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Questions
Use Gauss' law to derive the expression for the electric field `(vecE)` due to a straight uniformly charged infinite line of charge density λ C/m.
Use Gauss' law to obtain an expression for the electric field due to an infinitely long thin straight wire with uniform linear charge density λ.
Solution
Field due to an infinitely long straight uniformly charged wire,
Consider a thin, infinitely long straight line charge of linear charge density λ.
Let P be the point at a distance a from the line. To find the electric field at point P, draw a cylindrical surface of radius ‘a’ and length l.
If E is the magnitude of the electric field at point P, then electric flux through the Gaussian surface,
Φ = E × Area of the curved surface of a cylinder of radius r and length l
As electric lines of force are parallel to the end faces (circular caps) of the cylinder, there is no component of the field along the normal to the end faces.
Φ = E × 2πal …(i)
According to Gauss's Theorem,
`phi = q/epsilon_0`
`∵ q = lambdal`
`:. phi = (lambdal)/epsilon_0` ...(ii)
From equations (i) and (ii), we get:
`E xx 2pial = (lambdal)/epsilon_0`
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