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Question
Find the equation of the equipotentials for an infinite cylinder of radius r0, carrying charge of linear density λ.
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Solution
To find the potential at distance r from the line consider the electric field. We note that from symmetry the field lines must be radially outward. Draw a cylindrical Gaussian surface of radius r and length l. Then
`oint E.dS = 1/ε_0 λ1`
Or `E_r.2pirl = 1/ε_0 λ1`
⇒ `E_r = lambda/(2piε_0r)`
Hence, if r0 is the radius,
`V(r) - V(r_0) = - int_(r_0)^r E.dl = λ/(2piε_0)ln r_0/r`
For a given V,
ln `r/r_0 = - (2piε_0)/λ [V(r) - V(r_0)]`
⇒ r = r0e –2πε0Vr0/λe + 2πε0V(r)/λ
The equipotential surfaces are cylinders of radius r = r0e –2πε0[V(r) – V(r0)]/λ
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