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प्रश्न
Derive an expression for electrostatic potential due to a point charge.
उत्तर
- Consider a positive charge ‘q’ kept fixed at the origin. Let P be a point at distance r from the charge ‘q’.
Electrostatic potential at a point P - The electric potential at the point P is
V = `int_∞^"r" (- vec"E")* "d"vec"r" = - int_∞^"r" vec"E" * vec"dr"` - Electric field due to positive point charge is
`vec"E" = 1/(4piε_0) "q"/"r"^2 hat"r"`
V = `- 1/(4piε_0) int_∞^"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/(4piε_0) int_∞^"r" "q"/"r"^2 hat"r" * "dr" hat"r" = - 1/(4piε_0) int_∞^"r" "q"/"r"^2 "dr"` - After the integration,
V = `- 1/(4piε_0) "q"{- 1/"r"}_∞^"r" = 1/(4piε_0) "q"/"r"` - Hence the electric potential due to a point charge q at a distance r is
V = `1/(4piε_0) "q"/"r"`
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