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Question

# Write the statement of Gauss' law for electrostatics.Draw a diagram and derive an expression for electric field due to a uniformly charged infinite plane sheet at a point near the sheet.For the given diagram, write the value of electric flux passing through the surface.

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Solution

## Step 1: Statement of Gauss's law Gauss Law for electrostatics states that the total electric flux passing through a closed surface equals the enclosed charge in the surface divided by the permittivity of the medium. ϕE=∮(→E⋅→dS)=QenclosedεoStep 2: Deriving an expression for electric field due to infinite plane sheet Consider an infinite plane sheet of positive charge with charge density σ as shown in the attached figure. Electric field lines will be directed orthogonal and away from the sheet of charge. Hence, a cylindrical closed surface with its base parallel to the sheet of paper as shown in the figure is a good choice of Gaussian surface.For the curved surface electric field is orthogonal to the surface area vector. Hence, flux linked to curved surface is zero.ϕs=0For the plane surface, applying Gauss's Law, we get:ϕs+ϕb=QenclosedεoEA+EA=QenclosedεoBut Q=σA by definition of surface charge density⟹E=σ2εo[Note : For an infinite line sheet of charge, electric filed does not depend on the distance of the point from the surface.]Step 3: Electric flux passing through surface in given diagramFor the diagram shown in the question, enclosed charge is:Q=2+(−1)=1 μCUsing Gauss's Law,Electric flux through the surface is:ϕE=Qεo =10−6×4π×9×109 ≈1.13×105 Vm

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