Electric Flux Equation for Rotating Loop?

[Rijad Hadzic]

Homework Statement

A circular loop with radius r is rotating with constant angular velocity w 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 0. Write an equation for the electric flux through the loop as a function of time in terms of r, E, and w.

Homework Equations

The Attempt at a Solution

So I use flux = EAcos(x)

A = pir^2

Epir^2 and cos x is going to vary with time, and this is where I should put w, angular velocity.

so flux = Epir^2cos(wt)

but my book is telling me its Epir^rsin(wt)

Why would it be sin? That's going against the flux equation..

 

[NFuller]

At time $t=0$ the flux is zero but your equation gives
$$E\pi r^{2}\text{cos}(\omega\cdot 0)=E\pi r^{2}\neq 0$$
This is because your angle $\theta$ which you call $x$ is defined incorrectly. At time zero, $\theta=\pi/2$, so how would you correct this?

 

[Rijad Hadzic]

At time $t=0$ the flux is zero but your equation gives
$$E\pi r^{2}\text{cos}(\omega\cdot 0)=E\pi r^{2}\neq 0$$
This is because your angle $\theta$ which you call $x$ is defined incorrectly. At time zero, $\theta=\pi/2$, so how would you correct this?

I see. So starting at t = 0, my equation would give me the flux if Area vector and E are pointing in the same direction, which would be the maximum flux, but the problem is saying at t= 0, E is perpendicular to the surface, so flux should be 0..

so using (E)(3.14...)(r)^2sin(wt), my inital flux would be 0, and this is the correct function.

Is my understanding right now?

 

[NFuller]

Yes, because sin is shifted over from cos by $\pi/2$.