e. VA ≈ 0.
Horizontal range
or
Range will be maximum when R2 is maximum, ie.., .
h = H – h or h = H/2 ∴ Newton's law of viscosity
Where η is a constant called coefficient of viscosity of the liquid.
Dimension of η : M1L–1T–1 SI Unit : 1 poiseuille (PI) = 1 = deca pose CGS Unit : Dyne-sec/cm2 or poise such that 1 decapoise = 10 poise. Poiseuille's Formula In case of steady flow of liquid of viscosity (η) in a capillary tube of length (L) and radius (r) under a pressure difference (P) across it, the volume of liquid flowing per second is given by :
Stoke's law and terminal velocity Stoke showed that if a small sphere of radius r is moving with a velocity v through a stationary
medium (liquid or gas) of viscosity η then the viscous force acting on the sphere is Fv = 6πηrv. It is called Stoke's law. Terminal Velocity :
where r = radius of body ρ = density of body σ = density of liquid η = coefficient of viscosity Graph : where variation of velocity with time (or distance) is shown in the adjascent graph.
(29) Must know Formula on Physics' for AIPMT
Reynolds Number(Re) The type of flow pattern (laminar or turbulent) is determined by a non-dimensional number called
Reynolds number (Re). Which is defined as where ρ is the density of the fluid having viscosity η and flowing with mean speed v, d denotes the diameter of tube.
tHeRMAL Properties of matter Scale of temperature
Hence, Linear expansion
α∆θ ⇒ ∆ α∆θ = 0(1 + ) = 0 Superficial (areal) expansion : β∆θ A = A0(1 + ) β = 2α
Volume expansion : γ∆θ V = V0(1 + )
γ = 3α
Anomalous expansion of water : Generally matter expands on heating and contracts on cooling. In case of water, it expands on heating if its temperature is greater than o4 C. In the range 0oC to 4oC, wa- ter contracts on heating and expands on cooling, i.e.,γ is netagive. This behaviour of water in the range from 0oC to 4oC is called anomalous expnasion.
(30) Must know Formula on Physics' for AIPMT
Thermal Capacity Thermal capacity = mass × specific heat Heating curve If to a given mass (m) of a solid, heat is supplied at constant rate and a graph is plotted between temperature and time as shown in figure.
Coefficient of Thermal Conductivity : The heat conducted by a rod is given by the relation:
Q = where K = Coefficient of thermal conductivity, A = Area normal to the heat flow θ θ 1 – 2 = Temperature difference,t = Time, L = Length of the rod Heat Current
We have Q = or Heat current, H = = , Where R = is called thermal resistance of the conductor.
Series Connection : Two rods A and B of thermal resistances R1 and R2 are joined as shown in figure.
Then the equivalent thermal resistance R = R1 + R2.
and hence heat current H =
Parallel Connection :Equivalent resistance R is given by or, R =
∴ Heat current H = = (T1 –T2)
Wiedmann-Franz law : “The ratio of thermal and electrical conductivities (K &σ) at a particular temperature (T K) is same for all metals (except(31) mica) Must know Formula on Physics' for AIPMT
i.e., = constant Growth of ice on ponds :
Kirchhoff's Law
At a given temperature for all bodies the ratio of their spectral emissive power (eλ) to spectral
absorptive power (aλ) is constant and this constant is equal to spectral emissive (Eλ) of the ideal black body at same temperature
= constant = constant eλ ∝ aλ Good absorbers are good emitters and bad absorbers are bad emitters. Stefan’s Law [or stefan-Boltzmann Law] The amount of radiation emitted per second per unit area by a black body is directly proportional to the fourth power of its absolute temperature. E ∝ T 4 T = temperature of ideal black body (in K) E = σT 4 E = amount of radiation emitted This law is true for only ideal black body. Newton's Law of Cooling
Wien’s Displacedment Law
Temperature of the Sun : From S = ,
T = Where S = solar constants. Equation of state for ideal gas
Where P is the absolute pressure, V is the volume, N is the total number of moles present in the gas, T is the temperature in absolute (kelvin) units and R is the universal gas constant. The value of R is same for all gases. R = 8.314 J/mol-K Gas laws
(32) Must know Formula on Physics' for AIPMT
4. Avogadro’s Law : - At same temperature and pressure equal volumes of all gases contains equal number of molecules.
N1 = N2 if P, V, and T are same 5. Dalton’s Law : - According to this law, the pressure exerted by a mixture of several gases equals the sum of the pressure exerted by each component gas present in the mixture i.e.
Pmix. = P1 + P2 + P3 ......
Graham’s Law of diffusion : -Accroding to this law, at same temperature and pressure, the rate of diffusion of gas is inversely proportional to the square root of the density of gas i.e.,
∝ Rate of diffusion So, Vrms rd [For H2 gas (Max)] Kinetic Interpretation of Temperature
This is average translational K.E. per mole Degree of freedom for different gases : - (Acc. to atomicity of gas) Relation between degree of freedom & specific heat of gas Energy related with each degree of freedom = RT/2 Energy related with all degree of freedom = fRT/2 Internal Energy of one Mole of Ideal gas (Total K.E.) U = fRT/2 [∆U = fR∆T/2]
(33) Must know Formula on Physics' for AIPMT
Different Kinetic Energy of Gas (Internal Energy)
(i) Translatory Kinetic Energy E( T)
Total internal energy of ideal gas is kinetic energy.
