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Advanced Physics (2nd term)

 2 main parts: (8c) and Quantum Mechanics (6c)  Applications: 6 seminars (2 hours each two week).

Bibliography:  Zemanski and Sears: University Physics  Halliday and Resnik: Fundamentals Of Physics  Teachers’s notes: web site http://spin.utcluj.ro/webphysics/AdvancedPhysics.html Structure

(I) Electromagnetism:  , electric forces, electric , . and . Electrokinetics (9c)  Current, resistance, .  and magnetic forces. Electromagnetic induction.  Magnetic materials and superconductors.  Electromagnetic waves. Set of 4 eq. (Maxwell): basis of electromagnetism

(II) Quantum mechanics: (intro-basis)  Limitations of Classical Physics and historical hypotheses.  The wave-particle duality. Heisenberg uncertainty.  The wave quantum mechanics. Wave function. Schrodinger equation. (5c)  Applications of QM (Qstep, Qwell, Qbox, tunnel effect, Quantum Harmonic Oscillator).  QM as basis: From atom to solid state electronics.  Introduction in Spintronics. Physics, nanotechnologies @ modern devices. (I) Electromagnetism ELECTRIC CHARGE AND

Electromagnetic interactions involve particles that have a property called electric charge, an attribute that is as fundamental as mass. Just as objects with mass are accelerated by gravitational forces, so electrically charged objects are accelerated by electric forces.

1. Electric Charge

 The ancient Greeks discovered as early as 600 B.C. that after they rubbed amber with wool, the amber could attract other objects. Today we say that the amber has acquired a net electric charge, or has become charged.  The word “electric” is derived from the Greek word elektron, meaning “amber”.

Experiments have shown that there are exactly two kinds of electric charge. Benjamin Franklin called them positive (+) and negative (-).

Two positive charges or two negative charges repel each other. A positive charge and a negative charge attract each other.

Benjamin Franklin (1706–1790) Experiments in electrostatics.

Positively charged objects Negatively charged Positively charged objects and negatively charged objects repel each other repel each other objects attract each other Application: Schematic diagram of the operation of a laser printer

 The printer’s light-sensitive imaging drum is given a positive charge.  As the drum rotates, a laser beam shines on selected areas of the drum, leaving those areas with a negative charge.  Positively charged particles of toner adhere only to the areas of the drum “written” by the laser.  When a piece of paper is placed in contact with the drum, the toner particles stick to the paper and form an image. 2. Electric Charge and the Structure of Matter

Question: What happens phenomenologically when you charge a rod by rubbing it (fur, silk) ? To answer this question, we must look more closely at the structure of atoms, the building blocks of ordinary matter.

• The negatively charged are held within the atom by the attractive electric forces exerted on them by the positively charged nucleus. • The and are held within stable atomic nuclei by an attractive interaction, called the strong nuclear force, that overcomes the electric repulsion of the protons. The strong nuclear force has a short range, and its effects do not extend far beyond the nucleus. • The and : combinations of other entities called quarks, which have fractionary charges: (±1/3 and ±2/3 times the charge).

• Isolated quarks have not been observed, and there are theoretical reasons to believe that it is impossible to observe isolated quarks.  The negative charge of the electron has exactly the same magnitude as the positive charge of the proton.  In a neutral atom the number of electrons equals the number of protons in the nucleus (Z= atomic number of the element) , and the net electric charge= the algebraic sum of all the charges =0

 If one or more electrons are removed from an atom, what remains is called a positive ion.  A negative ion is an atom that has gained one or more electrons.  This gain or loss of electrons is called ionization. Charging electrostatically an object  creating ions (+) or (-)

The magnitude of charge of the electron or proton is a natural unit of charge.

Every observable amount of electric charge is always an integer multiple of this basic unit.

Charge of an object= multiple (N) of elementary charges Q=Ne (charge quantification*) -19 e = 1,602 X 10 C *Millikan 1909, Nobel Price (1923)

Principle of conservation of charge (universal conservation law)

The algebraic sum of all the electric charges in any closed (isolated) system is constant.

 Hence the total electric charge on the two bodies together does not change.  In any charging process, charge is not created or destroyed; it is merely transferred from one body to another.

If we rub together a plastic rod and a piece of fur, both initially uncharged, the rod acquires a negative charge (since it takes electrons from the fur) and the fur acquires a positive charge of the same magnitude (since it has lost as many electrons as the rod has gained). triboelectric series

 Electrons in a material are not all equally bonded. Some substances lose electrons quite easily while others will tend to steal electrons from others.  So when we rub the two materials => transfer of electrons from one material to another. 3. Conductors, insulators, induced charges Some materials permit electric charge to move easily from one region of the material to another, while others do not => conductors and insulators.

