Define potential difference between two points in an electrostatic field. Find an expresion for it. Define its SI unit.

Explanation

Potential difference : Electrostatic potential difference between any two points in an electric field is defined as amount of work done in moving unit positive charge from one : point to the other against the electrostatic force due to electric field.
Expression for electric potential difference : Consider that a point charge + q is placed at point O and points A and B are any two points in the electric field of the point charge + q. Let `r_(A)` and `r_(B)` be distance of points A and B from the point charge + q (Fig. below). Suppose that `V_(A)` and `V_(B)` are electric potentials at the points A and B respectively.
If `W_(AB)` the amount of work done in moving a vanishingly small positive test age `q_(0)` from point A to B, then
`V_(B)-V_(A)=(W_(AB))/(q_(0))" "......(1)`
Suppose that at any instant, the test charge qo is at point P. If `vecE` is electric field at point P due to charge + a placed at point O, then Force on the test charge qo due to the electric field of the source charge `=q_(0)vecE`.
To move the test charge against this electrostatic force without acceleration, the external force to be applied is given by

`vecF=-q_(0)vecE`
Suppose the test charge is moved through infinitesimally small displacement = `vec(dr)`, then small work done to move the test charge from point P to Q,
`dW=vecF.vec(dr)=(-q_(0)vecE).vec(dr)=-q_(0)Edrcos0^(@)=-q_(0)Edr`
Now, electric field at point P due to the charge +q
`E=(1)/(4piin_(0)).(q)/(r^(2))`
`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 go from point A to B is given by
`W_(AB)=int_(A)^(B)dW=int_(r_(A))^(r_(B))-(1)/(4piin_(0)).(qq_(0))/(r^(2))dr=-(1)/(4piin_(0)).qq_(0)int_(r_(A))^(r_(B))r^(-2)dr`
`=-(1)/(4piin_(0)).qq_(0)|(r^-1)/(-1)|_(r_(A))^(r_(B))=(1)/(4piin_(0)).qq_(0)|(1)/(r)|_(r_(A))^(r_(B))`
or `W_(AB)=(1)/(4piin_(0)).qq_(0)((1)/(r_(B))-(1)/(r_(A)))" "......(2)`
From the equations (1) and (2), we get
`V_(B)-V_(A)=(1)/(4piin_(0)).q((1)/(r_(B))-(1)/(r_A))`
Unit of electrostatic potential difference : The SI unit of potential difference is volt (V). In SI, work and charge are measured in joule and coulomb respectively. Therefore, from equation (2), the SI unit of potential difference is also given by
1 volt (V) `=(1" joule")/(1" coulomb")=1" joule coulomb"^(-1)(JC^(-1))`
The electrostatic potential difference between two points is said to be 1 volt, if 1 joule of work is done in moving a charge of I coulomb from one point to the other (without acceleration) against the electrostatic force due to electric field.