Electric Potential

Electric Potential (V)

Electric potential at any point is equal to the work done per unit positive charge in carrying it from infinity to that point in electric field.

Electric potential,

$V=\frac{W}{q}$

Its SI unit is J/C or volt and its dimension is [ML²T^–3A^–1]

It is a scalar quantity.

Electric potential due to a point charge at a distance r is given by

$V=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}$

Potential due to System of Charges

Let there be a number of point charges q₁,q₂,q₃……q_n at distances r₁,r₂,r₃…….r_n respectively from the point P, where electric potential is given by

$V=\frac{1}{4\pi {\epsilon }_0}\sum _{i=1}^n\frac{q_i}{r_i}$

Potential Gradient

The rate of change of potential with distance in electric field is called potential gradient.

Potential gradient = $\frac{dV}{dr}$

Its unit is V/m.

Relation between potential gradient and electric field intensity is given by

$E=-\left(\frac{dV}{dr}\right)$

Equipotential Surface

Equipotential surface is an imaginary surface joining the point of same potential in an electric field. So, we can say that the potential difference between any two points on an equipotential surface is zero. The electric lines of force at each point of an equipotential surface are normal to the surface.

(i) Equipotential surface may be planer, solid etc. But equipotential surface can never be point size.

(ii) Electric field is always perpendicular to equipotential surface.

(iii) Equipotential surface due to an isolated point charge is spherical.

(iv) Equipotential surface are planer in an uniform electric field.

(v) Equipotential surface due to a line charge is cylindrical.

Electric Dipole

An electric dipole consists of two equal and opposite point charge separated by a very small distance. e.g. a molecule of HCl, a molecule of water etc.

Electric Dipole Moment p = q × 2a

Its SI unit is 'coulomb-metre' and its dimension is [LTA].

It is a vector quantity and its direction is from negative charge towards positive charge.

Electric Field Intensity and Potential due to an Electric Dipole

(i) On Axial Line

Electric field intensity $E=\frac{1}{4\pi {\epsilon }_0}\frac{2pr}{{\left(r^2-a^2\right)}^2}$

If $Ifr>>2a,thenE=\frac{1}{4\pi {\epsilon }_0}\frac{2pr}{r^3}$

Electrical potential $V=\frac{1}{4\pi {\epsilon }_0}\frac{p}{\left(r^2-a^2\right)}$

If r >> 2a, then $V=\frac{1}{4\pi {\epsilon }_0}\frac{p}{r^2}$

(ii) On Equatorial Line

Electric field intensity

$E=\frac{1}{4\pi {\epsilon }_0}\frac{p}{{\left(r^2+a^2\right)}^{3/2}}$

If r > > 2a, then $E=\frac{1}{4\pi {\epsilon }_0}\frac{p}{r^3}$

Electric potential V = 0

(iii) At any point along a Line Making $\theta$ Angle with Dipole Axis

Electric field intensity $E=\frac{1}{4\pi {\epsilon }_0}\frac{p\sqrt{\left(1+3{\cos }^2\theta \right)}}{r^3}$

Electric potential $V=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{\left(r^2-a^2{\cos }^2\theta \right)}$

If r > > 2a, then $V=\frac{1}{4\pi {\epsilon }_0}\frac{p\cos \theta }{r^2}$

Torque

Torque acting on an electric dipole placed in uniform electric field is given by

$\tau =Ep\sin \theta or\tau =p\times E$

When $\theta =90,$ then ${\tau }_{\max }=Ep$

When electric dipole is parallel to electric field, it is in stable equilibrium and when it is anti-parallel to electric field, it is in unstable equilibrium.

Work Done

Work done is rotating an electric dipole in a uniform electric field from angle ${\theta }_{1}to{\theta }_2$ is given by

$W=Ep\left(\cos {\theta }_1-\cos {\theta }_2\right)$

If initially it is in the direction of electric field, then work done in rotating through and angle $\theta$

$W=Ep\left(1-\cos \theta \right)$

Potential Energy

Potential energy of an electric dipole in a uniform electric field is given by $U=-pE\cos \theta$

Dipole in Non-uniform Electric Field

When an electric dipole is placed in a non-uniform electric field, then a resultant force as well as a torque act on it.

Net force on electric dipole = (qE₁–qE₂) along the direction of greater electric field intensity.

Therefore electric dipole undergoes rotational as well as linear motion.

Potential Energy of Charge System

Two point charge system, contains charges q₁ and q₂ separated by a distance r is given by $U=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r}$

Three point charge system

$U=\frac{1}{4\pi {\epsilon }_0}\left[\frac{q_1{q}_2}{r_1}+\frac{q_2{q}_3}{r_2}+\frac{q_3{q}_2}{r_3}\right]$

Behaviour of a Conductor in an Electrostatic Field

1. Electricity field at any point inside the conductor is zero.
2. Electric field at any point on the surface of charged conductor ins directly proportional to the surface density of charge at that point but electric potential does not depend upon the surface density of charge.
3. Electric potential at any point inside the conductor is constant and equal to potential.