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Question

Physics Question on Gauss Law

An infinitely long wire is charged uniformly with a charge density λ and placed in the air, the electric field at a distance r from the wire will be:

A

14πε0λr\frac1{4πε_0} \frac{λ}{r}

B

14πε0λr2\frac1{4πε_0} \frac{λ}{r^2}

C

λ2ε0 \frac{λ}{2ε_0}

D

λ2πε0r \frac{λ}{2\piε_0 r}

Answer

λ2πε0r \frac{λ}{2\piε_0 r}

Explanation

Solution

The correct answer is (D): λ2πε0r \frac{λ}{2\piε_0 r}

Consider an infinity long straight conductor of linear charge density
An infinitely long wire is charged uniformly with a charge density λ and placed in the air, the electric field at a distance r from the wire will be:
Consider a cylindrical Gaussian surface of radius r and length l.
charge enclosed by the surface
q=λlq=λl

By Gauss law
ϕE⃗.ds=qε0=λlε0\phi \vec{E}.ds = \frac{q}{ε_0} = \frac{λl}{ε_0}

E⃗ϕds=λlε0\vec{E}\phi ds= \frac{λl}{ε_0}

E⃗×2πrl=λlε0\vec{E} \times 2\pi rl = \frac{λl}{ε_0}

E=λ2πrε0 E = \frac{λ}{2\pi rε_0}