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 y−axis 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=kλ∫a+Ladx√x2+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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