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## Homework Statement

A charge distribution with spherical symmetry has the density [tex] \rho = \rho _0 r/R [/tex] for 0≤ r ≤ R. Determine the total charge content of the sphere.

## Homework Equations

[tex] \rho = Q / V [/tex]

## The Attempt at a Solution

I started by thinking of the charge dQ of a small volume dV, since

[tex] \rho = dQ / dV [/tex]

I used spherical coordinates to define dV, and said that dV would be

[tex] dV = (r d \varphi )(r d \theta) dr[/tex]

Where [tex] \varphi [/tex] goes from 0 to 2*pi, [tex] \theta[/tex] goes from -pi/2 to pi/2, and r goes from 0 to R, thus covering the entire sphere.

So:

[tex] dQ = \rho dV = \rho r^2 d \varphi d \theta dr = \frac{\rho _0 r}{R} r^2 d \varphi d \theta dr = \frac{\rho _0 r^3}{R} d \varphi d \theta dr [/tex]

I then integrated over this as:

Q = [tex]\int ^ {2 \pi} _ {0} \int ^ {\pi /2} _ {-\pi /2} \int ^ {R} _ {0} \frac{\rho _0 r^3}{R} d \varphi d \theta dr [/tex]

[tex] Q = 2 \pi (\pi /2 + \pi /2) \frac{\rho _0 }{R} \int ^ {R} _ {0} r^3 dr = 2 \pi ^2 \frac{\rho _0 }{R} \frac{R^4}{4} = \frac{1}{2} \pi ^2 \rho _0 R^3[/tex]

And this is not correct, and I can't figure out where I've gone wrong. (It's supposed to be only [tex] \pi \rho _0 R^3[/tex]