 Insulators could be charged because the charges remained on the object.  We cannot charge the conductors if we held them in our hand, because the charges move from the object towards our hand.  The facility with which the charges move in a material is described by the relaxation time = the time required for the charges to reach their equilibrium in an object.

Material Relaxation time (s) Copper 10-12 s Glass 2 s Amber 4000 s Polystyrene 1010s (300 years)

The relaxation time determined by the way the atoms are bounded in the object.  In metals, electrons are shared by very many atoms (metal bonds), which allow the electrons to move quite easily.  In other substances, in which there are ionic or covalent bonds, the of the electrons is much more difficult. Charging by contact

The ball acquire the same charge (charge redistribution) Charging by Induction

the plastic rod can give another body a charge of opposite sign without losing any of its own charge.

=> Induced charges Electric Forces on Uncharged Objects

The charges within the molecules of an insulating material can shift slightly. As a result, a comb with either sign of charge attracts a neutral . By Newton’s third law the neutral insulator exerts an equal-magnitude attractive force on the comb. 4. ’s law

. Charles Augustin de Coulomb (1736–1806) studied the interaction forces of charged particles (1784). . He used a torsion balance similar to the one used 13 years later by Cavendish to study the much weaker gravitational interaction.

direction of the force For point charges:

Permitivity of vacuum

The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Electric force versus gravitational force

This is always true for interactions of atomic and sub-nuclear particles. 1040 But within objects the size of a person or a planet, the positive and negative charges are nearly equal in magnitude, and the net electric force is usually much smaller than the gravitational force. Superposition of Forces

 Coulomb’s law as we have stated it describes only the interaction of two point charges.  When two charges exert forces simultaneously on a third charge, the total force acting on that charge is the vector sum of the forces that the two charges would exert individually.

=> Principle of superposition of forces (holds for any collection of charges)

See problems seminary 5. Electric field

When two electrically charged particles in empty space interact, how does each one know the other is there? => Necessity to introduce the concept of electric field.

Any charge Q modifies the properties of the space around it => electric field A charged body creates an electric field in the space around it.

The electric force on a charged body is exerted by the electric field created by other charged bodies.

Unit: [N/C] The charge q0 can be either positive or negative.

q0 (positive +) => and have same direction

⃗ q0 (negative -) => and have opposite direction

⃗

Correspondence to gravitational force:

= (intensity of the) electric field

Electric Field of a Point Charge

 The location of the charge = the source point  The point where we are determining the field = the field point

is produced by q and lead to a force 0 on a test charge q0

A point charge produces an electric field at all points in space

The field produced The field produced by a positive point by a negative point charge points away charge points toward from the charge. the charge. Since can vary from point to point, it is not a single vector quantity but rather an infinite set of vector quantities, one associated with each point in space => vector field.   E() r E(x , y , z )      E  Ex ( x , y , z ) i  Ey (,,) x y z j  Ez (,,) x y z k    r  xi  yj  zk Vector field (e.g. electric field, magnetic field, gravitational field..)

Scalar field (e.g. temperature field)

In any point the temperature In any point the field is described by a represents a value (scalar) vector (magnitude+ orientation) In some situations the magnitude and direction of the have the same values everywhere throughout a certain region  the field is uniform

E1: in electrostatics the electric field at every E2: the electric field within the two plates of a point within the material of conductor is zero plane is constant

Vector field representation 6. Electric-Field Calculations The Superposition of Electric Fields

Consequence of force superposition principle

Total field in a point P = vector sum of individual fields produced by

individual charges qi at

See seminary

Electric field of a continuous charge distribution

Total E obtained by integration E1. Calculating the electric field on the axis of a ring of positive charge dQ  dS

=dQ/dS surface (C/m2)

To find Ex we integrate this expression over the entire ring—that is, for s from 0 to 2a (the circumference of the ring). E2. Calculating the electric field of a positive charge line segment

linear charge density (C/m) E3. Field of a uniformly charged disk Surface charge density (C/m2)

On a ring of area dA we have the charge dQ

We use dQ in place of Q in the expression for the field due to a ring that we found in E1, and replace the ring radius a with r.

Then the field component dEx at point P due to this ring is:

If the disk is very large (or if we are very close to it), so R>>x we can neglect with respect to 1 the term:

the electric field produced by an infinite plane sheet of charge is independent of the distance from the sheet. The field direction is everywhere perpendicular to the sheet, away from it. 7. Electric-Field lines An electric is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric-field vector at that point.

