How to Calculate Charge Content of a Sphere with Spherical Symmetry?

[Hannisch]

Homework Statement

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

Homework Equations

$$\rho = Q / V$$

The Attempt at a Solution

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

$$\rho = dQ / dV$$

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

$$dV = (r d \varphi )(r d \theta) dr$$

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

So:

$$dQ = \rho dV = \rho r^2 d \varphi d \theta dr = \frac{\rho _0 r}{R} r^2 d \varphi d \theta dr = \frac{\rho _0 r^3}{R} d \varphi d \theta dr$$

I then integrated over this as:

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

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

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

 

[cupid.callin]

consider any spherical shell inside the sphere of radius x(<R) and thicknedd dx
find charge on it and then integrate it dx from 0 to R

 

[ehild]

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

$$dV = (r d \varphi )(r d \theta) dr$$

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

Check dV. It is wrong.

ehild