(ii) Energy per unit Volume or Energy Density(Ev)
(iii) Molar K.E. or Mean Molar K.E. (E)
for N0 molecules or Mw(gram) E = RT = N0kT (iv) Molecular Kinetic energy or Mean Molecular K.E. ( )
Average velocity Because molecules are in random motion in all possible direction in all possible velocity. Therefore, the average velocity of the gas in molecules in container is zero.
rms speed of molecules
Mean speed of molecules
By maxwell’s velocity distribution law vM or = vmean
Most probable speed of molecules (vmp) At a given temperature, the speed to which maximum number of molecules belongs is called as
most probable speed (vmp)
(34) Must know Formula on Physics' for AIPMT first law of thermodynamics
First law of thermodynamics is in accordance with law of conservation of energy. According to Clausius statement of first law of thermodynamics, the heat given to a system (dQ) is used up in two ways : (i) In increasing the temperature of system and hence in increasing internal energy of the sys- tem. It is represented by dU. (ii) In doing work against external pressure. It is represented by dW. ∴ dQ = dU + dW where dW = PdV Specific heat Capacity
Molar specific heat at constant pressure( Cp) : It is defined as the amount of heat required to raise the temperature of one mole of a gas by 1°C at constant pressure.
Cp =
(ii) Specific heat at constant volume s( v) : It is defined as the amount of heat required to raise the temperature of unit mass of a gas by 1°C at constant volume.
sV =
Molar specific heat at constant volume (CV) : The amount of heat required to raise the temperature on one mole of a gas by 1°C at constant volume is called molar specific heat at constant volume.
CV =
If Mw be the molecular mass of the gas, then
Relation between CP and CV ...... (Mayer’s relation) Isothermal Process T = constant or ∆T = 0 ⇒ PV = constant ∆ µ ∆ In this process U = CV T = 0
Q = W = µRT Adiabatic Process PVγ = constant or TγP1 – γ = constant or TVγ – 1 = constant In this process Q = 0 and W = – ∆U
⇒ ( PV = µRT) Slope of adiabatic is greater than the slope of isotherm
Isobaric Process
(35) Must know Formula on Physics' for AIPMT ⇒ ∝ µ ∆ ∆ µ ∆ µ ∆ P = constant V T; Q = CP T, U = CV T, W = P (V2 – V1) = R T Isochoric Process
V = constant ⇒ P ∝ T; Q = ∆U = µCV∆T, W = 0 Polytropic Process : PVx = constant
Work done Molar heat capacity C = CV + Heat Engine
Efficiency of heat engine η =
Refrigerator
For Carnot reversible refrigerator
Oscilation
(36) Must know Formula on Physics' for AIPMT
VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION If y = Asinωt ∴
or,
a = a = ENERGY IN SIMPLE HARMONIC MOTION The restoring force F = – kx is analogous to the spring force which is conservative. The associated potential energy
..... The kinetic energy
..... Total enery
(37) Must know Formula on Physics' for AIPMT
.....
∴
Figure shows the graphs of total energy (E), potential energy (U) and the kinetic energy (K) versus the displacement [x] from the equilibrium postion
The frequency of oscillations of each of U and K is twice the frequency
of the SHM.
From the figure given below, it is clear that |K – U| com- pletes its one cycle (maximum to zero and zero to maximum) in time
, so its frequency is four times the frequency of SHM.
Spring Block System
(1) If ms be the mass of the spring, then the expression of time period is modified as
(2) If a spring of force constant k is divided into n equal parts and one such part is attached to a mass m, then the time period is given by
, because spring constant of each part is nk.
(3) If two springs of force constant k1 and k2 are connected in parallel and a mass m is attached to
(38) Must know Formula on Physics' for AIPMT them, then the time period is given by
where k = k1 + k2
(4) If two springs of force constants k1 and k2 are connected in series and a mass m is attached to them, then the time period is given by
where = or
(5) If two masses m1 and m2 are connected to the two ends of a spring, then the time period is given by
where Here m is known as reduced mass. Some more points about a Pendulum (1) The period T does not depend of the amplitude if it is small.
(2) The formula is true only for << R (radius of the Earth)
(3) From the formula
i.e., For a change of x% in , the change in T is % (4) When a girl sitting on swing stands up, then her centre of gravity rises up, and hence the effective length of the swing decreases. Therefore, periodic time also decreases.
(5) Due to thermal expansion, where a is coefficient of thermal expansion and∆θ = change in temperature
(39) Must know Formula on Physics' for AIPMT
∴ or
Hence a pendulum clock will be slow by per day. (6) At a height h (<< R) above the Earth’s surface
so or Hence a pendulum clock will be slow by]
If h = 1 km, loss of time = 13.6 s per day.