Electric field lines show the direction of at each point their spacing gives a general idea of the magnitude of at each point: strong lines bunched closely together; weaker lines are farther apart. At any particular point, the electric field has a unique direction, so only one field line can pass through each point of the field. In other words, field lines never intersect. 8. Electric An electric is a pair of point charges with equal magnitude and opposite sign (a positive charge +q and a negative charge -q) separated by a distance d.

The product of the charge q and the separation d is the magnitude of a quantity called the electric dipole , denoted by p.

Force and on an Electric Dipole When placed in an electric field the net force on this electric dipole is zero, but there is a torque directed into the page that tends to rotate the dipole clockwise. of an Electric Dipole

When a dipole changes direction in an electric field, the electric-field torque does on it, with a corresponding change in potential energy.

In a finite displacement from 1 to 2

But the work is minus the change of the potential energy

The potential energy has its minimum when =o so p and E are parallel

In an external electric field an electric dipole rotates to always orient along the field direction Heating food with microwaves

Food can be warmed and cooked in a microwave oven if the food contains because water molecules are electric dipoles.

Microwaves produced by a magnetron source are directed and adsorbed by the water contained in the food. The molecule of water (dipole) tries to follow the electric field component of the microwave which oscillates by changing orientation with f=2.45GHz.

As the water molecule rotates they bump into other molecules surrounding them and transfer some kinetic energy, dissipative friction, breaking of hydrogen bonds => field heating

GAUSS’s Law

In Physics, an important tool for simplifying problems is the symmetry properties of systems.

Many physical systems have symmetry:  a cylindrical body doesn’t look any different after you’ve rotated it around its axis  a charged metal sphere looks just the same after you’ve turned it about any axis through its centre.

Gauss’s law is part of the key to using symmetry considerations to simplify electric-field calculations. Carl-Friedrich-Gauss

Given any general distribution of charge

 we surround it with an imaginary surface that encloses the charge.

 then we look at the electric field at various points on this imaginary surface.

Gauss’s law is a relationship between the field at all the points on the surface and the total charge enclosed within the surface. 1. Charge and

“Given a charge distribution, what is the electric field produced by that distribution at a point P ?”

Superposition/ integration: The total field at P is then the vector sum of the fields due to all the point charges – hard task :)) => Looking for alternative formalism based on symmetry

Pb: How can you measure the charge inside a box without opening it?

To determine the contents of the box, we actually need to measure only on the surface of the box

Electric Flux and Enclosed Charge Flux comes from Latin (=flow)

The electric field on the surface of boxes containing a charge q

Three cases in which there is zero net charge inside a box and no net electric flux through the surface of the box.

An empty box immersed in a uniform electric field. Flux of a Uniform Electric Field

The direction of is always chosen to be outward    E>0 outward electric flux E<0  inward electric flux

Flux of a Nonuniform Electric Field

Surface integral Gauss’s law Formulated by (1777–1855), one of the greatest mathematicians.

 Is an alternative to Coulomb’s law.  While completely equivalent to Coulomb’s law, Gauss’s law provides a different way to express the relationship between electric charge and electric field.

Gaussian surfaces are imaginary. Remember that the closed surface in Gauss’s law is imaginary; there need not be any material object at the position of the surface. We often refer to a closed surface used in Gauss’s law as a Gaussian surface.

Interpretation of Gauss law (1st eq. Of Maxwell –set of 4: the electric charge = source of electric field Applications of Gauss’s Law

Gauss’s law is valid for any distribution of charges and for any closed surface. Gauss’s law can be used in two ways.  If we know the charge distribution, and if it has enough symmetry to let us evaluate the integral in Gauss’s law, we can find the field.  Or if we know the field, we can use Gauss’s law to find the charge distribution, such as charges on conducting surfaces.

E1. Electric Field of Point Charge

Considering a Gaussian surface in the form of a sphere at radius r, the electric field has the same magnitude at every point of the sphere and is directed outward. The electric flux is then just the electric field times the area of the sphere.

The electric field at radius r is then given by:

If another charge q is placed at r, it would experience a force: => Coulomb’s law E2. Field of a sphere of Uniform Charge

 Sphere of uniform charge density and total charge Q

 Considering a Gaussian surface in the form of a sphere at radius r > R, the electric field has the same magnitude at every point of the surface and is directed outward. The electric flux is then just the electric field times the area of the spherical surface.

The electric field outside the sphere (r > R) is seen to be identical to that of a point charge Q at the center of the sphere.

