Talk:QB/d cp2.6

This solution to problem 6 will NOT be given on your formula sheet $$\rho (\tilde r)=a\tilde r^n$$ describes charge density as a function of distance to the origin. Since the Gaussian sphere is fixed at radius $$r$$ we use a tilde to denote the variable of integration used to calculate the charge enclosed by the Gaussian sphere.

If the field point $$r<R$$, the radius of the nonuniformly (but symmetrically charged sphere), then Gauss' law is:

$$\varepsilon_0\oint\vec E\cdot d\vec A = \int \underbrace{a\tilde r^n}_{\rho (\tilde r)} \underbrace{4\pi \tilde r^2 d\tilde r}_{dVol} =4\pi a \int_0^r \tilde r^{n+2}d\tilde r= \frac{4\pi a}{n+3}r^{n+3}$$

But also,

$$\varepsilon_0\oint\vec E\cdot d\vec A = 4\pi r^2\varepsilon_0E$$ (using the surface area of a sphere)