A right circular cylinder has radius R and length L and a no

A right circular cylinder has radius R and length L, and a nonuniform volume charge density rho(r). If the z axis is chosen to coincide with the axis of the cylinder, the charge density is rho(r) = rho(z z) = rho_0 + beta_z. This means that the charge density varies linearly along the length of the cylinder. Find the force on a point charge q placed at the center of the cylinder.

Solution

The total force on the charge is the sum of the froces from each charge density. Note that the charge densities are not all equal. The porblem states that if were to look at the charge density as we move along the z axis it would change linearly.

F=dF where dF is The individual force equations can be written as.

dF=[(dq²)/r(₁₂)²] ^r(₁₂)/r(₁₂)

dF=[((o)+z)(dV)]^r(₁₂)/(h²+l²)^3/2

Now integrate both sides. dF=[((o)+z)(dV)]^r(12)/(h²+l²)^3/2

F⁼[((o)+z)(dV)]^r(12)/(h²+l²)^3/2