IIT JEE , AIEEE Forum - Physics , Electricity - pls-------------- help potential problem

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The electric field due to an infinitely long cylinder of radius r, at a distance x from its center can be found by Gauss' Law, and is found to be
E =

r/

x away from the cylinder.
Also, the electric field at any point inside the cylinder is zero (using Gauss' Law)

Consider an origin O at any point on the axis of the +vely charged cylinder. Let the X axis start from this point O, perpendicular to the axes of both the cylinder.
Hence, at a point at a distance x from the origin O (outside the cylinder),
E =

r/

x i₊ (-

)r/

(a-x)(-i) =

r/

[1/x + 1/(a-x)]i

Potential difference between the axes of +ve and the -ve cylinder is

V = - _a

⁰ E.dr = - _a

^a-r 0.dr - _a-r

^r E.dr - _r

⁰ 0.dr

V = - _a-r

^r

r/

[1/x + 1/(a-x)]i . dxi = - (

r/

) _a-r

^r [1/x + 1/(a-x)]dx

V = (2

r/

)ln[(a-r)/a]

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