(7) At a depth d below the Earth’s surface, so
(8) A simple pendulum with period 2 seconds, is called seconds pendulum. Its length is about 1 m on the earth’s surface. (9) If the length of the pendulum is comparable to the radius (R) of the Earth, the formula for time period is
(10) If > > R, i.e., for a pendulum of infinite length so
Putting the values of R and g, This is maximum limit of the time period for a pendulum. (11) If = R, (12) If a simple pendulum has an acceleration then g in the formula for T will be re- placed by
(13) If is vertically upward. geff = g + a
∴
(14) If is vertically downward, geff = g – a, so
(15) If is horizontal,
(16) In a freely falling lift, geff = 0, so T = ∞
(40) Must know Formula on Physics' for AIPMT
i.e., the pendulum will not oscillate. This is also true inside a satellite because geff = 0. So, a pen- dulum clock cannot work iniside a satellite.
(17) In addion to gravity, if other force (e.g., electrostatic force) also acts on the bob of mass m, then
(18) If a simple pendulum is suspended from the ceiling of a cart sliding on a smooth inclined plane
of inclination q,
Torsional pendulum
Motion is angular simple harmonic. Hence the period Wave Motion Equation of a Plane Progressive Wave
y = a sin Relation between particle velocity and wave velocity :
Wave equation : y = A sin(ωt – kx), Particle velocity v = = Aω cos(ωt – kx).
Wave velocity Relation between Phase difference, Path difference & Time difference
Phase (φ) 0 π 2π 3π
Wave length (λ) 0 λ λ
Time-period (T) 0 T
Path difference = Phase difference VELOCITY OF TRANSVERSE WAVE
(41) Must know Formula on Physics' for AIPMT
Velocity of transverse wave in any wire v = or where m is linear mass density and T = tension in the wire SPEED OF LONGITUDINAL (SOUND) WAVES
Newton Formula vmedium = (Use for every medium)
Where E = Elasticity coefficient of medium & = Density of medium
For solid medium = vsolid = Where E = Y = Young's modulus
For solid medium vliquid = Where E = B, where B = volume elasticity coefficient of liquid Laplace Correction
Adiabatic Elasticity = γP so that v =
And from kinetic-theory of gases vrms = Effect of various quantities (a) Effect of temperature
For a gas γ & Mw is constant v ∝ (b) Effect of Relative Humidity
With increase in humidity, density decreases so in the light of v = We conclude that with rise in humidity velocity of sound increase. (c) Effect of Pressure
As velocity of sound v = So pressure has no effect on velocity of sound in a gas as long as temperature remain constant. Vibration of air columns in open organ pipe (OOP)
When open organ pipe vibrate in mth overtone then = (m + 1) so λ = ⇒ n = (m + 1)
Hence frequency of overtones are given by the relation n1 : n2 : n3 ...... = 1 : 2 : 3 : ...... Vibration of air column in closed organ pipe(COP)
When closed organ pipe vibrate in mth overtone then = (2m + 1)
So
Hence freuency of overtones is given by n1 : n2 : n3 ...... = 1 : 3 : 5 ...... DOPPLER EFFECT
1. If medium (air) is also moving with vm velocity in direction of source and observer. Then velocity
of sound relative to observer will be v ± vm(–ve sign, if vm is opposite to sound velocity).
So, n' = n
(42) Must know Formula on Physics' for AIPMT 2. If medium moves in a direction opposite to the direction of propagation of sound, then
n' = 3. Source in motion towards the observer. Both medium and observer are at rest. n'= {v/(v –
vs)}/n So, when a source of sound approaches a stationary abserver, the apparent frequency is more than the actual frequency.
4. Source in motion away from the observer. Both medium and observer are at rest.n'={v/(v + vs)}/n So, when a source of sound moves away from a stationary observer, the apparent frequency is less than actual frequency.
5. Observer in motion towards the source. Both medium and source are at rest. n' = {(v + vo)/v}n So, when observer is in motion towards the source, the apparent frequency is more than the actual frequency.
6. Observer in motion away from the source. Both medium and source are at rest. n' = {(v – vo)/v} n So, when observer is in motion away from the source, the apparent frequency is less than the actual frequency. 7. Both source and observer are moving away from each other. Medium at rest.
n' = {(v – vo)/(v + vs)}n Electrostatics COULOMB’S LAW
F = where, k = , if charges are placed in vacuum.
and k = , when charges are placed in medium with relative permittivityε r.
where, εr = (ε is the absolute permittivity of medium) Principle of superposition The resultant force acting on a charge due to a group of charges is equal to the vector sum of individual forces.
⇒ ELECTRIC FIELD
Field of isolated point charge :
Electric field Electric field due to charged ring
(43) Must know Formula on Physics' for AIPMT
E =
Electric flux φ = Electric Flux through various surfaces φ π 2 φ φ π 2 φ φ 2 (i) | in| = R E (ii) out = in = R E (iii) in = out = Ea φ π 2 φ | out| = R E total = 0.
