Statement#

A solid sphere of radius $R$ has total charge $Q$ uniformly distributed throughout its volume.

Using Gauss's Law, find the electric field $\mathbf{E}(\mathbf{r})$ as a function of distance $r$ from the center:

1. Inside the sphere: $r < R$
2. Outside the sphere: $r > R$

Required Topics#

• Gauss's law
• Electric fields
• Spherical symmetry
• Charge distributions
• Volume integrals
• Flux calculations

Gauss's Law#

$\oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}$

where the integral is over a closed surface $S$.

Setup#

Charge density (uniform): $\rho = \frac{Q}{\frac{4}{3}\pi R^3} = \frac{3Q}{4\pi R^3}$

Gaussian surface: Choose spherical surfaces of radius $r$ centered at the origin.

What to Find#

1. For $r < R$ (inside):

   • Calculate $Q_{\text{enclosed}}(r)$
   • Apply Gauss's law
   • Find $E(r)$

2. For $r > R$ (outside):

   • $Q_{\text{enclosed}} = Q$ (all charge)
   • Apply Gauss's law
   • Find $E(r)$

3. Show continuity at $r = R$

Expected Results#

• Inside: $E(r) \propto r$ (linear)
• Outside: $E(r) \propto 1/r^2$ (like a point charge)

Solution#

Solution coming soon.