Faraday's Law of Induction — Question 4

Question 4

Expanding a circular loop induces an emf proportional to its instantaneous radius

A circular conducting loop in a constant perpendicular field B has radius $r(t)=r_0+ut$, with u positive. Its positive normal is parallel to B. Model its total resistance as constant R and neglect self-inductance.

1. Find emf and current.

2. Find the charge transferred while its radius grows from r0 to 2r0.

Question 4 - Solution

Expanding a circular loop induces an emf proportional to its instantaneous radius

(a) Method and calculation

$$\Phi=B\pi(r_0+ut)^2\Rightarrow\mathcal E=-2\pi Bu(r_0+ut).$$$$\boxed{i(t)=-2\pi Bu(r_0+ut)/R}.$$

(b) Interpretation and further calculation

$$Q=-\frac{\Delta\Phi}R=-\frac{B\pi[(2r_0)^2-r_0^2]}R.$$$$\boxed{Q=-3\pi Br_0^2/R}.$$

Physical check

The stated constant resistance is a model assumption; stretching a real wire could also change its resistance.