Coulomb force on a line charge

[Kudaros]

Homework Statement

We have a source charge that is a uniform sphere with a radius a (centered at origin) and uniform charge density, $$\rho$$. There is a line charge with a length L that begins at Z₀ and ends at Z₀ + L (lies on the Z axis). This line charge has a uniform charge density of $$\lambda$$.

Homework Equations

Ill combine this with my attempt.

The Attempt at a Solution

First, I am resolving the source charge as a point. The sphere is of a uniform charge density and centered on the origin. So the 'q' for this source charge is $$\frac{4}{3}\pi$$*a²*$$\rho$$.

Second, I am calling the line charge L*$$\lambda$$.

My solution so far : F_{q'onq(sphere/point on line charge)}= $$\frac{\lambda*\rho*a^{3}*L}{3*\epsilon}*\frac{\vec{R}}{R^{3}}$$
My problem (assuming the above is correct) is that I am uncertain how to express the vector, R, from the source charge to the line charge.

In general, I am unsure of how to express a vector from a point to a continuous distribution of charges, or even from a continuous distribution to another.

 

[Redbelly98]

I don't think you can treat the line charge as a point charge, as you can do with spheres.

 

[Kudaros]

Ok I've got it.My line charge was indeed wrong.

Basically, I chose my vector to be simple. The position with respect to the source was Z*Z(hat) and I ended up integrating from Z⁰ to Z₀+L with 1/Z² as the integrand ( lambda is constant, was pulled out.)

Some quick simplification results in the answer in the back of the book.

I was thinking too hard about the vector I suppose.

Thank you!