Question

A uniform magnetic field B exists in a cylindrical region, shown dotted in figure. The magnetic field increases at a constant rate dBdt. Consider a circle of radius r coaxial with the cylindrical region. (a) Find the magnitude of the electric field E at a point on the circumference of the circle. (b) Consider a point P on the side of the square circumscribing the circle. Show that the component of the induced electric field at P along ba is the same as the magnitude found in part (a)
Figure

Answer 1

Magnitude of Electric Field is $\frac {rdB}{2dt}$

Explanation:

(a) Work done per unit test charge =$\phi$ E. dl (E = electric field)

$\phi$ E. dl = eE$\phi$ dl = $\frac {d\phi}{dt}$

⇒ $E\ 2\pi r = \frac {dB}{dt} \times A$

⇒ E = $2\pi r = \pi r^2 \frac {dB}{dt}$

E = $\frac{\pi r^2}{2\pi} \frac {dB}{dt}$

= $\frac {rdB}{2 dt}$

(b) When the square is considered, $\phi$ E dl = e

$E \times 2r \times 4 = \frac {dB}{dt} (2r)^2$

E = $\frac {dB4r^2}{dt8r}$

E = $\frac {rdB}{2dt}$