Question

A time-dependent magnetic field $\overrightarrow{B}\left(t\right)$ is produced in a cylindrical region of radius R. The magnetic field is given as $\overrightarrow{B}\left(t\right)=B_{0}t\hat{z}$ for $0\le r<R$ and zero for r > R, where $B_0$ is a positive constant, $\hat{r}$ is the unit vector in the radially outward direction and $\hat{\theta }$ is the unit vector in the tangential direction, such that $\hat{r}\times \hat{\theta }=\hat{z}$. The electric field at $r>$ R is

• A. $\frac{B_o{R}^2}{r}\hat{r}$
• B. $\frac{B_o{R}^2}{2r}\hat{\theta }$
• C. $-\frac{B_o{R}^2}{r}\hat{r}$
• D. $-\frac{B_o{R}^2}{2r}\hat{\theta }$

Answer 1

Answer: (D)

$-\frac{B_o{R}^2}{2r}\hat{\theta }$

Emf Generated $=$ Rate of change of Flux ( $-$ sign due to the opposing nature of induced emf )
$\int E.dl=-\frac{d(\int BdA)}{dt}$
$E\left(2\pi r\right)=-B_0\pi R^2$
$E=-\frac{B_0{R}^2}{2r}$