Lectures (Ch. 29) Electromagnetic Induction 1.The law of EM induction 2.The Lenz’s rule 3. Motional emf: slide wire generator, Faraday’s disc dynamo, Ac and dc current generators

Electromagnetic Induction, Joseph Henry ( ) Michael Faraday ( ) Change of magnetic flux through the loop of wire induces current (i.e. emf) in the loop.

Law of EM Induction (Faraday’s law) Flux can be changed by change of B or A or angle between B and dA. In order to find the direction of the induced current it is convenient to write the faraday’s law in the form It is convenient to choose in such direction that where is the unite vector of a circulation in the loop which direction is connected with by a RHR: Wilhelm Weber (1804 – 1891)

Examples. Find direction of the induced current. Induced current is in the direction ofInduced current is in the direction opposite to

Lenz’s law Heinrich Lenz (1804 –1865 ) Magnetic field produced by induced current opposes change of magnetic flux

Examples

Example

Motional emf Slide-wire generator Origin of this emf is in separation of charges in a rod caused by its motion in B. Motional emf exists in the conductor moving in B. It does not require the existence of the closed circuit.

The secondary magnetic force External force is required to keep constant velocity of the rod m’m’

Example. Find motional emf in the rod. I L d Example. Find induced current in the loop with resistance R. I V V

Example A single rectangular loop of wire with the dimensions inside a region of B=0.5 T and part is outside the field. The total resistance of the loop is 0.2Ω. The loop is pulled from the field with a constant velocity of 5m/s. 1)What is the magnitude and direction of the induced current? 2) In which part of the loop an induced emf is developed? 3) Find the force required to pull the loop at a constant velocity. 4) Explain why such force is required. x Bv 0.1m 0.5m 0.75m

Example 30cm 40cm v=2cm/s x B=1T Find emf in each side of the loop and the net emf when the loop is the region: a)all inside the region of B b)partly outside of this region c)all outside of this region

Faraday’s disk dynamo v + - FmFm F’ m

AC –current generator (alternator) Induced current results in torque which slows down a rotation. External torque is required to maintain the rotation with a constant frequency.

DC-current generator

Applications

2007 Nobel Prize in Physics Peter Grünberg Albert Fert For the discovery of the giant magnetic resistance Tiny magnetic field triggers large change in electrical resistance. Better read-out heads for pocket-size devices: miniaturization of PC, ipods, etc. Big R B Small R B Resistance strongly depends on the direction of the spin in the first ferromagnetic layer. When it is the same as in the next ferromagnetic layer R is small, when it’s opposite to it R is big. I

Eddy currents

Eddy currents responsible for levitation and Meisner effect in superconductors Meisner’s effect B ind B0B0 v N S

Eddy currents limit efficiency of transformers

Induced nonelectrostatic electric field Origin of emf? No motion, moreover no B outside solenoid, i.e. in the region of a wire loop. Then it should be E which results in induced current. Nonconcervative force Nonelectrostatic field

B(t) should induce E by independently on the presence of the loop of the wire! Let’s find E(r). E R r R

Displacement current 1 2 dB/dt produces E. Let’s show that dE/dt produces B! Consider the process of charging the capacitor. Calculate B in front of the plate of capacitor at r>R. Using the plane surface 1 we get Using the bulging surface 2 we get We come to contradiction! What is wrong ? 1.Now we get the same answer for both surfaces 1 and 2! 2. B≠0 between the plates!

General form of Amper’s law Let’s find B between the plates. B r R μ= K m μ 0, ε=Kε 0, In free space K=1, K m =1

Maxwell’s equations James Clerk Maxwell (1831 –1879) Two Gauss’s laws + Faraday’s law +Amper’s law Maxwell introduced displacement current, wrote these four equations together, predicted the electromagnetic waves propagating in vacuum with velocity of light and shown that light itself is e.m. wave Maxwell’s theory 1887 Hertz’s experiment 1890 Marconi radio (wireless communication)