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Question

The diagram shows a sphere of radius $$R$$ that carries a charge uniformly distributed throughout its volume. The volume charge density is $$\rho$$. A Gaussian sphere of radius $$a$$ is imagined that is concentric to the charged sphere.
If $$a>R$$, what is the charge enclosed in the Gaussian surface?
495716_1e27819704774dcb87b4d5c3badc0396.png


A
Qenc=4πρR2
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B
Qenc=43πρa3
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C
Qenc=4πρa2
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D
Qenc=43πρR3
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E
Qenc=πρa2
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Solution

The correct option is B $$Q_{enc} = \cfrac{4}{3}\pi \rho R^3$$
The volume charge density is given by ,
                                 $$\rho=$$charge on sphere / volume of sphere
                                 $$\rho=Q/\frac{4}{3}\pi R^{3}$$
or                             $$Q=\frac{4}{3}\pi \rho R^{3}$$
we can see that the charge enclosed by gaussian surface is equal to $$Q$$ i.e.$$Q_{enc}=Q$$
therefore   $$Q_{enc}=\frac{4}{3}\pi \rho R^{3}$$

Physics

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