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Question

The electric field in a certain region is acting radially outward and is given by E = Ar. A charge contained in a sphere of radius= a centred at the origin of the field, will be given by:


A

ϵ0Aa3

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B

4π ϵ0Aa2

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C

A ϵ0a2

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D

4π ϵ0Aa3

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Solution

Step 1: Given that:

The electric field at a certain region is given by

E=Ar

the radius of the sphere placed in the electric field E = a

Step 2: Formula used:

According to Gauss' law;

The electric flux linked with a closed surface enclosing a charge q is equal to 1ε0 times the charge enclosed by the surface.

Mathematically,

E.dS=qε0

Step 3: Calculation of the charge enclosed by the sphere:

Since, the sphere is placed in the electric field E=Ar ,

The electric field at the surface of the sphere will be calculated by placing r=a that is the radius of the sphere;

Thus, The electric field at sphere is given as;

E=A×a

Using Gauss' law

E.dS=qencε0

EdS=qencε0

As,theelectricfieldandthesurfaceareavectorofspherearedirectedinsamedirection.

EdS=qencε0

A×a.4πa2=qencε0

qenc=4πε0Aa3

Thus,

Option D) 4πε0Aa3 is the correct option.


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