A long, cylindrical non-conductor of radius R and length L is placed with it lon

Question

A long, cylindrical non-conductor of radius R and length L is placed with it long axis along the Z-axis as shown. The cylinder has a total charge Q distributed non-uniformly throughout its volume. The charge density is only a function of the radial distance "r" from the cylinder axis and varies as rho(r) = a/r^2 + 9R^2 where alpha is a constant a) Show alpha = 3.02 middot Q/L We will Now assume the cylinder length is very large compared to our field distances and we are far from ends of the cylinder. We will approximate the charge distribution as an infinitely with the bore given volume charge density. Use "l" as the length of any Gaussian surface.

Explanation / Answer

net charge = Q
consider a small volume of cylincer, concentric, of width dr at radius r
charge on this volume = rho*dV
rho = volume charge density, rho(r) = alpha/(r^2 + 9(R^2))
dV = 2*pi*r*h*dr ( h is height of cylinder )
integrating from r = 0 to r = R

Q ={integrate} alpha*2*pi*r*h*dr/(r^2 + 9(R^2)) = alpha*pi*h*ln(10/9)
h = L
alpha = Q/L*pi*ln(10/9) = 3.022 Q/L

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