A 1.2-m radius cylindrical region contains a uniform electric field along the cylinder axis. It is increasing uniformly with time. To obtain a total displacement current of 2.0*10 ^- ⁹ A through a cross section of the region, the magnitude of the electric field should change at a rate of:

Question

Quizplus · Accepted Answer

According to Ampere's law, 
$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$, where 
$I_{enc} = \epsilon_0 \frac{d\Phi_E}{dt}$ is the enclosed displacement current.

For a cylindrical region of radius $r$ and length $L$, the left-hand side of the equation becomes $\oint \vec{B} \cdot d\vec{l} = B(2\pi r L)$. 
Setting $I_{enc} = 2.0	imes 10^{-7}$ A, we can solve for the required rate of change of the electric field: 
$\frac{dE}{dt} = \frac{c^2}{\mu_0} \cdot \frac{I_{enc}}{\pi r^2}$
Substituting $r = 1.2$ m, $c=3.0	imes 10^8$ m/s, and $\mu_0 = 4\pi	imes 10^{-7}$ Tm/A, we get 
$\frac{dE}{dt} = \frac{(3.0	imes 10^8)^2}{4\pi	imes 10^{-7}} \cdot \frac{2.0	imes 10^{-7}}{\pi (1.2)^2} \approx 50$ V/m/s. 
Therefore, the correct answer is D.

A 12-M Radius Cylindrical Region Contains a Uniform Electric Field Along