φ φ ⇒ φ circular = curved total = 0
φ φ π 2 (iv) T = 0 (v) T = 2 R E
ELECTRIC DIPOLE
Dipole placed in uniform electric field
Electric field due to an electric dipole (i) At a point on the axis of a dipole :
E = E1 – E2
E = ( p = q × 2)
(44) Must know Formula on Physics' for AIPMT
If r >>> E = (ii) At a point on equatorial line of dipole :
E =
If r >>> E = At any point
Enet =
tan α = ELECTROSTATIC PRESSURE
P = ELECTRIC POTENTIAL (V)
V = Potential due to group of charges
V = V1 + V2 + ...... +Vn = Potential difference
Vab = Electric potential gradient
(45) Must know Formula on Physics' for AIPMT
Electric Field = – Electric potential due to charged ring
V =
Electric potential due to hollow or conducting sphere (i) At outside sphere
V =
(ii) At surface
V = (iii) Inside the surface
Inside the surface E = 0, = 0, or V = constant [ E = – ]
so V = Electric potential due to dipole
Net electric potential V If r > > then V = (ii) At equatorial point
Net potential V = V1 + V2 = = 0 So V = 0 (iii) Electric potential due to an electric dipole at any point
(46) Must know Formula on Physics' for AIPMT
2 2 2 VP = kq ( cos θ <<< r , P = q2) VP= Potential Energy of a System of charges
Energy of a system of two charges, U =
Energy of a system of three charges, U =
Geometrical point charge system
(a) Wsys = Usys = – B.E. = –Usys = +
(b) usys =
=
= K (3-pairs)
(c) NP = Number of pairs = → Number of charges. Capacitor and Capacitance Capacitance C = Q/V πε Capacitance of an isolated spherical conductor of radius R, C = 4 0R Energy stored in a charged conductor/capacitor
Parallel Plate Capacitor (i) Capacitance
ε ε If medium between the plates is air or vacuum, then r = 1, C0 = 0A/d
(where εr = K = dielectric constant) (ii) Force between the plates
(47) Must know Formula on Physics' for AIPMT
Magnitude of force F =
Force per unit area or energy density of electrostatic pressure = Spherical capacitor (i) Outer sphere is earthed
π ε ε C = 4 r 0 (ii) Inner sphere is earthed
C = = Capacitor in series :
5.2 Capacitors in parallel
Cp = C1 + C2 + C3
When dielectric is completely filled between plates If a dielectric slab is fills completely the gap between the plates, capacitance increases byK times i.e.,C
(48) Must know Formula on Physics' for AIPMT
=
When dielectric is partially filled between the plates
C =
C' = C' = COMBINATION OF DROPS (49) Must know Formula on Physics' for AIPMT Suppose we have n identical drops each having Radius r, Capacitance c, Charge q, Potential v and Energy u. If these drops are combined to form a big drop of Radius R, Capacitance C, Charge Q, Potential V and Energy U then (i) Charge on big drop : Q = nq (ii) Radius of big drop : Volume of big drop = n × volume of a single drop
i.e., , R = n1/3r (iii) Capacitance of big drop : C = n1/3c
(iv) Potential of big drop : V = ⇒ V = n2/3v
(v) Energy of big drop : U = ⇒ U = n5/3u CHARGING AND DISCHARGING OF CAPACITOR Charging of a capacitor
∞ At t = , Q = Q0, then capacitor gets fully charged
Q = CV, Q0 = CE
Current at any instant At t = τ = RC (time constant) –1 Q = Q0 [1 – e ] = 0.632 Q0 Time constant is that time in which the charge increases to 0.632 or 63.2% of the maximum charge.
Discharging of a capacitor :
at t = ∞ I = 0, q = 0, then capacitor gets fully discharged
at t = RC (time constant) Q = Time constant is that time in which the charge decays to 0.37 or 37% of the maximum charge.
ELECTRIC CURRENT
Average current iav = ; Instantaneous current i = CURRENT DENSITY (J)
(50) Must know Formula on Physics' for AIPMT
The current density at a point in a conductor is the ratio of the current at that point in the conductor to the area of cross-section of the conductor of that point.
, dA = Cross section area It is a vector quantity. It’s direction is the direction of motion of the positive charges at that point.
Unit = ∆i = , i = Electric current is the flux of current density. DRIFT VELOCITY
Relaxation Time(τ) Average time elapsed between two successive collisions.It is of the order of 10–14 s It is a temperature dependent characteristic of the material of the conductor. It decreases with increases in temperature. Mean Free Path(λ) The distance travelled by a conduction electron during relaxation time is known as mean free path. Mean free path of conduction electron = Thermal velocity × Relaxation time RELATION BETWEEN CURRENT DENSITY, CONDUCTIVITY & ELECTRIC FIELD
In vector σ depends only on the material of the conductor and its temperature. OHM’S LAW At constant temperature, current is directly proportional to the applied potential difference. This law is called ohm’s law and substance which obey it are called ohmic or linear conductors.
V = Ri R is constant, it’s unit is ohm (Ω).
Resistance depends on (1) length of the conductor (R ∝ ) (2) area of cross - section of the conductor (R ∝ 1/A)
(3) nature of material of the conductor R = α∆ (4) temperature Rt = R0 (1 + t)
Where Rt = Resistance at t°C
R0 = Resistance at 0°C
(51) Must know Formula on Physics' for AIPMT ∆t = Change in temperature α = Temperature coefficient of resistance Resistivity depends on (i) Nature of material (ii) Temperature of material ρ does not depend on the size and shape of the material because it is the characteristic property of the conductor material.
(i) Series Combination Same current passes through each resistance
R = R1 + R2 + R3 Where R = equivalent resistance (ii) Parallel Combination There is same drop of potential across each resistance.
