Question

Derive an expressions for electric field due to infinitely long charged wire

Answer 1

Consider a uniformly charged infinitely long thin wire of surface charge density lambda. We wish to find its electric field at a distance of r from it.

Let a cylinder of radius r and height l be the Gaussian surface.

According to Gauss's law :

EA = q/(epsilon not)

Now at the top and bottom phases of cylinder i.e. its base and top E and dS are perpendicular to each other hence it does not contributes to electric flux.

Now,

E * 2πrl = q/(epsilon not)

lambda = q / l

Simplifying

E = lambda / 2π(epsilon not)r.

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Answer 2

Electric field due to an infinite line of charge

Ф$_{E}$ = ∅ E . dA = q/∈₀

Ф$_{E}$ = ∅ EdAcosθ = q/∈₀

⇒ E ∅dA = q/∈₀

Let q is the amount of charge in that body. We want to find the electric field at point P. Let the length of the cylinder is l and the radius be r.

Linear charge density (λ) = q/l ...... Equation 1

∅ E.dA = q/∈₀

E∅dA=q/∈₀ ( θ = 0° and E=constant)

E(2πrl) = q/∈₀

E= q/2πrl∈₀

Putting the value of q/l from equation 1

⇒ E = λ/2πr∈₀

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