Solve Gauss's Law: Electric Flux Through Sphere from Line Charge

[sisigsarap]

An infinitely long line charge having a uniform charge per unit length (half-life - if anyone could tell me the correct name for that symbol it would be great too lies a distance d from the point O, where the point O is the center of a sphere. Determine the total electric flux through the surface of the sphere of radius R centered at O resulting from this line charge. Consider the case where R > d.

So my take on this is, if R is larger than d, then the infinitely long line will strike through the sphere.

I set this up as electric flux(f) = EA = Charge in sphere(q)/permittivity of free space(e)

so f = EA = q/e
The area of the sphere is 4(pi)r^2
q = (halflife)(length)

So f = E4(pi)r^2 = (halflife)(length)/e

so E = (halflife)(length)/e4(pi)r^2

I know the correct answer is ((2halflife)/e)(square root of (R^2 - d^2)

If anyone can help me out it would be greatly appreciated, I think I am on the right track, but I just cannot figure out how to go from the answer I have to the answer in the back of the book. I appreciate anyones help who takes the time to read this!

Josh

 

[StatusX]

First of all, E isn't perpendicular to the sphere at all points, so you can't pull it outside the surface integral. But in any case, you're doing this the hard way. You don't need to know the electric field at all. You said it yourself: the total flux through the sphere is equal to the charge enclosed over e. And that half life symbol is called lambda.

 

[sisigsarap]

Ok so I have Flux = (lambda * length)/permittivity of free space(e)

when I set (lambda * length)/e = (2 * lambda/e) * (square root of R^2 - d^2)

I get length = 2 * (square root of R^2 - d^2)

I am having an awefully difficult time seeing the relationship between these two.

I know that is the length of the line charge cutting through the sphere, but I am lost.

I know its something easy and I am just not seeing it!

 

[StatusX]

It's just geometry. How much of a line that d away from the center of a sphere of radius R is inside the sphere? Draw a picture and it should be clear. Use the pythagorean theorem with one of the legs going from the center of the sphere to the midpoint of the line, and one from the midpoint of line to the surface of the sphere, where the line cuts through it. What do you know about the length of the hypotenuse of this triangle?

 

[sisigsarap]

I don't know why I didnt see that right off, I must be working too hard :P

I wasnt seeing R as the hypotenuse for some reason. I feel like a dummy for making that second post now heh

Thank you StatusX you have been a great help !