Faraday's Law of Induction — Question 9

Question 9

A specified emf waveform reconstructs the field up to its initial value

A stationary N-turn loop of area A has normal +z and records $\mathcal E(t)=kt$ for $0<t<T$. Its field is uniform along z, initially B0.

1. Find B(t) and the time when it first reaches zero, if that time is within the interval.

2. Find the total charge through resistance R during the interval, neglecting self-inductance.

Question 9 - Solution

A specified emf waveform reconstructs the field up to its initial value

(a) Method and calculation

$$\mathcal E=-NA\dot B=kt\Rightarrow\dot B=-kt/(NA).$$$$\boxed{B(t)=B_0-kt^2/(2NA),\quad t_0=\sqrt{2NAB_0/k}}.$$

(b) Interpretation and further calculation

$$Q=\int_0^T\frac{kt}{R}dt.$$$$\boxed{Q=kT^2/(2R)}.$$

Physical check

The zero-crossing time is relevant only if it lies within the supplied waveform interval; no extrapolation beyond T is implied.