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## Homework Statement

19. A nonconducting spherical shell, with an inner radius of 4.0 cm and an outer radius of 6.0 cm, has charge spread nonuniformly through its volume between its inner and outer surfaces. The

*volume charge density*[itex]\rho[/itex] is the charge per unit volume, with the unit coulomb per cubic meter. For this shell [itex]\rho = b/r[/itex], where r is the distance in meters from the center of the shell and b = 3.0 [itex]\mu C/m^2[/itex]. What is the net charge in the shell.

## Homework Equations

[tex]

\vec{F}_{12} = \frac{k_{e}q_{1}q_{2}}{{r_{12}}^2}\hat{r}_{21}

[/tex]

[tex]

|\vec{F}_{12}| = \frac{k_{e}|q_{1}||q_{2}|}{{r_{12}}^2}

[/tex]

## The Attempt at a Solution

This problem is unlike most problems I have solved before so I am not sure if my approach is correct.

From what was given in the problem I conclude the following equations.

V = Volume

A = Area

[tex]

\rho = \frac{q}{V}

[/tex]

[tex]

\rho = \frac{b}{r}

[/tex]

[tex]

b = \frac{q}{A}

[/tex]

Though, the problem mentions that the charge has spread

**nonuniformly**through its volume between its inner and outer surfaces of the spheres.

I am unsure as to how I am supposed to use that to solve the problem.

I assume I can treat the inner and outer spheres: [itex]sphere_{1}[/itex] and [itex]sphere_{2}[/itex]; separately and use the previous equations to come up with,

[tex]

\rho_{1} = \frac{q_{1}}{V_{1}}

[/tex]

[tex]

\rho_{2} = \frac{q_{2}}{V_{2}}

[/tex]

Volume of a sphere is,

[tex]

V = \frac{4}{3} \pi r^3

[/tex]

--------------------------------------------------------------------------

[tex]

q_{1} = \left(\frac{4}{3} \pi r_{1}^3\right)\left(\frac{b}{r_{1}}\right)

[/tex]

[tex]

sig. fig. \equiv 2

[/tex]

[tex]

q_{1} = 20 nC

[/tex]

[tex]

q_{2} = \left(\frac{4}{3} \pi r_{2}^3\right)\left(\frac{b}{r_{2}}\right)

[/tex]

[tex]

sig. fig. \equiv 2

[/tex]

[tex]

q_{2} = 45 nC

[/tex]

Although, the answers, [itex]q_{1} = 20 nC[/itex] [itex]q_{2} = 45 nC[/itex]. Are somewhat close to the book answer, they’re wrong. And because the problem asks for the net charge in the shell, I thought maybe I could take the average of these two figures and get the right answer, however (though it was close) it was still the wrong answer for the net charge in the shell.

Yea, so I am stuck on how to find the net charge in the shell.

Any help would be appreciated.

-PFStudent

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