Finding the electric flux through a sphere

[Timebomb3750]

Homework Statement

A sphere of radius R is placed in a uniform electric field of E=233 N/C i. Find the electric flux into and out of the sphere.

Homework Equations

I understand that Gauss's Law is shown as...
∫E dot dA = Q/epsilon not

The Attempt at a Solution

Well, since we're dealing with a sphere with E being constant, I figured you could pull the E out of the integral and be left with E∫dA where dA is =4pi*r^2. So you'd be left with E(4pi*R). But this doesn't look right to me. Keep in mind, there is no charge in or out of the sphere. And the field lines are perfectly horizontal through the sphere.

 

[Doc Al]

Well, since we're dealing with a sphere with E being constant, I figured you could pull the E out of the integral and be left with E∫dA where dA is =4pi*r^2. So you'd be left with E(4pi*R).

You forgot about the dot product. You need the component of surface area in the direction of the field. (Or vice versa.)

 

[Timebomb3750]

I'm confused about what you're asking for. Are you saying there should be a dot somewhere in my flux=E(4pi*R) equation. I understand what the dot product is (I use it a lot in my Calc 3 class) I'm confused as to what you mean by "the component of the surface area". I understand that the field lines only have an x-component or i, so they're horizontal through the sphere.

 

[Doc Al]

What do you think E*dA means? What's the significance of the dot product?

(You were treating E*dA as if it were the same as EdA.)

 

[Timebomb3750]

What do you think E*dA means? What's the significance of the dot product?

(You were treating E*dA as if it were the same as EdA.)

Oops. I see. So, I should be left with E*(4piR) R=radius. But what I'm now stumped on is the component of the surface area. As I said before. I understand the E has an i-component. But how do I find the components of the surface area?

 

[Doc Al]

Well, you can do it the hard way. (By setting up the integral.) Or you can think about it a bit and maybe it'll dawn on you. Imagine a hemisphere with its axis along the x-axis. What will be the x-component of its surface area?

This might be an even better way to visualize it. What flux of E field will be intercepted by the sphere? (Who cares about the shape of the surface?)

 

[Timebomb3750]

Wouldn't the total flux through the hemisphere be zero? Thus, meaning the flux through a sphere would be zero as well? I'm talking about total flux meaning the sum of the positive and negative flux.

 

[Doc Al]

Wouldn't the total flux through the hemisphere be zero? Thus, meaning the flux through a sphere would be zero as well? I'm talking about total flux meaning the sum of the positive and negative flux.

The total flux through the sphere will be zero*. (Since whatever goes into it must go out of it.) But not through the hemisphere.

*That should be clear from Gauss's law.