KIRCHHOFF’S LAW First law In an electric circuit, the algebraic sum of the current meeting at any junction in the circuit is zero. This is based on law of conservation of charge. Second law In any closed circuit the algebraic sum of e.m.f.’s and algebraic sum of potential drops is zero.
This law is based on law of conservation of energy. CELL Cell convert chemical energy in to electrical energy. Combination of Cells (i) Series combination
equivalent internal resistance r = r1 + r2 + r3 + .... equivalent emf = E = E1 + E2 + E3 + ......
Current i = If all n cell are identical then i =
If nr >> R, i = current from one cell; If nr << R, i = n × current from one cell (ii) Parallel combination If m identical cell connected in parallel then total internal resistance of this combination
(52) Must know Formula on Physics' for AIPMT
Total e.m.f. of this combination ET = E
Current in the circuit i = If r << mR If r >> mR
i = = Current from one cell i = = m × current from one cell (iii) Mixed combination Total number of identical cell in this circuit is nm. If n cells connected in series and their are m such branches in the circuit. The internal resistance of the cells connected in a row = nr.
The internal resistance of the circuit rnet = ( There are such m rows)
ET = nE
i =
Current in the circuit is maximum when external resistance in the circuit is equal to the total internal
resistance of the cells R =
Wheat Stone Bridge The configuration in the adjacent figure is called Wheat Stone Bridge.
If current in galvanometer is zero (Ig = 0) then bridge is said to be balanced
(53) Must know Formula on Physics' for AIPMT
VD = VB or
If then VB > VD and current will flow from B to D
If then VB < VD and current will flow from D to B Metre Bridge It is based on principle of wheatstone bridge. It is used to
find out unknown resistance of wire.
Potentiometer Comparision of emf of two cells
Internal resistance of a given primary cell
Joules’s Law of Heating When a current i is made to flow through a passive or ohmic resistance R for time t, heat Q is
(54) Must know Formula on Physics' for AIPMT
produced such that Q = i2 Rt = P × t = V i t = SI unit : joule; Practical Units : 1 kilowatt hour (kWh) 1 kWh = 3.6 × 106 joule = 1 unit 1 BTU (British Thermal Unit) = 1055 J Power
P = Vi = SI unit : Watt Combination of Bulbs (i) Series combination of resistors (bulbs)
Total power consumed Ptotal = . If n bulbs are identical Ptotal = In series combination of bulbs, Brightness ∝ Power consumed by bulb ∝ V ∝ R ∝
Bulb of lesser wattage will shine more. For same current P = i2 R P ∝ R R↑ ⇒ P↑ (ii) Parallel combination of resistors (bulbs)
Total power consumed Ptotal = P1 + P2
If n bulbs are identical Ptotal = np In parallel combination of bulbs,
Brightness ∝ Power consumed by bulb ∝ i ∝ . Bulb of greater wattage will shine more.
For same V more power will be consumed in smaller resistance as P ∝ . MAGNETIC EFFECT OF CURRENT AND MAGNETISM Magnetic Flux (φ) The number of magnetic field lines which are crossing through given area of cross section is called magnetic flux of that area.
Biot-Savart’s law
In SI units,
In cgs system,
(55) Must know Formula on Physics' for AIPMT
In vector form, we may write or Applications of Biot Savart’s Law I. Magnetic Field Due to Straight Conductor Carrying Current
B = Special Cases (i) When the conductor XY is of infinite length and the point P lies near the centre of the conductor φ φ then 1 = 2 = 90°
So, φ (ii) When the conductor XY is of infinite length but the pointP lies near the end Y (or X), then 1 = 90° φ and 2 = 0°.
So, 2. Magnetic Field at a Point on the Axis of a Circular Coil Carrying Current
∴
Magnetic field at the centre of current carrying circular arc :
Where α is always in radian and B0 = ampere’s circuiTaL law (acl)
The line integral of the resultant magnetic field along a closed plane curve is µ equal to 0 times the algebraic of the currents crossing the area bounded by the closed curve if the
(56) Must know Formula on Physics' for AIPMT electric field inside the loop remains constant. i.e.,
Magnetic field due to infinite long solid cylinderical wire (a) Solid cylinderical wire
Current density J = = constant, where A = πR2 Magnetic field at specific positions :
π r > R Bout (2 r) = µ0I
⇒ ∝ Bout
r = 0
(57) Must know Formula on Physics' for AIPMT
(b) Hollow cylinderical wire
The magnetic field at an axial point P (figure) of a solenoid of any length is given by
Case I: If the length of solenoid is infinite and P is well-inside the solenoid then,
So, Case II: If the length is infinite but point P is near an end then,
φ1 = 0 and φ2 = Magnetic field due to toroid A toroid can be considered as a ring shaped closed solenoid also called end less solenoid. Magnetic
field inside a toroid by A.C.L. given as Where
n (turn density) = N = total number of turns
Rm (mean radius of toroid) =
R1 & R2 = internal and external radius of toroid respectively.
