JEE Physics Electromagnetic Induction Notes Set 05

Important Notes and Formulas

Faraday's and Lenz's Laws

A change in the magnetic flux linked with a circuit induces an emf (and a current in a closed circuit) as long as the flux keeps changing. Faraday's law: \( e=-\frac{d\phi}{dt} \) with \( \phi=BA\cos\theta \), so an emf can be produced by changing \( B \), \( A \) or \( \theta \).
Lenz's law: the induced current opposes the cause that produces it. This follows from conservation of energy and gives the minus sign.
Induced current and charge: \( i=\frac{e}{R}=-\frac{1}{R}\frac{d\phi}{dt} \) and the charge that flows is \( q=\frac{|\Delta\phi|}{R} \), independent of how fast the flux changes.

Motional emf

A conductor of length \( l \) moving with speed \( v \) perpendicular to a field \( B \) (and to itself) has the emf \( e=Bvl \).
If it forms part of a circuit of resistance \( R \): current \( i=\frac{Bvl}{R} \), magnetic force opposing the motion \( F=Bil=\frac{B^2v^2l^2}{R} \) and the power needed to keep the speed constant \( P=Fv=\frac{B^2v^2l^2}{R} \), which equals the heat produced.
A rod of length \( L \) rotating with angular speed \( \omega \) about one end in a perpendicular field has \( e=\frac12B\omega L^2 \). The emf is the same if more spokes are added, because the spokes are in parallel.

Example 1. A 0.5 m rod moves at 4 m/s perpendicular to a field of 0.2 T on rails that complete a circuit of resistance 2 \( \Omega \). Find the emf, the current, the force needed and the power.
Solution:
\( e=Bvl=0.2\times4\times0.5=0.4 \) V and \( i=\frac{0.4}{2}=0.2 \) A.
\( F=Bil=0.2\times0.2\times0.5=0.02 \) N and \( P=Fv=0.08 \) W, which equals \( ei=0.4\times0.2 \).

Example 2. A metal rod of length 1 m rotates at 20 rad/s about one end in a perpendicular field of 0.5 T. Find the emf between its ends.
Solution:
\( e=\frac12B\omega L^2=\frac12\times0.5\times20\times1=5 \) V.

Eddy Currents

Circulating currents induced in a piece of metal kept in a changing magnetic field (or moving in a field) are called eddy currents. They are reduced by using a laminated core or cutting slots in the block. They are useful in magnetic brakes, induction furnaces and dead-beat galvanometers.

Self Induction

A change of current in a coil induces an emf in the same coil that opposes the change. The flux linked is \( \phi=Li \), where \( L \) is the self inductance (unit henry, dimensions \( [ML^2T^{-2}A^{-2}] \)) and \( e=-L\frac{di}{dt} \).
For a solenoid of \( N \) turns, length \( l \) and area \( A \): \( \phi=BAN=(\mu_0ni)AN \), so \( L=\mu_0n^2Al \) with \( n=\frac Nl \). If the number of turns is doubled on the same length, \( L \) becomes four times as large because \( L\propto n^2 \).

Mutual Induction

A change of current in the primary coil induces an emf in a neighbouring secondary coil. The flux in the secondary is \( \phi=Mi \), where \( M \) is the mutual inductance (unit henry). For two coaxial solenoids with \( n_1 \) and \( n_2 \) turns per unit length, area \( A \) and length \( l \): \( M=\mu_0n_1n_2Al \).