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Questions
Derive an expression for electrostatic potential due to a point charge.
Derive an expression for electric potential due to a point charge.
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Solution
Consider a positive charge ‘q’ kept fixed at the origin. Let P be a point at a distance r from the charge ‘q’.
Electrostatic potential at a point P
The electric potential at the point P is:
V = `int_infty^r (-vec E) * d vec r`
= `-int_infty^r vec E * d vec r`
The electric field due to a positive point charge is:
`vec E = 1/(4 pi epsilon_0) q/r^2 hat r`
V = `-1/(4 pi epsilon_0) int_infty^r q/r^2 hat r * d vec r`
The infinitesimal displacement vector,
`d vec r = dr hat r` and using `hat r hat r` = 1, we have,
V = `-1/(4 pi epsilon_0) int_infty^r q/r^2 hat r * dr hat r`
= `-1/(4 pi epsilon_0) int_infty^r q/r^2 dr`
After the integration,
V = `-1/(4 pi epsilon_0) q {-1/r}_infty^r`
= `1/(4 pi epsilon_0) q/r`
Hence, the electric potential due to a point charge q at a distance r is:
V = `1/(4 pi epsilon_0) q/r`
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