Definition of Electric Potential

For defining electric potential we take the reference of zero potential at infinity.

The electric potential at any point in an electric field is defined as the work done in bringing per unit (infinitesimal) positive test charge from infinity to given point. Therefore, if W is the work done is bringing test charge q₀ from infinity to given point in an electric field, then electric potential

q₀  is taken as infinitesimal so that it may not affect the original electric field; hence it is better to write

Definition of Electric Potential Difference : The electric potential difference between two points in an electric field is defined as the work done in bringing per unit infinitesimal positive test charge from one point to another against the direction of electric field.

If  is the work done in bringing the positive test charge  from B to A, then electric potential difference between A and B is

Units and Dimensions of Electric Potential

The electric potential 

The S.I. unit of work ( W ) is joule and of charge ( q₀ ) is coulomb, so S.I. unit of electric potential is joule/coulomb or volt.

Dimensional formula for work is  [ML²T^-2]and of q₀ is [AT], so dimensional formula for electric potential is

Relation between electric field and potential

Let dV = potential difference between A and B.

= work done in taking unit +ve charge from  A → B  or from B → A against the field

=  E. dr

Negative gradient of potential is electric field.

Physical Meaning of Electric Potential

 

The electric potential is that physical quantity which determines the direction of flow of charge from one body to another when brought in electrical contact.

If two charged bodies A and B have charges Q_A  and QB and potentials V_A and V_A respectively such that V_A > V_A , then when bodies A and B are brought in electrical contact, the charge will always flow from body A to body B till then potentials become equal; whatever the initial charges on bodies A and B may be.

Electric Potential as Line Integral of Electric Field

The electric potential at any point in an electric field is given by the negative line integral of electric field from infinity to given point in electric field.

If P is a point in electric field of a distance r and  is electric field at intermediate length element  (at point A ) between  and point P, then electric potential at P is

This line integral is independent of path followed between infinity and point P.

Electric Potential due to a point charge

The electric potential due to a point charge at a any point is defined as the work done in bringing per unit (infinitesimal and positive) test charge from infinity to given point. Let q be the charge and P a point at a distance r from q at which the potential is to be found.

Suppose test charge  is being brought from infinity to point P . When test charge  is at a distance x, the electrostatic coulomb force on the test charge is

where  is unit vector along  .

Work done by external force against electric force during small displacement dx is

Work done in bringing test charge  from infinity to point P is

 The electric potential at point P (distant r from point charge q ) is

Clearly, the potential due to a point charge varies inversely with distance r from the point charge (fig.)

Electric Potential due to a system of Charges

The electric potential at a point due to a system of point charges is equal to the algebraic sum of potentials due to all individual charges of the system at that point. Therefore, if  V₁,V₂,V₃,……..,V_n are the electric potentials at given point due to charges  q₁,q₂,q₃,……. q_n of the system, then the net electric potential at that point will be

 

 

While calculating potential at a point it should be taken into account what type of charge is it. If it is positive charge, potential is positive. If it is negative charge, potential is negative.

Electric potential due to an Electric Dipole

The  is electric dipole moment  of dipole formed of charge ? q and + q at separation 2 a , then the electric potential at a distance r from the centre of dipole at difference positions are given by

(i) At Axial point, C

(ii) At equatorial point D ,

V =0     (zero)

(iii) At any general point P having polar coordination  (r,θ)

 .

Equipotential Surface

The surface having same electric potential at its each point is called an equipotential surface. The surface of a charged conductor is always an equipotential surface.

Charactersitics :

(i) The electric field lines are always perpendicular to every point of equipotential surface.

(ii) The work done in moving a charge between any two points on equipotential surface is always zero.

Electric Potential Energy

The electric potential energy of a system of charges is defined as the work done in assembling the system of charges assuming them initially at infinity from one another.

(i) For a system of two charges : If two charges q₁  and q₂  are at a distance r_12,  then electric potential energy is given by

(ii) For a system of three charges :

(iii) For a system of n -charges :

Electric Potential Energy of an Electric Dipole in a Uniform Electric Field

The electric potential energy of an electric dipole in an electric field is defined as the work done by external force in bringing the dipole from infinity to its present position and orientation. If p is dipole moment and E is the electric field and θ is angle between the directions of  and  , then the electric potential energy of dipole

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