For a radius r < R, a Gaussian surface will enclose less than the total charge and the electric field will be: Q 4 r3 r3 Q  V   Q int 4 R3 3 R3 3

3 2 Qint Q r EA  E 4 r   3  0 0 R E3. Electric Field of Conducting Sphere

Considering a Gaussian surface in the form of a sphere at radius r > R , the electric field has the same magnitude at every point of the surface and is directed outward. The electric flux is then just the electric field times the area of the spherical surface.

The electric field is seen to be identical to that of a point charge Q at the center of the sphere.

Since all the charge will reside on the conducting surface, a Gaussian surface at r< R will enclose no charge, and by its symmetry E can be seen to be zero at all points inside the spherical conductor E4. Uniformly charged wire Gaussian surface = cylinder

E6. Plane capacitor (2 uniformly charged plates with +Q and –Q) E5. Uniformly charged surface ELECTRIC POTENTIAL

This chapter is about energy associated with electrical interactions

Work and Energy defined in the context of mechanics will be applied to electric charge, electric forces, and electric fields.

When a charged particle moves in an electric field, the field exerts a force that can do work on the particle. This work can always be expressed in terms of electric potential energy. Just as gravitational potential energy depends on the height of a mass above the earth’s surface, electric potential energy depends on the position of the charged particle in the electric field.

We’ll describe electric potential energy using a new concept called electric potential, or simply potential.

In circuits, a difference in potential from one point to another is often called .

The concepts of potential and voltage are crucial to understanding how electric circuits and many other devices work. 1. Electric Potential Energy

First, when a force acts on a particle that moves from point a to point b, the work done by the force Wa->b is given by a line integral: b a

If the force is conservative the work done by F can always be expressed in terms of a potential energy U.

Work-energy theorem: Wab  K  Kb  Ka

total mechanical energy (kinetic plus potential) is conserved Electric Potential Energy in a Uniform Field analogy

Wab  mgh

U  mgy U  q0 Ey

For charge moving by field: Wab=Ua-Ub>0 => Ub the charge moves to decrease the potential energy

Whether the test charge is positive or negative, the following general rules Same for mass m falling to decrease U along apply: U decreases if the test charge moves in the direction of the electric the direction of the gravitational force force . U increases if the charge moves in the direction opposite to the force ⃗ = ⃗ = Gravitational vs Electric field analogy

U  mgy U  qEy y y

Mass m falling to decrease U along the U decreases if the test charge moves in the direction of the electric direction of the gravitational force force . U increases if the charge moves in the direction opposite to the force ⃗ = ⃗ = Electric Potential Energy of Two Point Charges

 The idea of electric potential energy isn’t restricted to the special case of a uniform electric field.  We can apply this concept to a point charge in any electric field caused by a static charge distribution

Simple analysis

Wab=Ua-Ub Depends only on the end points a and b.

valid no matter what the signs of the charges q and q0 Graphs of the potential energy of two point charges and versus their separation r.

Repulsion (r increase) to decrease U Attraction (r decrease) to decrease U

Similar sign charges repel Opposite sign charges attract Potential energy is always defined relative to some reference point where U=0

(*) U  0 if r =

  qq  1 1  0 U a  Wa  F() r dr  q0 E() r dr From: Wab      Ua Ub   a a 4 0  ra rb  rb =   Ub =0 U() r  W  F() r dr  q E() r dr r   0 r a

Therefore U in a point r represents the work that would be done on the test charge by the field forces (field produced here by q) if moved from the initial distance r to infinity.

We emphasize that the potential energy U is a shared property of the two charges.

Equation (*) also holds if the charge q0 is outside a spherically symmetric charge distribution with total charge q; the distance r is from the center of the distribution. Electric Potential Energy with Several Point Charges

the potential energy associated with the test charge

q0 at point a is the algebraic sum (not a vector sum):

gives only the potential energy in point a associated with the presence of the test charge

q0 in the field E produced by q1, q2, q3,…

But there is also potential energy involved in assembling these charges (total potential energy U of the system q1, q2, q3,… ):

Pair interaction potential energy:

This sum extends over all pairs of charges Interpreting Electric Potential Energy

The potential-energy difference Ua-Ub equals the work Wa->b that is done by the electric force when the particle moves from a to b.

Wa->b = Ua-Ub

When Ua is greater than Ub the field does positive work on the particle as it “falls” from a point of higher potential energy (a) to a point (b) of lower potential energy

Alternative (equivalent definition) :

The potential energy difference Ua-Ub is defined as the work that must be done by an external force to move the particle slowly from b to a against the electric force. 2. Electric Potential

we want to describe the potential energy U on a “per unit charge” basis, just as electric field describes the force per unit charge on a charged particle in the field.

the concept of electric potential, often called simply potential.

