Electrostatic Potential and Capacitance

 1. Electric Potential

The electric potential is the physical quantity which determines the direction of charge flow between two bodies when brought in contact. The positive charge always flows from a body at higher potential to that at lower potential.

Definition: The electric potential at any point in an electric field is defined as the work done in bringing a unit positive test charge from infinity to that point without acceleration.

If W is the work done in bringing infinitesimal positive test charge q0 from infinity to given point, then electric potential

V=Wq0

Electric potential at any point is also defined as the negative line integral of electric field from infinity to given point (independent of path followed).

i.e., V=–∫r∞E⃗ .dl→

The unit of electric potential is joule/coulomb or volt and its dimensional formula is [ML2 T –3 A–1].

2. Potential Difference

The potential difference between two points in an electric field is defined as the work done in bringing unit positive charge from one point to another.

3. Formulae for Electric Potential

(a) Due to a point charge q at a point distant r is V=14πε0qr

(b) Due to a short electric dipole at a distance r from its centre

(i) at its axis is V=14πε0pr2

(ii) at its equatorial position, V = 0

(iii) at a general point having polar coordinates (r, θ) with respect to centre of dipole is

V=14πε0pcosθr2

(c) due to a system of charges is

V=V1+V2+...+VN=∑Ni=114πε0qiri=14πε0[q1r1+q2r2+...+qNrN]

4. Equipotential Surface

An equipotential surface is the surface having the same potential at each point. The surface of a charged conductor in equilibrium is a equipotential surface.

5. Electric Potential Energy of a System of Point Charges

If q1 and q2 are point charges at separation r12, then electric potential energy U=14πε0q1q2r12.

If there are n point charges q1, q2,.... qn in system at separation rij between ith and jth charge
(i=1, 2,..., n, j=1, 2,...n) then potential energy of system

U=14πε0∑i∑j>iqiqjrij (i=1, 2,..., n, j=1, 2,...n)

6. Electric Potential Energy of a Dipole in Uniform Electric Field

Potential energy of dipole in uniform electric field is

U=–pE cos θ =– p⃗ . E⃗ 

Work done in rotating the dipole in uniform electric field from inclination θ1 to θ2

W=U2–U1=pE (cos θ1– cos θ2)

If dipole is initially in stable equilibrium position (θ1=0) and finally its inclination is θ, then

W= pE (1– cos θ)

7. Conductors and Insulators

Conductors are those substances which contain free charge carriers and so allow easy flow of current.

Insulators are those substances which contain practically no free charge carriers and do not allow the flow of current.

8. Free and Bound Charges Inside a Conductor

The electrons are free charge carriers inside a metallic conductor while positive ions fixed in lattice are bound charge carriers.

9. Dielectrics and Electric Polarisation

The insulators are often referred as dielectrics. Each dielectric is formed of atoms/molecules. In some dielectrics the positive and negative charge centres coincide, such dielectrics are said to be non-polar dielectrics. While in some other dielectrics the centres of positive and negative charges do not coincide, such dielectrics have permanent electric dipole moment and said to be polar dielectrics. The example of polar dielectric is water, while example of non-polar dielectric is carbon dioxide (CO2).

When a dielectric is placed in an external electric field, the centres of positive and negative dipoles get separated (in non-polar dielectrics) or get farther away (in polar dielectrics), so that molecules of dielectric gain a permanent electric dipole moment; this process is called polarisation and the dipole is said to be polarised.

The induced dipole moment developed per unit volume in an electric field is called polarisation density. Numerically it is equal to surface charge density induced at the faces which are perpendicular to the direction of applied electric field.

10. The behaviour of a conductor and dielectric in the presence of external electric field.

Conductor	Dielectric
	where K is dielectric constant
1. No electric field lines travel inside conductor.	1. Alignment of atoms takes place due to electric field.
2. Electric field inside a conductor is zero.	2. This results in a small electric field inside dielectric in opposite direction. Net field inside the dielectric is EK.

11. Capacitor and Capacitance

A capacitor contains two oppositely charged metallic conductors at a finite separation. It is a device by which capacity of storing charge may be varied simply by changing separation and/or medium between the conductors.

The capacitance of a capacitor is defined as the ratio of magnitude of charge (Q) on either plate and potential difference (V ) across the plate, i.e.,

C=QV

The unit of capacitance is coulomb/volt or farad (F).

12. Combination of Capacitors in Series and Parallel

(a) Series Combination: When capacitors are connected in series, then net capacitance C is given by

1C=1C1+1C2+1C3

Net charge Q=q1= q2 = q3 (remain same)

Net potential difference V=V1+V2+V3

(b) Parallel Combination: When capacitors are connected in parallel, then the net capacitance

C = C1 + C2 + C3

In parallel combination net charge Q = q1 + q2 + q3

Net potential difference V = V1 = V2 = V3 (remain same)

13. Capacitance of Parallel Plate Capacitor

A parallel plate capacitor consists of two parallel metallic plates separated by a dielectric. The capacitance, charge and voltage of parallel plate capacitor is given by

C=Kε0Ad, Q=KC0V,V'=VK

where K is dielectric constant, A = area of each plate and d = separation between the plates.

Special Cases:

(i) When there is no medium between the plates, then K=1, so

Cvacuum=ε0Ad=C0

(ii) When space between the plates is partly filled with a medium of thickness t and dielectric constant K, then capacitance

C=ε0Ad–t+tK=ε0Ad–t(1–1K)

Clearly, C>C0, i.e., on introduction of a dielectric slab between the plates of a parallel plate capacitor, its capacitance increases.

14. Charge Induced on a Dielectric

q′=–q(1–1K) where q is free charge on the capacitor plates.

15. Energy stored in a Charged Capacitor

U=12CV2=Q22C=12QV

This energy resides in the medium between the plates.

The unit is joule (J) .The energy stored per unit volume of a charged capacitor is given by

u=UV=12εE2

where E is electric field strength. The unit is joule/m3(J/m3)