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Magnetic Forces, Fields, and Faraday’s Law ISAT 241 Fall 2003 David J. Lawrence

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Magnetic Fields Every magnet, regardless of its shape, has two “poles”, called “north” and “south”. These poles exert forces on each other in a manner analogous to electric charges. The poles received their names from the behavior of a magnet in the earth’s magnetic field. Difference: Electric charges can be “isolated” while magnetic poles are always found in pairs.

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Serway & Jewett, Principles of Physics, 3 rd ed. Figure 22.1

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Magnetic Fields Recall: the gravitational field g at some point in space is the gravitational force acting on a “test mass” divided by the test mass, i.e., g = F g /m o The electric field E at some point in space is the electric force acting on a “test charge” divided by the test charge, i.e., E = F E /q o

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Magnetic Fields The magnetic field vector B (or “magnetic induction” or “magnetic flux density”) is now defined at some point in space in terms of the magnetic force acting on an appropriate “test object”. “Test Object” = a charged particle moving with velocity v.

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Magnetic Fields This is the test object in a magnetic field B. The magnetic force on the test object, F B, depends on q, v, and B according to the equation v B q “ ” does not denote normal multiplication. More about this later.

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Magnetic Fields The magnitude of the magnetic force, F B, is proportional to the charge q, the speed v = | v | of the particle, and the magnetic field B. | F B | also depends on v B q

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Example Problem A proton moves with a speed of 8.0 x 10 6 m/s along the x axis. It enters a region where there is a magnetic field of 2.5 T in the xy plane, directed at an angle of 60 o to the x axis. Calculate the initial magnetic force on and acceleration of the proton. x y z v B p+p+ 60 o Redraw this diagram with the x and y axes in their “normal” directions.

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Magnetic Fields The direction of the magnetic force F B depends on the sign of the particle’s charge, the direction of its velocity, and on the direction of the magnetic field. The direction of the magnetic force F B is given by the Right Hand Rule.

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Serway & Jewett, Principles of Physics, 3 rd ed. Figure 22.4

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Serway & Jewett, Principles of Physics, 3 rd ed. Figure 22.3

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Serway & Jewett, Principles of Physics, 3 rd ed. Figure 22.6

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Magnetic Fields When the charged particle moves parallel to the magnetic field B, then F B = 0. When the velocity vector v makes an angle with the magnetic field B, the magnetic force acts in a direction perpendicular to both v and B. The magnetic force on a negative charge is in the direction opposite to the force on a positive charge moving in the same direction.

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Magnetic Force on a Charge If the velocity vector makes an angle with the magnetic field, the magnitude of the magnetic force is proportional to sin . All of these observations can be summarized by using a special vector notation to write the magnetic force: The product denoted by is called the cross product.

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Vector Cross Product The vector cross product yields a vector that is perpendicular to both of the vectors in the cross product. Quick Reference Table for Cross Products of Unit Vectors i k -j j -i -k

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Magnetic Fields v B is perpendicular to both v and B. The direction is given by the right hand rule. The magnitude of the magnetic force is F B = |F B | = |q| v B sin When v is parallel to B ( = 0 or 180 o ) then F B = 0. When v is perpendicular to B ( = 90 o ) then F B has its maximum value F B = |q|vB. The equation F B = q v B serves to define the magnetic field B.

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Example Problem A proton moving at 4.0 x10 6 m/s through a magnetic field of 1.7 T experiences a magnetic force of magnitude 8.2x10 -13 N. What is the angle between the proton’s velocity and the magnetic field?

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Magnetic Fields Differences between electric and magnetic forces on charged particles. The electric force on a charged particle is independent of the particle’s speed. The magnetic force only acts on a charged particle when the particle is in motion. The electric force is always along or opposite to the electric field. The magnetic force is perpendicular to the magnetic field. (F E = q E vs. F B = q v B)

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Magnetic Fields Differences between electric and magnetic forces on charged particles. The electric force does work in displacing a charged particle. The magnetic force does no work when a charged particle is displaced. A magnetic field can change the direction but not the speed of a moving charged particle.

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Magnetic Fields Units of B The SI unit of B is weber/sq. meter = Wb/m 2 = tesla = T = N/(C m/s ) = 10 4 G (gauss) Earth’s magnetic field ~ 0.5G = 0.5 x 10 -4 T

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Magnetic Force on a Wire A force is exerted on a single charged particle when it moves through a magnetic field: F B = |q| v B sin An electric current is a collection of many charged particles in motion, e.g., electrons moving through a metal wire. a current-carrying wire also experiences a force when it is placed in a magnetic field. Recall that electric current is denoted I.

