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Question

(a) Define electric flux. Is it a scalar or a vector quantity?
A point charge q is at a distance of d/2 directly above the centre of a square of side d, as shown in the figure. Use Gauss' law to obtain the expression for the electric flux through the square.
(b) If the point charge is now moved to a distance d from the centre of the square and the side of the square is doubled, explain how the electric flux will be affected.
800650_2e1def6afd39448d805108fa9666ffe6.png

Solution
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Part (a)

Step 1: Definition of electric flux

Qualitative Definition: The Electric flux is proportional to the number of field lines passing a given area in a unit of time. This definition cannot be used for calculating the exact value of flux, it is used only for the comparison of flux through two surfaces.

Quantitative Definition: Magnitude of Electric flux is given by
Δϕ=E.dS
For plane surface and uniform Electric field, the above equation becomes:
Δϕ=E.S=EScosθ

It is a scalar quantity.

Step 2: Electric flux through square
Suppose a cube enclosing the charge q of side d.
Hence according to Gauss's law, the electric flux through the cube is qϵo

The given case can be considered as one of the face of such a cube.
By symmetry, the flux through the face is 16th of the flux through the cube.
Hence flux through the square face = q6ϵo

Part (b)

Electric flux through square when the side is doubled
Now, the charge is placed at a distance d from the centre of square. Also, the side of the square is 2d in this case.

In this case, the charge will be at the center of a cube of side 2d and the square will be one of its face.
According to a similar analysis as shown in Step 2 the electric flux through the face is: q6ϵo
Hence there will be no change in the electric flux.

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