U correlated to

V=U/ q0 correlated to /q0 Potential is potential energy per unit charge. ⃗ = ⃗ Potential energy and charge are both scalars, so potential is a scalar Unit: 1V = 1J/1C (Joule/Coulomb)

the work done per unit charge by the electric force when a charged body moves from a to b is equal to the potential at a minus the potential at b. The potential Vab of a with respect to b equals the work: done by the electric field force when e a UNIT charge moves from a to b.

… that must be done by an external force to move slowly a UNIT charge moves from b to a against the electric field force.

The instrument that measures the difference of potential between two points = Voltmeter. 3. Calculating Electric Potential

r is the distance from the point charge q to the point at which the potential is evaluated. Obs: V()=0.

the potential due to a collection of point charges: V=V +V +V +… q 1 2 i 1 r1 r P q 2 the electric potential due to a collection of point charges is the scalar sum of 2 r i the potentials due to each charge.

qi

along a line, over a surface, or through a volume

r P P dq r Finding Electric Potential from Electric Field

dividing by q0

 If b= a=r; V()=0 V() r   E() r dr r

Electron Volts From: If q=e=1.6 10-19 C

1 electron volt

multiples meV, keV, MeV, GeV, and TeV

When a particle with charge e moves through a potential difference of 1 volt, the change in potential energy is 1 eV. Calculating Electric Potential

E1. charged conducting sphere

Gauss Inside the car: protected because E=0

Pb. When you step out  (discontinuity in E). V() r   E() r dr r 4. Equipotential Surfaces

 Field lines help us visualize electric fields.

 In a similar way, the potential at various points in an electric field can be represented graphically by equipotential surfaces.

By analogy to contour lines on a topographic map, an equipotential surface is a three- dimensional surface on which the electric potential V is the same at every point.

If a test charge q0 is moved from point to point on such a surface, the electric potential energy remains constant.

Equipotential Surfaces and Field Lines

Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work on such a charge.

is perpendicular on the equipotential surface at every point so that = is perpendicular to the displacement ⃗ Field lines and equipotential surfaces are always mutually perpendicular

Equipotentials and Conductors

 When all charges are at rest, the surface of a conductor is always an equipotential surface.

 When all charges are at rest, the entire solid volume of a conductor is at the same potential.

 E field just outside the conductor is always perpendicular to the surface.

which is impossible because the electric force is conservative 5. Potential Gradient Electric field and potential are closely related.

but

Electric field = minus the Gradient gradient of the potential

if is radial with respect to to a point or axis and r is the distance

q dV q Ex. Point charge V() r   E() r    2 4 0r dr 40r 6. Poisson and Laplace equations

  Q Qint Gauss theorem:  E dA  int   0  V

Surface  including the volume V Gauss-Ostrogradski theorem (math):   E  dA    EdV Transforms a surface integral in volume integral  V

Divergence of a vector field: E E E   E  div() E  x  y  z x y z

       Nabla operator   i  j  k  x y z      Vector field  E  Ex i  Ey j  Ez k

Measures the expansion or the contraction of a vector field Physical meaning of divergence

We may regard the divergence of a vector field at a given point as a measure of how much the field diverges or emanates from that point

source sink

div( )>0 div( )<0 div( )=0

The Divergence describes the net flow going in vs going out for a vector field. dQ    Qint Q  dV   E dA    EdV  int  dV Volume charge density   V  V  0

 1  1      EdV  dV  E   dV  0   E        V 0 V V   0  0 Gauss law for the electric field in the differential formulation  Because: E  V  grad() V

2  ( V )  2V  V  V  V   Poisson equation 0 2 2 2 2    2 2 2     2  2  2  V(,,) x y z  V(,,) x y z  V(,,) x y z  x y z 2  2  2   x y z  0 = Laplace operator: 2nd order differential operator

Knowing (x,y,z), by solving (numerically, analytically) the Poisson eq. => the electric potential V(x,y,z)

If ( )= (x,y,z)=0  no charge density source 2V  V  0 Laplace equation ⃗ 2V(,,) x y z 2V(,,) x y z 2V(,,) x y z    0 x2 y2 z 2 Finite Difference Methods Matlab 2V(,) x y 2V(,) x y (,)x y   V(,) x y  V(,) x y      ( , E  V   i  j  x2 y2   x y  0 numerically ) Vector field reprezentation Plane capacitor Charge uniformly distributed in a square Uniformly charged sphere Cylindrical capacitor