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## Homework Statement

A sphere of radius ## R ## carries charge density ## \rho = ar^5 ## where ## a ## is a constant. Find the flux ## \Phi ## of its electric field through a surface of a circle with radius ## R ## if the circle lies in a plane tangent to the sphere and its center touches the sphere.

## Homework Equations

Gauss' law to calculate ## \vec{E} ## of the charged sphere: $$\oint \vec{E}\cdot d\vec{A} = \frac {Q_{enc}}{ε_0}$$

The total charge of the sphere (assuming volumetric charge density, wasn't specified which type in the original prob.): $$ Q_{enc} = \int_V \rho(r)\,dV $$

And of course, the electric flux: $$ \Phi_E = \int \vec{E}\cdot d\vec{A} $$

## The Attempt at a Solution

I found total charge enclosed by integrating all over the volume of the sphere and obtained: $$ Q_{enc} = \frac{1}{2} \pi a R^8 $$

Then I used Gauss' law to find ## \vec{E} ## and obtained: $$ \vec{E} = \frac{aR^8}{8ε_0r^2} \hat{r} $$

The surface over which I wanted to find the flux, take it to be at the right of the sphere and touching it at its center. I will call its differential area ## dA' = r' d r' d \theta ##.

The only contributions of ##\vec{E}## come from its component normal to ##\vec{dA'}##. Thus we need to find the magnitude corresponding to that component using: $$ cos(α) = \frac{R}{r} $$ where ## r= \sqrt{r'^2 + R^2} ##. I apologize if you cannot visualize it or understand how I am approaching this, but bear with me. Basically α is the angle between the radius ## R ## of the sphere and ## r ##, the distance of the ## \vec{E} ## component.

So now I take the flux:

$$ \Phi_E = \int_S \vec{E}\cdot d\vec{A'} = \int_S E cos(α) dA' $$

And then substitute in my previous expressions for ## E ## and ## cos(α) ##, and integrating all over the flux surface:

$$ \Phi_E = \int_0^{2\pi} \int_0^R (\frac{aR^8}{8ε_0r^2})(\frac{R}{r}) r' dr' d \theta' = \int_0^{2\pi} \int_0^R (\frac{aR^8}{8ε_0(r'^2 + R^2)})(\frac{R}{\sqrt{r'^2 + R^2}}) r' dr' d \theta' $$

(I don't know how to write the equation in steps using Latex, if someone could please inform me how to on the forums that would be appreciated). My final solution is:

$$ \Phi_E = \frac{\pi a R^8}{4ε_0} (1 - \frac{1}{\sqrt{2}}) $$

I was hoping to check if my methodology was correct and if I am doing something wrong. This problem took me a while to do and I want to at least check that it seems proper. Feel free to ask and I will clarify. Thank you!