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Magnetic Force on a Wire The magnetic force on the wire is given by B L I where L is a vector in the direction of the current and | L | = L= length of wire in the magnetic field.

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Magnetic Force on a Wire The magnetic force has its max magnitude F B = I L B when L is perpendicular to B ( = 90 o ). B L I

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Serway & Jewett, Principles of Physics, 3 rd ed. Figure 22.15

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Example Problem A wire having a mass per unit length of 0.50 g/cm carries a 2.0 A current horizontally to the south. What are the direction and magnitude of the minimum magnetic field needed to lift this wire vertically upward? 2 A North

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Total Force If we have a gravitational field g, an electric field E, and a magnetic field B all at the same point in space, then a particle with mass m, charge q, and velocity v will experience all three forces. The total force is given by: F tot = F g + F E + F B = m g + q E + q v B

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Magnetic Fields Hans Oersted (and earlier, Gian Dominico Romognosi) observed that an electric current in a wire deflected a nearby compass needle an electric current produces a magnetic field. I=0 I

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Serway & Jewett, Principles of Physics, 3 rd ed. Figure 22.27

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Faraday’s Law of Induction (Henry’s Law of Induction) H A loop of wire is connected to a galvanometer (an instrument for measuring electric current). Figure 31.1, page 981. g If a magnet is moved toward or away from the loop, a current is measured. g If the magnet is stationary, no current exists. g If the magnet is stationary and the wire loop is moved, a current is measured. A current is set up in the loop as long as there is relative motion between the magnet and the loop.

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Faraday’s Law of Induction (Henry’s Law of Induction) H The current is called an “induced current”. H If instead we measure the voltage, we call the the measured voltage an “induced voltage” or “induced electromotive force” or “induced emf”. H We can produce a current or voltage without a battery.

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Faraday’s Law of Induction H An electric current and a voltage (emf) are produced in a wire loop or coil whenever the “magnetic flux” through the loop changes. H So what is this thing called magnetic flux? ?

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Faraday’s Law of Induction H Suppose that we have a uniform magnetic field B directed into the page. We have a wire loop of area A in the page. We define the magnetic flux threading through the loop as B = |B|A = BA. Since B has units of Wb/m 2 or T, has units of Wb (Webers) or T-m 2. area of loop = A B

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Faraday’s Law of Induction H Faraday’s law of induction states that the voltage (emf) induced in the loop is directly proportional to the time rate of change of magnetic flux through the loop, i.e., V

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Faraday’s Law of Induction H If instead of a single loop of wire, we have a “coil” consisting of N loops or N turns then

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Example Problem H A coil is wrapped with 200 turns of wire on the perimeter of a square frame (18 cm on a side). A uniform magnetic field perpendicular to the plane of the coil is applied. If the field changes linearly from 0 to 0.5 Wb/m 2 in 0.8 s, find the magnitude of the induced emf in the coil while the field is changing.

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Faraday’s Law of Induction H To produce an emf (voltage), the magnetic flux through the wire loop must change with time. H How can this happen? H There are several ways: g The magnitude of B can vary with time. g The area of the loop can change with time. The angle between B and the normal to the loop can change with time. g Any combination of the above can occur.

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Faraday’s Law of Induction H Another manifestation of Faraday’s law is the so-called “motional emf,” which is the voltage (emf) induced in a wire moving through a magnetic field. + v B

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Faraday’s Law of Induction H When v, the velocity of the conductor (m/s) is perpendicular to B, the magnetic field (Wb/m 2 or T), then the motional emf, V, is given by V = E = B l v, where l is the length of the wire. + v B l V

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Faraday’s Law of Induction H If we complete the circuit, the current induced is equal to I = V/R = ( B l v)/R The power produced by the force moving the bar is equal to the power in the circuit: P = F app v = (I l B) v = (B 2 l 2 v 2 )/R = V 2 /R and it is dissipated as heat in the circuit.

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Example Problem A Boeing-747 jet with a wing span of 60 m is flying horizontally with a speed of 300 m/s over Phoenix. Assume the magnetic field is perpendicular to the velocity and has a magnitude of 50.0 T. H What is the voltage generated between the wing tips?

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