Electrostatic Potential and Capacitance — Study Notes

1. Introduction

The concept of electric potential gives us a scalar description of the electric field, which greatly simplifies calculations in electrostatics. Whereas the electric field is a vector, potential is a scalar, so adding potentials is far easier than adding field vectors. This chapter develops the potential due to point charges, the potential energy of charge systems, and the behaviour of conductors and dielectrics in electrostatic fields.

The chapter then introduces capacitance - the ability of a conductor system to store charge - and the capacitor, one of the most widely used components in electronics. We study the parallel plate capacitor, the effects of dielectrics, the combinations of capacitors, and the energy stored in a capacitor. Energy stored in an electric field is a key concept that connects electrostatics with energy conservation.

Understanding potential and capacitance is essential not only for this chapter but also for current electricity, where potential difference drives current, and for oscillating circuits in later chapters. The idea that energy is stored in electric fields has deep significance in electromagnetic theory.

2. Electric Potential and Potential Difference

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

V = W / q0

The SI unit of potential is the volt (V), where 1 V = 1 J/C. Potential is a scalar quantity. The potential difference between two points A and B is the work done per unit charge in moving a charge from A to B.

The potential at a point due to a point charge Q at a distance r is:

V = (1/(4 pi epsilon_0)) * Q / r = k Q / r

For a positive charge, the potential is positive and decreases with distance; for a negative charge, the potential is negative and increases (becomes less negative) with distance. The potential due to a system of charges is the algebraic sum of the potentials due to each charge, since potential is a scalar.

3. Potential Due to a Dipole

The potential at a point due to an electric dipole depends on the angle theta between the position vector and the dipole axis. For a point at distance r, making angle theta with the dipole axis, and for r much greater than the separation 2a:

V = (1/(4 pi epsilon_0)) * p cos(theta) / r^2

The potential is maximum along the axial line (theta = 0) where V = k p / r^2, and zero on the equatorial plane (theta = 90 degrees) where cos(theta) = 0. The dipole potential falls off as 1/r^2, faster than the 1/r potential of a point charge.

The electric field of a dipole can be obtained from the potential using E = -dV/dr, giving the axial field E = 2k p / r^3 and the equatorial field E = k p / r^3, matching the results of the previous chapter.

4. Equipotential Surfaces

An equipotential surface is a surface on which the potential at every point is the same. Since the potential is constant on such a surface, no work is done in moving a charge along it. Consequently, the electric field is always perpendicular to equipotential surfaces, directed from higher to lower potential.

The equipotential surfaces of a point charge are concentric spheres centred on the charge. For a uniform field, the equipotential surfaces are parallel planes perpendicular to the field. The spacing between equipotential surfaces indicates the field strength: the field is strong where the surfaces are closely spaced, since E = -dV/dr.

Equipotential surfaces never intersect, because at a point of intersection there would be two different potentials, which is impossible. These surfaces are useful because they give a visual picture of the field without drawing field lines.

5. Potential Energy of a System of Charges

The potential energy of a system of charges is the work done in assembling the system by bringing the charges from infinity one by one. For two charges q1 and q2 separated by a distance r, the potential energy is:

U = (1/(4 pi epsilon_0)) * q1 * q2 / r

For three charges q1, q2, and q3, the total potential energy is the sum over all pairs:

U = (1/(4 pi epsilon_0)) * (q1 q2 / r12 + q2 q3 / r23 + q3 q1 / r31)

The potential energy is positive for like charges (they repel, so work is done in bringing them together) and negative for unlike charges. When a charge q is moved between two points differing in potential by V, the work done is W = q V.

6. Conductors in Electrostatic Equilibrium

When a conductor is placed in an electric field, the free electrons redistribute until the field inside is zero. In electrostatic equilibrium, the following facts hold:

1. The electric field inside a conductor is zero.
2. The electric field just outside the surface is perpendicular to the surface.
3. The net charge resides entirely on the surface.
4. The potential is constant throughout the conductor and its surface.
5. The field just outside a charged conductor is E = sigma/epsilon_0.

These properties lead to electrostatic shielding: a conductor can shield its interior from external electric fields. The surface charge density is greater where the surface is more sharply curved, which is why lightning conductors are pointed.

7. Dielectrics and Polarization

A dielectric is an insulating material whose molecules polarize when placed in an electric field. Polarization is the partial alignment of molecular dipole moments along the field. Two types of dielectrics exist: polar molecules, which have a permanent dipole moment (like water), and non-polar molecules, which acquire an induced dipole moment in an external field.