Magnetic Force on Moving Charge
Consider a charge ±q moving with velocity in uniform magnetic
(58) Must know Formula on Physics' for AIPMT
field , such that makes an angle θ with , then magnetic force
experienced by the charge is given by Motion of Charge in Uniform Transverse Magnetic Field Radius of circular path (r) : Radius of circular path of charge in uniform transverse magnetic field given as
Time period (T)
, Frequency , Angular frequency Motion of Charge in Uniform Magnetic Field at an Acute Angle (0° < θ < 90°)
Radius of circular path Time period of circular motion :
Pitch of helix (p) : The linear distance travelled by the charge particle in one rotation along external magnetic field direction is called ‘pitch of helix’.
p = (v cosθ)T, where T = ⇒ FORCE ON A CURRENT CARRYING CONDUCTOR PLACED IN A MAGNETIC FIELD
F = I l B sin θ where θ is the smaller angle between and
. Special Cases Case I. If θ = 0° or 180°, sin θ = 0. F = I lB (0) = 0 (Minimum) Case II. If θ = 90°, sin θ = 1, F = I lB × 1 = I lB (Maximum) Magnetic Force Between two Long Parallel Current Carrying Conductors or Wires
(59) Must know Formula on Physics' for AIPMT
Magnetic force per unit length of each wire given as :
torque on a current carrying coil in a magnetic field where, n IA = M = magnitude of the magnetic dipole moment of the rectangular current loop
∴ Cases: 1. If the coil is set with its plane parallel to the direction of magnetic field B, then θ = 0° and cos θ = 1 ∴ Torque, τ = nIBA (1) = nIBA (Maximum) This is the case with a radial field. 2. If the coil is set with its plane perpendicular to the direction of magnetic field B, then θ = 90° and cos θ = 0 ∴ Torque, τ = nIBA (0) = 0 (Minimum) MAGNETIC DIPOLE IN MAGNETIC FIELD
Torque on Magnetic Dipole
(a) Bar Magnet τ = force × perpendicular distance between force couple τ = (mB) (sinθ), where M = m
magnitude form
Vector form Work done in Rotating a Magnetic Dipole θ θ so work done in rotating a dipole from angular position 1 to 2 with respect to the magnetic field
Potential Energy of Magnetic Dipole The potential energy of dipole defined as work done in rotating the dipole through an angleθ ‘ ’ with respect to a direction perpendicular to the field : ⇒ θ U = Wθ – W90° U = MB (1 – cos ) – MB
Tangent Law When the magnet is in the equilibrium position,
(60) Must know Formula on Physics' for AIPMT
or when a magnet is simultaneously acted upon by two uniform fields at right angle to each other, it will be deflected through an angle θ, such that the tangent of the angle of deflection gives the ratio of the two fields.
Magnetising field or Magnetic intensity Field in which a material is placed for magnetisation called as magnetising field. Magnetising field ; Unit of : Ampere/meter
(ii) Intensity of magnetisation When a magnetic material is placed in magnetising field then induced dipole moment per unit
volume of that material is known as intensity of magnetisation. ;
Unit of : Ampere/meter χ (iii) Magnetic susceptibility ( m)
[It is a scalar with no units and di- mensions] Physically it represent the ease with which a magnetic material can be magnetised.A material with χ more m, can be change into magnet easily. (iv) Magnetic permeability (µ)
It measures the degree to which a magnetic material can be penetrated (or permeated) by the magnetic field lines.
Unit of [ φ = LI ∴ Weber = Henry – Ampere]
Relative permeability µr = (It has no units and dimensions) (v) Relation between permeability & susceptibility
(vi) Curie’s Law and Curie Temperature :
(61) Must know Formula on Physics' for AIPMT
χ ∝ (T = absolute temperature) (vii) Curie Temperature : The magnetic susceptibility of these substances decreases on increasing the temperature, a fer- romagnetic substance behaves like a paramagnetic substance. This particular temperature is called the curie temperature of the substance. Curie temperature of iron and nickel is 770oC and 370oC. (viii) Curie-Weiss law : At temperature above curie point, the magnetic susceptibility of ferromagnetic substance is in-
versely proportional to (T–Tc)
χ ∝
(TC = Curie Temperature) Faraday’s Laws of Electromagnetic Induction First law Whenever the amount of magnetic flux linked with a closed circuit changes, an emf is induced in the circuit. The induced emf lasts so long as the change in magnetic flux continues. Second law The magnitude of emf induced in a closed circuit is directly proportional to rate of change of magnetic flux linked with the circuit. If the change in magnetic flux in a time dt is dφ then
Lenz’s Law
MOTIONAL EMF FROM LORENTZ FORCE
Motional emf in wireacb in a uniform magnetic field is the motional emf in an imaginary wireab .
Thus eacb = eab = (length of ab) = the component of velocity perpendicular to both and ab. From right hand rule b is at higher potential and a at lower potential.
