Question

A solid nonconducting sphere of radius R has a charge Q uniformly distributed throughout its volume. A Gaussian surface of radius r with r < R is used to calculate the magnitude of the electric field E at a distance r from the
center of the sphere. Which of the following equations results
from a correct application of Gauss’s law for this situation? And Why?

1. E (4 p r^2) = (Q r^3)/( epsilon sub0 * R3)
2. E(4pr^2)= (Q 3r^3)/ (epsilon sub0 * 4 p R)
3. E (4 p r^2) = (Q) / (epsilon sub0)

Answer 1

Answer:

$E(4 \pi r^{2})={\frac{ Q r^{3}}{R^{3}\epsilon _{0}}}$

or

$E(4 \pi)={\frac{ Q r}{R^{3}\epsilon _{0}}}$

Explanation:

We know that Gauss's law states that the Flux enclosed by a Gaussian surface is given by

$E.S=\frac{q}{\epsilon_{0}}$

Here , E is electric field and S is surface are and q is charge enclosed by the surface and e is electrical permeability of the medium.

Here the Gaussian is of radius r<R so area of surface is

$S=4 \pi r^{2}$

Also, charge enclosed by the surface is

$Charge =\frac{Total \: Charge }{Total \:Volume} \times Volume \: of \: Gaussian \: surface$

therefore,

$q=\frac{Q}{\frac{4}{3} \pi R^{3} }\frac{4}{3} \pi r^{3} =\frac{ Q r^{3}}{R^{3}}$

Here Q is total charge,

Insert values in Gauss's law

$E(4 \pi r^{2})=\frac{\frac{ Q r^{3}}{R^{3}}}{\epsilon _{0}}$

Rearrange them

$E(4 \pi r^{2})={\frac{ Q r^{3}}{R^{3}\epsilon _{0}}}$

on further solving

$E(4 \pi)={\frac{ Q r}{R^{3}\epsilon _{0}}}$

This is the required form.

Answer 2

The correct equation result from the Gauss law is, $E(4 \pi r^{2})=\dfrac{Qr^{3}}{R^{3} \epsilon_{0}}$. Hence, option (1) is correct.

The given problem is based on the concepts of Gauss's law. According to the Gauss's law, the flux enclosed by a Gaussian surface is given by,

$E.S=\dfrac{q}{\epsilon_{0}}$

Here,

E is the electric field.

S is the Gaussian surface, and its value is,

$S=4\pi r^{2}$

$\epsilon_{0}$  is the permittivity of free space.

q is the charge enclosed by the surface.

Then,

$E(4 \pi r^{2})=\dfrac{q}{\epsilon_{0}}$ .................................................................(1)

And the expression for the charge enclosed by the surface is,

$q = \dfrac{Q}{V} \times V'$

Here,

Q is the total charge.

V is the Total Volume.

V' is the volume of Gaussian Surface.

Therefore,

 $q = \dfrac{Q}{\dfrac{4}{3} \times \pi \times R^{3}} \times \dfrac{4}{3} \pi \times r^{3}\\\\\\q=\dfrac{Qr^{3}}{R^{3}}$

Substitute the values in equation (1) as,

$E(4 \pi r^{2})=\dfrac{Qr^{3}}{R^{3} \epsilon_{0}}$

Thus, we can conclude that the correct equation result from the Gauss law is, $E(4 \pi r^{2})=\dfrac{Qr^{3}}{R^{3} \epsilon_{0}}$. Hence, option (1) is correct.

learn more about the Gauss law here:

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