Gauss' Law with Calculus Video Lessons

Concept

Problem: A hollow non-conducting spherical shell has inner radius R1 = 8cm and outer radius R2 = 15 cm. A charge Q = - 45 nC lies at the center of the shell. The shell carries a spherically symmetric charge density ρ = Ar for R1 &lt; r &lt;R2 that increases linearly with radius, where A= 29 μC/m4 Write an equation for the radial electric field in the region r&lt;R1 in terms of Q, r and Coulomb's constant k. You may take the positive direction as outward.

FREE Expert Solution

Gauss' law:

$\overline{){\mathbf{\oint }}{\mathbf{E}}{\mathbf{·}}{\mathbf{d}}{\mathbf{A}}{\mathbf{=}}\frac{\mathbf{Q}}{{\mathbf{\epsilon }}_{\mathbf{0}}}}$

Area of a sphere:

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

The Gaussian surface is a sphere with a radius of r. We'll use  -Q for the charge enclosed.

The closed integral of dA = A

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Problem Details

A hollow non-conducting spherical shell has inner radius R= 8cm and outer radius R2 = 15 cm. A charge Q = - 45 nC lies at the center of the shell. The shell carries a spherically symmetric charge density ρ = Ar for R1 < r <R2 that increases linearly with radius, where A= 29 μC/m4

Write an equation for the radial electric field in the region r<R1 in terms of Q, r and Coulomb's constant k. You may take the positive direction as outward.