A uniform electric field perpendicular to a spherical surface of radius 5.00 cm is increasing at a rate of 4.0 V/m · µs.Find the displacement current through the spherical surface.

Question

Quizplus · Accepted Answer

The displacement current is given by:$$I_d = \epsilon_0 \frac{dA}{dt} \oint E \cdot d\ell$$where:* $\epsilon_0$ is the permittivity of free space* $\frac{dA}{dt}$ is the rate of change of the area of the spherical surface* $E$ is the electric field* $d\ell$ is an element of length along the circumference of the spherical surfaceIn this problem, the electric field is perpendicular to the spherical surface, so the dot product $E \cdot d\ell$ is equal to $Ed\ell$. The rate of change of the area of the spherical surface is given by:$$\frac{dA}{dt} = 4\pi r^2 \frac{dr}{dt}$$where $r$ is the radius of the spherical surface.Substituting these expressions into the equation for the displacement current, we get:$$I_d = \epsilon_0 4\pi r^2 \frac{dr}{dt} E$$Plugging in the given values, we get:$$I_d = (8.85 	imes 10^{-12} 	ext{ F/m})(4\pi (0.05 	ext{ m})^2)(4.0 	ext{ V/m} \cdot \mu	ext{s})$$$$I_d = 1.11 \mu	ext{A}$$Therefore, the correct answer is A.

A Uniform Electric Field Perpendicular to a Spherical Surface of Radius