Hence, Vba = Vb – Va = (ab) (v cosθ) (B) Induced E.M.F. Due to Rotation of a Conductor Rod in a Uniform Magnetic Field
SELF INDUCTANCE
(62) Must know Formula on Physics' for AIPMT
L = Self- Inductance of a Plane Coil Total magnetic flux linked with N turns,
Self-Inductance of a Solenoid Let cross-sectional area of solenoid = A, Current flowing through it = I
Lm = Grouping of Coils (a) Coils in Series
(b) Coils in Parallel
MUTUAL INDUCTION
where M : is coefficient of mutual induction. Coefficient of mutual inductance between two solenoids
Coefficient of mutual inductance of two plane concentric coils
For two magnetically coupled coils : here ‘K’ is coupling factor between two coils and its range 0 ≤ K ≤ 1 ⇒ For ideal coupling Kmax = 1 Mmax = (where M is geometrical mean of L1 and L2)
For real coupling (0 < K < 1) M = Inductance in Series and Parallel Two coil are connected in series : Coils are lying close together (M) ≠ If M = 0, L = L1 + L2 If M 0, L = L1 + L2 + 2M
(a) When current in both is in same direction, then L = (L1 + M) + (L2 + M)
(b) When current flow in two coils are mutually in opposite directions, then L = L1 + L2 – 2M
(63) Must know Formula on Physics' for AIPMT Two coils are connected in parallel :
(a) If M = 0 then or L = (b) If M ≠ 0 then Magnetic Energy Per Unit Volume or Energy Density
GROWTH AND DECAY OF CURRENT IN AN INDUCTOR Growth of current.
where
If τ = L/R, then
I = I0(1 – 0.368) = 0.632 I0 = 63.2% I0. We may define time constant ofLR circuit as the time in which current in the circuit grows to 63.2% of the maximum value of current. –t/τ ∞ Again, we find that for I = I0, e = 0 or t = Decay of current.
where
τ –1 If = L/R, then I = I0e = Hence we may define time constant of LR circuit as the time in which current decayss to 36.8% of the maximum value.
Again, I = 0 e–t/τ = 0 or t = ∞ Efficiency of the d.c. motor.
(64) Must know Formula on Physics' for AIPMT
Transformer
N2/N1 is called turn ratio (r) of the transformer.
If r > 1, N2 > N1 will be a step-up transformer and
If r > 1, N2 < N1 it will be a step-down transformer. ∴ Input power = Output power
V1I1 = V2I2
∴ From equation (2) and (3), we get
Efficiency of a transformer It is defined as the ratio of the output power of a transformer to the input power supplied to the transformer. Mathematically
Alternating Current AVERAGE VALUE OR MEAN VALUE
average value of current for half cycle < I > = ROOT MEAN SQUARE (rms) VALUE
(65) Must know Formula on Physics' for AIPMT
This shows that capacitor blocks the flow of d.c. but provides an easy path for a.c.
(66) Must know Formula on Physics' for AIPMT
(67) Must know Formula on Physics' for AIPMT
(68) Must know Formula on Physics' for AIPMT
RAY OPTICS Reflection from plane mirror If there are two plane mirror inclined to each other at an angle θ the number of image of a point object formed are determined as follows :
(i) If is even. No. of image (n) = m – 1
(ii) If is odd. There will be two case (a) When object is not on bisector, then n = m (b) When object is at bisector, then n = m – 1 Relation between f and R for the spherical mirror
Mirror formula
Magnification
Other formulae of magnification
Velocity of the image of an object :
Area of image :
Power of a mirror The power of a mirror is defined as
refractiVE INDEX
wµg = Refractive index of glass with respect to water and light is travelling from water to glass.
Refractive index depend on wavelength, nature of medium and on temperature.
(69) Must know Formula on Physics' for AIPMT Apparent depth and normal shift
Object in a rarer medium is seen from a denser medium
Lateral Shift
total internal reflection Conditions
(i) Angle of incident > Critical angle [i = θC] (ii) Light should travel from denser to rare medium ⇒ Glass to air, water to air, Glass to water
Snell’s Law at boundary x - y
µD sin θC = µR sin 90
When light ray travel from µ refractive index medium to air then µR = 1, µD = µ
sin θC = 1/µ sin θC = ∝ λ ⇒ θC (Red) > θC (violet) For TIR
i > θC sin i > sin θC sin i > 1/µ A point object is situated at the bottom of tank filled with a liquid of refractive index µ upto height h it is found light from the source come out of liquid surface through a circular portion above the object then radius and area of circle
Angle which the eye of fish make = 2θC = 2 × 49° = 98°. This angle does not depend on depth of liquid. Refraction from curved surface
Lens-maker’s formula :
(70) Must know Formula on Physics' for AIPMT
Linear magnification produced by a lens :
Power of lens Power of a lens is defined as the ability of the lens to converge a beam of light falling on the lens. It is measured as the reciprocal of focal length of the lens
i.e. Focal Length of equivalent lens
Newton's Formula :
f =
x1 = distance of object from first focus.
x2 = distance of image from second focus. Displacement Method :
Achromatism
Refractive Index of the prism
when prism is thin. Angular dispersion It is the difference of angle of deviation of violet colour and red colour
Angular dispersion θ = δV – δR = (µV – 1)A – (µR – 1)A = (µV – µR)A
It depends on prism material and on the angle of prism. Dispersive Power (ω)
It is ratio of angular dispersion (θ) to mean colour deviation (δy)
Refractive index of mean colour (µy) = (µv + µR)/2 Deviation without dispersion( θ = 0) (71) Must know Formula on Physics' for AIPMT
Dispersion without deviation :( δ = 0)
Simple microscope
Magnifying power (MP) =
(Minimum Magnification)
(Maximum Magnificaiton) Compound Microscope When final image is formed at minimum, distance of distinct vision.
