Magnetic Induction Chapter 27 27.1 Induced currents 27.2 Induced EMF and Faraday’s Law 27.3 Lenz’s Law, Motional EMF 27.4 Inductance 27.5 Magnetic energy 27.6 Induced electric fields

Two symmetry situations We found that: current loop + magnetic field  torque If there is no current and we turn loop by hand, will the opposite occur? torque + magnetic field  current ? Faraday’s Law of induction YES

Experiment 1 A current appears only of there is relative motion between the loop and the magnet; the current disappears when the relative motion ceases. Faster motion produces greater current Current direction depends on which magnetic pole is moving towards (or away) Induced current and induced emf

Experiment 2 If we close switch S, which turns on current in right-hand loop, the meter registers a current very briefly and suddenly in the left-hand loop. If we open the switch, another sudden and brief current is measured, but in the opposite direction. i.e. there is an induced current (and thus an induced emf) only when the current in the right-hand loop is changing, and not when it is constant. In both experiments an induced current and induced emf are apparently caused by something changing

What is this something that is changing? Faraday discovered that: An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing. The values of the induced emf and induced current are determined by the rate at which the number of field lines is changing (not the actual number).

Magnetic Flux We need a way to calculate the amount of magnetic field that passes through a loop. We define a magnetic flux. Suppose a loop enclosing area A is placed in a magnetic field B. As for electric fields we define a vector area, A. When B makes an angle  with the normal to the area, the flux through the loop is If dA is an element of area on surface S, the magnetic flux through S is

Magnetic Flux For the special case when the magnetic field is perpendicular to the loop, and the magnetic field is uniform SI unit for magnetic flux is the Weber: 1 Wb = 1 T.m2 We can now state Faraday’s law as: “ The magnitude of the emf induced in a conducting loop is equal to the rate at which the magnetic flux through the loop changes with time ” EXERCISE: show that 1 Wb/s = 1 V

Faraday’s Law of Induction To change the magnetic flux we can change: the magnitude B of the magnetic field within the coil the area of the coil, or the portion of that area that lies within the magnetic field (eg expanding the coil or moving it in or out of the field) the angle between the direction of the field B and the area of the coil (eg by rotation of the coil) To change the magnetic flux we can change: the magnitude B of the magnetic field within the coil the area of the coil, or the portion of that area that lies within the magnetic field (eg expanding the coil or moving it in or out of the field) the angle between the direction of the field B and the area of the coil (eg by rotation of the coil) If we change the magnetic flux through a coil of N turns, an induced emf appears in every turn and the total emf induced in the coil is the sum of these individual emfs. It is

CHECKPOINT: The graph gives the magnitude B(t) of a uniform magnetic field that exists throughout a conducting loop, perpendicular to the plane of the loop. Rank the five regions of the graph according to the magnitude of the emf induced in the loop, greatest first. t Answer: b first d and e tie a and c tie (zero) Clue: It is the special case where and A is constant

CHECKPOINT: Answer: out of the page If the circular conductor undergoes thermal expansion while it is in a uniform magnetic field, a current will be induced clockwise around it. Is the magnetic field directed into the page or out of the page? CHECKPOINT: To change the magnetic flux we can change: the magnitude B of the magnetic field within the coil the area of the coil, or the portion of that area that lies within the magnetic field (eg expanding the coil or moving it in or out of the field) the angle between the direction of the field B and the area of the coil (eg by rotation of the coil) Answer: out of the page (The induced magnetic field is into the page, opposing the increase in flux outwards through the loop.)

To change the magnetic flux we can change: the magnitude B of the magnetic field within the coil the area of the coil, or the portion of that area that lies within the magnetic field (eg expanding the coil or moving it in or out of the field) the angle between the direction of the field B and the area of the coil (eg by rotation of the coil) EMF is E = E0sin(2πft) This is the principle of an alternating-current generator See Wolfson page 471

EXAMPLE: A uniform magnetic field makes an angle of 60° with the plane of a circular coil of 300 turns and a radius of 4 cm. The magnitude of the magnetic field increases at a rate of 85 T/s while its direction remains fixed. Find the magnitude of the induced emf in the coil. PICTURE THE PROBLEM: The induced emf equals N times the rate of change of the flux through a single turn. Since B is uniform, the flux through each turn is simply B = BAcos, where A=r2. NB what is the angle ? EXERCISE: if the resistance of the coil is 200 , what is the induced current?

27.3 Induction and energy: Lenz’s Law Conservation of energy gives us the direction of the induced current – loop acts as a magnet – it is hard to move a N pole towards another. If we move a bar magnet towards a wire loop, an induced current flows and energy is dissipated as heat in the wire. Where did the energy come from?

If we move a bar magnet away from the wire loop, an induced current flows and energy is dissipated as heat in the wire. This time work is needed to pull the magnet away

NB The induced flux of Bi always opposes the change in the flux of B, but this does not mean that it necessarily points in the opposite direction to B. When flux through loop is decreasing [(b) and (d)], the flux of Bi must oppose this change and therefore Bi and B are in the same direction.

Two adjacent circuits As switch is opened, I1 decreases and flux through circuit 2 changes. Induced current I2 then tends to maintain the flux through circuit 2. Just after switch is closed, I1 increases in direction shown, inducing I2. The flux through circuit 2 due to I2 opposes the the change in flux due to I1.

CHECKPOINT: The figure shows three situations in which identical circular conducting loops are in uniform magnetic fields that are either increasing or decreasing in magnitude at identical rates. Rank them according to the magnitude of the current induced in the loops, greatest first. Answer: a and b tie, then c (zero)

Lenz’s law is related to the conservation of energy. False True True or False? The induced emf in a circuit is proportional to the magnetic flux through the circuit. There can be an induced emf at an instant when the flux through the circuit is zero. Lenz’s law is related to the conservation of energy. False Induction False True True