Define electric potential. What is the SI unit of potential? Obtain an expression for electric potential at a distance r from isolated unit positive charge.

Explanation

Electric potential : Electric potential at a point in the electric field is defined as the amount of work done in bringing unit positive charge from infinity to that against the electrostatic force due to electric SI unit of potential is volt (V).
Electric Potential at a point due to an isolated unit positive charge :
Consider a point charge + q placed at point O. Suppose that `V_(A)` is electric potential at point A, whose distance from the source charge + q is `r_(A)`.
If `W_(ooA)` is work done in moving a vanishingly small positive test charge `q_(0)` from infinity point A, then
`V_(A)=(W_(ooA))/(q_(0))" "......(1)`
Suppose that at any instant, the test charge is at point P. The electric field `vecE` due to the charge +q exerts force `q_(0)vecE` on the test charge placed at point P. The test charge can be moved against this force without acceleration by applying an external force `vecE=-q_(0)vecE` on it. If the test charge `q_(0)` is moved through infinitesimally small displacement OP. given by

or`dW=vecF.vec(dr)=(-q_(0)vecE).vec(dr)=-q_(0)Edrcos0^(@)=-q_(0)Edr`
or `dW=-q_(0)Edr`
Since `E=(1)/(4piin_(0)).(q)/(r^(2))`, we get
`dW=-q_(0)((1)/(4piin_(0)).(q)/(r^(2)))dr=-(1)/(4piin_(0)).(qq_(0))/(r^(2))dr`
Hence, work done in moving the test charge from infinity to point A is given by
`W_(ooA)=int_(oo)^(A)dW=int_(oo)^(r_(A))-(1)/(4piin_(0)).(qq_(0))/(r^(2))dr=(1)/(4piin_(0))qq_(0)int_(oo)^(r_(A))r^(-2)dr`
`=-(1)/(4piin_(0)).qq_(0)|(r^(-1))/(-1)|=(1)/(4piin_(0)).qq_(0)|(1)/(r)|_(oo)^(r_(A))`
`=-(1)/(4piin_(0)).qq_(0)((1)/(r_(A))-(1)/(oo))=(1)/(4piin_(0)).qq_(0)((1)/(r_(A))-0)`
or `W_(ooA)=(1)/(4piin_(0)).(qq_(0))/(r_(A))`
`(W_(ooA))/(q_(0))=(1)/(4piin_(0))(q)/(r_(A))`
`V_(A)=(1)/(4piin_(0)).(q)/(r_(A))` (Using (1))
In case, the distance of point A is from the charge + q is denoted by r (in place of `(r_(A))`, then subscript A may be dropped. If V denotes the electrostatic potential at any point at distance r from the charge +q, then
`V=(1)/(4piin_(0)).(q)/(r)`