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 byE=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=qencε0As,theelectricfieldandthesurfaceareavectorofspherearedirectedinsamedirection.E∫dS=qencε0A×a.4πa2=qencε0qenc=4πε0Aa3 Thus,Option D) 4πε0Aa3 is the correct option.

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