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Magnetic Fields and Forces

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Magnetic Fields and Forces Magnetic Fields and Forces (1) Magnetic Fields are produced by moving charges. (2) Magnetic Fields exert forces on moving charges. (3) Stationary Charges do not produce magnetic fields. (4) Stationary Charges are not affected by stationary magnetic fields. Permanent Magnets There are naturally occurring magnetic substances. These were long used for navigation. As a result, we speak of “north” and “south” magnetic poles, instead of positive and negative magnetic charges. Permanent magnets can also be made by people from raw materials. These magnetic poles also respect an “opposites attract” rule: N S N S attraction N S S N repulsion Earth: North and South Revisited The ancients defined the “north” end of a permanent magnet as the end that is attracted to the “north” pole of the earth. points north S N S Therefore the north geographic pole of earth is a south magnetic pole. N Magnetic Field of a Bar Magnet This is a magnetic dipole field, very similar to the electric dipole field of two opposite electric charges. Note that magnetic field lines are all closed loops, unlike electric field lines that must start and stop on a charge. Important: There is no magnetic “monopole.” N Such a thing has never been observed. North and South magnetic poles always come in pairs of equal strength. Properties of the magnetic force on a charge moving in a B field 1. The magnetic force is proportional to the charge q and speed v of the particle 2. The magnitude and direction of the magnetic force depend on the velocity of the particle and on the magnitude and direction of the magnetic field. 3. When a charged particle moves in a direction parallel to the magnetic field vector, the magnetic force F on the charge is zero. 4. When the velocity vector makes an angle with the magnetic field, the magnetic force acts in a direction perpendicular to both v and B; that is, F, is perpendicular to the plane formed by v and B. 5. The magnetic force on a positive charge is in the direction opposite the direction of the force on a negative charge moving in th same direction. 6. If the velocity vector makes an angle with the magnetic field, the magnitude of the magnetic force is proportional to sin Force on a moving charge exerted by a magnetic field A magnetic field exerts a force on a charge if • the charge is moving, and • the velocity of the charge has a component perpendicular to the magnetic field. The direction of the force is perpendicular to both the magnetic field and the velocity of the charge. We describe the force by this r r cross product : F = qvr × B The magnitude of the force r is given by F = qvBsinθ. θ The right-hand rule Magnetic Field Units The standard unit is the Tesla = (N/C)(s/m). Note that this has the units of an electric field divided by a velocity. A smaller, more convenient unit, in very common use, is the Gauss = 10−4 Tesla. The earth’s magnetic field, at the surface of the earth, is about one-half Gauss. Example A proton moves with a speed of 8 X 106 m/s along the x axis. It enters a region where there is a field of magnitude 2.5T directed at an angle of 60° to the x-axis and lying in the xy plane. Calculate the initial magnetic force and acceleration of the proton. Magnetic Force on a Current Carrying Conductor • It should be no surprise that a current carrying conductor experiences a force when placed in a magnetic field. • This follows from the fact that the current represents a collection of many charged particles in motion – The resultant force on the wire id the result of the sum of the individual forces on the charged particles. Force on a current in a magnetic field Consider a wire of length L and cross sectional area A carrying a current I. We place the wire in a magnetic field B at some angle θ with respect to the wire. r r The force on each charge q is F = qvr × B . If it takes t seconds for the charge to travel the length L, then v = L/t. Then the force is r ⎛ q⎞ r F = ⎜ ⎟() vr t × B ⎝ t ⎠ r r r or F = IL × B . • A curved wire carrying a currrent I; the wire is located Case I in a uniform external magnetic field B. –Because B is uniform it can • From the law of vector be taken outside of the integral addition the sum of all the ds vectors equals the vector L which is directed from a to b • An arbitrarily shaped, closed loop carrying a current I is Case II placed in a uniform external magnetic field B. –Because B is uniform it can again be taken outside of the integral • The vector sum of the displacements must be taken over the closed loop Motion of charges in uniform electric and magnetic fields (I) Uniform Electric Field: The force is constant in direction and magnitude. The charge continues to accelerate and increase in speed. Final motion is primarily along the field direction. v0 r E Motion of charges in uniform electric and magnetic fields (II) Uniform Magnetic Field: The force is not constant in direction but changes with the direction of the velocity. The particle does not speed up, because the force is always perpendicular to the velocity. The magnetic field does no work on the particle. The magnitude of the velocity remains constant, so the magnitude of the force remains constant. v0 The result is a r B circular motion about the field lines, with a constant speed. Circular orbit of a charged particle in a uniform magnetic field The magnitude of the force is F = qvB. The force is always directed towards the center of the orbit. A circular orbit requires a centripetal force: mv 2 F = c r mv 2 mv ⇒ qvB = ⇒ r = r qB Note : momentum p = mv = qBr Velocity Selector Using crossed E and B fields, we can create a device which separates charged particles according to velocity. r r r If qE = qvB, then F = qE + qvr × B = 0, E v = is the selected velocity and the particle will pass through undeflected. B Mass Spectrometer Device which separates particles of the same charge by mass. These are used in a variety of scientific and medical settings. 1 kinetic energy qV = mv 2 2 2qV velocity of particles ⇒ v = m entering the field 14 4 2 4 4 3 detector plate mv 2mV radius of orbit r = = qB qB2 qB2r2 mass of particles ⇒ m = 2V reaching the detector The Hall Effect (I) A conducting slab of width w carrying a current I is placed in a magnetic field B oriented perpendicular to the r E direction of the current. The magnetic force qvdB on the charge carriers pushes them to one edge of the slab. This separation of charge produces an electric field, which opposes any further separation of charge. Eventually these two forces come into equilibrium: qE = qvd B The electric field produces a potential difference V = Ew : V q = qv B ⇒ V = wv B the "Hall voltage" measures v w d H d d The Hall Effect (II) But suppose that the charge carriers are negative instead of positive. The charge separation set up will be in the opposite sense as before, and so the sense of the electric field and potential difference will be reversed. The magnitude of the Hall voltage is sensitive to the drift velocity. The sense of the Hall voltage tells the sign of the charge carriers..
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