When a dielectric is placed between the plates of a capacitor, the induced charges on its surfaces reduce the effective field inside. The ratio of the field in vacuum to the field in the dielectric is the dielectric constant K:

E_dielectric = E_vacuum / K

The dielectric constant of a material also equals the relative permittivity, so the permittivity of the medium is epsilon = K epsilon_0. Dielectrics increase the capacitance of a capacitor and also allow it to sustain a higher voltage.

8. Capacitance and Capacitors

The capacitance of a conductor is defined as the ratio of the charge on it to its potential:

C = Q / V

The SI unit of capacitance is the farad (F), where 1 F = 1 C/V. The farad is a very large unit; practical capacitors are rated in microfarads (mu F = 10^-6 F) and picofarads (pF = 10^-12 F). The capacitance of a parallel plate capacitor in vacuum is:

C = epsilon_0 * A / d

where A is the plate area and d the separation. When a dielectric of constant K fills the space between the plates:

C = K epsilon_0 A / d

A capacitor stores energy. The work done to charge a capacitor is stored as electrical potential energy:

U = (1/2) Q V = (1/2) C V^2 = Q^2 / (2 C)

This energy is stored in the electric field between the plates, with energy density u = (1/2) epsilon_0 E^2 per unit volume.

9. Combination of Capacitors

In series combination, the capacitors are connected end to end, so the charge on each is the same while the potential differences add. The equivalent capacitance is:

1/C_series = 1/C1 + 1/C2 + 1/C3 + ...

The series combination has a total capacitance smaller than the smallest individual capacitor. In parallel combination, the potential difference across each capacitor is the same while the charges add:

C_parallel = C1 + C2 + C3 + ...

The parallel combination has a total capacitance larger than the largest individual capacitor. Series combinations reduce capacitance but increase the voltage rating, while parallel combinations increase capacitance at the same voltage.

Quick Revision Tables

Mind Map

Important Diagrams (SVG)

Diagram 1: Equipotential Surfaces and Field of a Point Charge

Diagram 2: Parallel Plate Capacitor with Dielectric

Common Mistakes

1. Confusing potential (V = kQ/r) with potential energy (U = qV); potential is per unit charge, while potential energy depends on the charge.
2. Forgetting that potential is a scalar; unlike field, potentials add algebraically, not as vectors.
3. Taking the potential of a dipole to be zero everywhere; it is zero only on the equatorial plane.
4. Believing the field inside a charged conductor is nonzero; in electrostatic equilibrium the field inside is always zero.
5. Using C = Q/V but confusing charge distribution when capacitors are in series, where the charge is the same on each capacitor.
6. Forgetting that for series combination the reciprocal of capacitances add, not the capacitances themselves.
7. Omitting the factor 1/2 in the energy stored; U = (1/2) CV^2, not CV^2.

Exam Tips

1. Define electric potential V = W/q0 and write V = kQ/r for a point charge, giving the volt as the unit.
2. Derive the potential of a dipole V = k p cos theta / r^2 and state that it is zero on the equatorial plane.
3. State the properties of equipotential surfaces and that the field is perpendicular to them.
4. Write the potential energy of two charges U = k q1 q2/r and of a system of three charges.
5. List the properties of conductors in electrostatic equilibrium, including E = 0 inside and E = sigma/epsilon_0 outside.
6. Derive the parallel plate capacitance C = epsilon_0 A/d and its modified form C = K epsilon_0 A/d with a dielectric.
7. Derive the energy stored U = (1/2) CV^2 and the energy density u = (1/2) epsilon_0 E^2.
8. Solve combination problems using 1/C = sum of 1/Ci for series and C = sum of Ci for parallel.

Conclusion

This chapter introduced electric potential as a scalar that simplifies electrostatic calculations. We derived the potential due to point charges and dipoles, studied equipotential surfaces, and learned that the field is always perpendicular to them. The potential energy of charge systems was developed, followed by the properties of conductors and the physics of dielectrics through polarization. We then studied capacitance, the parallel plate capacitor, series and parallel combinations, and the energy stored in a capacitor. The central results - V = kQ/r, C = K epsilon_0 A/d, and U = (1/2) CV^2 - are used throughout electricity and electronics, from simple circuits to energy storage systems.