Question

A circular loop with radius r is rotating with constant angular velocity ω in a uniform electric field with magnitude E. The axis of rotation is perpendicular to the electric field direction and is along the diameter of the loop. Initially, the electric flux through the loop is at its maximum value. Write an equation for the electric flux through the loop as a function of time in terms of r, E, and ω. Find the expression for electric flux through the loop as a function of time.

Answer 1

Answer:

$\Phi_{E} = E\pi r^2 \omega t$

Explanation:

The electric flux is defined as the multiple of electric field and the area that the electric field passes through, such that

$\Phi_{E} = \vec{E}\vec{A}$

When calculating the electric flux, the angle between the directions of electric field and the area becomes important, especially if the angle is changing with time.

The above formula can be rewritten as follows

$\Phi_{E} = EA\cos(\theta)$

where θ is the angle between the electric field and the area of the loop. Note that, the direction of the area of the loop is perpendicular to the plane of the loop.

If the loop is rotating with constant angular velocity ω, then the angle can be written as follows

$\theta = \omega t$

At t = 0, cos(0) = 1 and the electric flux through the loop is at its maximum value.

Therefore the electric flux can be written as a function of time

$\Phi_{E} = E\pi r^2 \omega t$