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