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Question

Calculate the electric potential at the point P due to the given charged rod of length L having charge per unit length λ.


A
V=kλ ln{a+La}
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B
V=2kλ ln{a+La}
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C
V=2kλ ln⎪ ⎪⎪ ⎪a+L+b2+(a+L)2a+b2+a2⎪ ⎪⎪ ⎪
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D
V=kλ ln⎪ ⎪⎪ ⎪a+L+b2+(a+L)2a+b2+a2⎪ ⎪⎪ ⎪
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Solution

The correct option is D V=kλ ln⎪ ⎪⎪ ⎪a+L+b2+(a+L)2a+b2+a2⎪ ⎪⎪ ⎪

Let the total charge on the rod is Q.
The charge per unit length, λ=QL
Let's take a small length of charged rod dx at a distance x from the vertical yaxis as shown below.


Total charge in dx element is dq
dq=QLdx=λdx
Let the distance between point P and dx is r.
r=x2+b2.

Now the electric potential at point P due to charge dq is given by

dV=kλdxr
The electric potential at point P due to entire rod is obtained by integration.

dV=kλa+Ladxx2+b2

V=kλ ln(x2+b2+xb)a+La

V=kλ ln⎪ ⎪⎪ ⎪(a+L)2+b2+a+Lb⎪ ⎪⎪ ⎪kλ ln{a2+b2+ab}

V=kλ ln⎪ ⎪⎪ ⎪a+L+b2+(a+L)2a+a2+b2⎪ ⎪⎪ ⎪

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