JEE Physics Electromagnetic Induction Notes Set 02

Faraday's and Lenz's Laws

1. Magnetic flux \(\phi=\vec B\cdot\vec A=BA\cos\theta\), where \(\theta\) is the angle between the field and the normal to the area. A field parallel to the surface gives zero flux.
2. Faraday's law: \(\varepsilon=-N\dfrac{d\phi}{dt}\). The induced emf depends on the rate of change of flux, not on the flux itself, so a coil can have zero flux and non-zero emf at an instant.
3. Lenz's law: the induced current opposes the change of flux that produces it. A flux increasing into the page is opposed by an anticlockwise current; a decreasing one by a clockwise current.
4. Charge that flows through a coil of resistance \(R\) is \(q=\dfrac{\Delta\phi}{R}\), independent of the time taken.
5. Eddy currents are induced in bulk conductors and oppose the motion that produces them (electromagnetic damping, magnetic braking).

Motional emf

1. A straight conductor of length \(l\) moving at velocity \(v\) in field \(B\): \(\varepsilon=(\vec v\times\vec B)\cdot\vec l\). If the rod makes an angle with the velocity, only the perpendicular components count (\(Blv\sin\theta\)).
2. A bent wire between two end points has the emf of the straight line joining its ends when it moves or rotates as a whole.
3. Rod rotating about one end: \(\varepsilon=\tfrac12B\omega l^2\); a disc rotating about its centre has the same emf across its radius.
4. Coil of N turns, area A rotating at \(\omega\): \(\varepsilon=NBA\omega\sin\omega t\), peak value \(NBA\omega\).
5. The end towards which \(\vec v\times\vec B\) points along the conductor becomes the positive end.

Inductance

Parallel inductors that start from zero current share the current in inverse proportion to their inductances, because they always have the same voltage. Parallel coils joined from one cut solenoid each have half the inductance and half the resistance.

LR Circuits

1. Growth: \(I=\dfrac{E}{R}\left(1-e^{-t/\tau}\right)\) with \(\tau=\dfrac{L}{R}\). The emf across the inductor is \(Ee^{-t/\tau}\), starting at \(E\) and falling to zero.
2. Decay: \(I=I_0e^{-t/\tau}\).
3. At the instant of switching on, an inductor acts like an open circuit (initial rate \(\dfrac{dI}{dt}=\dfrac{E}{L}\)); in the steady state it acts like a plain wire.
4. For a network, replace the inductor by a wire in the steady state to find the final current, and find the equivalent resistance seen by the inductor for the time constant.

Transformers

1. \(\dfrac{V_s}{V_p}=\dfrac{N_s}{N_p}\); ideal transformer: \(V_pI_p=V_sI_s\).
2. Efficiency \(=\dfrac{V_sI_s}{V_pI_p}\). A step-up transformer raises voltage and lowers current. The frequency of the output equals the frequency of the input.

Worked Examples

1. Flux \(\phi=5t^2-6t+9\): emf magnitude \(|10t-6|\) gives \(6\ \text{V}\) at \(t=0\) and \(1\ \text{V}\) at \(t=0.5\ \text{s}\).
2. An aeroplane with wing-span \(50\ \text{m}\) at \(100\ \text{m/s}\) in a vertical field of \(2\times10^{-4}\ \text{T}\): emf \(=2\times10^{-4}\times50\times100=1\ \text{V}\).
3. A step-down transformer delivering \(2400\ \text{W}\) at 80% efficiency draws \(3000\ \text{W}\) from a \(1000\ \text{V}\) line, so the line current is \(3\ \text{A}\).