Gauss' Law with Calculus Video Lessons

Concept

# Problem: An insulating sphere of radius a, centered at the origin, has a uniform volume charge density  ρ.A spherical cavity is excised from the inside of the sphere. The cavity has radius  a/4 and is centered at position  h, where |h| &lt; 3a/4 , so that the entire cavity is contained within the larger sphere. Find the electric field  inside the cavity.Express your answer as a vector in terms of any or all of  ρ, ε0, r, and h.

###### FREE Expert Solution

Gauss' law:

$\overline{)\begin{array}{rcl}\mathbf{\oint }\mathbf{E}\mathbf{·}\mathbf{d}\mathbf{A}& {\mathbf{=}}& \frac{{\mathbf{q}}_{\mathbf{enc}}}{{\mathbf{\epsilon }}_{\mathbf{0}}}\\ \mathbf{E}\mathbf{·}\mathbf{A}& {\mathbf{=}}& \frac{\mathbf{q}}{{\mathbf{\epsilon }}_{\mathbf{0}}}\end{array}}$

The cavity is inside the sphere, the sphere is charged, and the electric field is to be found inside the cavity.

A Gaussian surface completely inside the cavity encloses no charge.

$\begin{array}{rcl}\mathbf{\oint }\mathbf{E}\mathbf{·}\mathbf{dA}& \mathbf{=}& \frac{\mathbf{0}}{{\mathbf{\epsilon }}_{\mathbf{0}}}\\ \mathbf{E}\mathbf{·}\mathbf{A}& \mathbf{=}& \mathbf{0}\end{array}$

Charge in a gaussian surface,

Area of a sphere, A:

$\overline{){\mathbf{A}}{\mathbf{=}}{\mathbf{4}}{\mathbf{\pi }}{{\mathbf{r}}}^{{\mathbf{2}}}}$

###### Problem Details

An insulating sphere of radius a, centered at the origin, has a uniform volume charge density  ρ.

A spherical cavity is excised from the inside of the sphere. The cavity has radius  a/4 and is centered at position  h, where |h| < 3a/4 , so that the entire cavity is contained within the larger sphere. Find the electric field  inside the cavity.

Express your answer as a vector in terms of any or all of  ρ, ε0, r, and h.