M
When final image is formed at infinity
Astronomical Telescope
If the final image is at infinity
length of the tube If the final image is at D
Length of the tube is WAVE OPTICS Analysis of interference of light
Resultant amplitude and Phase angle θ = tan–1
Constructive Interference When both waves are in same phase. So phase difference is an even multiple of π
(72) Must know Formula on Physics' for AIPMT ⇒ φ = 2nπ; n = 0, 1, 2,.... Destructive interference When both the waves are in opposite phase. So phase difference is an odd multiple of π. φ = (2n – 1)π; n = 1, 2,.... Young's Double Slit Experiment (YDSE) Condition for bright and dark fringes Bright Fringe
th The distance of n bright fringe from the central bright fringe xn = n Dark Fringe The distance of the mth dark fringe from the central bright fringe
xm = Fringe width
β = Angular Fringe width
Colours in thin films For minima or destructive interference
When path differnece is odd multiple of λ/2 ⇒ So the film will appear dark if 2µt cos r = nλ For transmitted system
Since No additional path difference the transmitted rays CT1 and ET2. So the net path difference between them is x = 2µt cos r
For maxima 2µt cos r = nλ, n = 0, 1, 2,.... For minima 2µt cos r = (2n + 1) , n = 0, 1, 2,.... diffraction of light
For minima where n = 1, 2, 3 ....
For maxima or where n = 1, 2, 3 .....
Linear width of central maxima wx = 2x ⇒
Angular width of central maxima Intensity curve of Fraunhofer’s diffraction
(73) Must know Formula on Physics' for AIPMT
Intensity of maxima in Fraunhofer’s diffraction is determined by
I0 = intensity of central maxima n = order of maxima
intensity of first maxima
intensity of second maxima Resolving Power of Microscope
µ sin θ is called numerical aperture of the microscope. To increase resolving power of microscope. We decrease λ i.e., use blue light of smaller wavelength. We cannot increase θ because in that case aperture of the lens would increase. Resolving Power of Telescope
Law of Malus
I ∝ cos2θ ⇒
Brewsters angle θB is given by
where n1 is the refractive index of the medium through which the incident and reflected rays travel
and n2 is the refractive index of the medium from which the light reflects. MODERN PHYSICS einstein’s photo electric equation ν φ h = (K.E.max) + 0
Quantum efficiency = Linear momentum of photon Momentum of photon
(74) Must know Formula on Physics' for AIPMT Effective mass of photon
Intensity of light
.... (i)
Here P = power of source, A = Area, t = time taken E = Energy incident in t time =Nh ν N = number of photon incident in t time De Broglie Wavelength Associated with Uncharged Particles
wavelength associated with the particle Orbital Speed of electron
Orbital Radius of electron
.... [Orbit radii] For hydrogen atom, Z = 1, so
..... [Orbit radii for H-atom]
Putting n = 1, we get the minimum radius, called the "Bohr radius", denoted by a0, i.e., ..... [Bohr radius] Hence,
.....[Orbit radius of H-atom]
and ....[Orbit radius of H-like ion] Energy
.... [nth state energy of H-atom] .... [Minimum energy of H-atom, i.e., ground state energy]
(75) Must know Formula on Physics' for AIPMT
.... [Ground state energy of H-like ions]
Wavelength of Radiation
Absorption of X-ray
When X-ray pass through x thickness then its intensity
I0 = Intensity of incident X-ray I = Intensity of X-ray after passing through x distance µ = absorbtion coefficient of material
Binding Energy (Eb) 2 Binding energy of a nucleus is the energy required to split it into its nucleons. ∆Eb = ∆mc Q value for α decay A → A – 4 4 ZX Z – 2Y + 2He
Q = kα + ky .... (i) Momentum conservation,
pY = pα .... (ii)
⇒
⇒ Rutherford and Soddy’s Law
Half Life (Th)
Mean or Average Life (Ta) It is the average of age of all active nuclei.
Activity(A) or Decay Rate (R) It is the rate of decay of a radioactive sample.
–λt or R = R0e unit of R is becqueral
(76) Must know Formula on Physics' for AIPMT 1 Bq = 1 decay/sec. 1 Ci(Curie) = 3.7 × 1010 decay/sec 1 Rutherford (1 Rd) = 106 decay/sec SEMICONDUCTOR ELECTRONICS:MATERIALS, DEVICES AND SIMPLE CIRCUITS
Conductivity of semi conductor Common Base Amplifier :
DC current gain
AC current gain
αac = AC voltage gain
AV = AC power gain
AC power gain = = × resistance gain Common Emitter Amplifier :
DC current gain
β = AC current gain
β ac =
(77) Must know Formula on Physics' for AIPMT Voltage Gain
Av = βac ×
AC power gain =
AC power gain = Relation between α and β
β = or α = Digital Electronics and Logic Gates OR GATE
AND GATE
NOT GATE
(78) Must know Formula on Physics' for AIPMT
The XOR gate
COMMUNICATION SYSTEM Size of antennas required for their efficient radiation would be large i.e. about 75 km as explained below: For efficient radiation of a signal, the minimum length of an antenna is one quarter wavelength
. Effective power (P) radiated by an antenna is given by P ∝
(79)