Electric Charges and Fields

In this chapter: properties of charge, conductors and insulators, charging by induction, Coulomb's law and superposition, electric field and field lines, electric flux, electric dipole (axial and equatorial field, torque in a uniform field), continuous charge distributions, Gauss's law and its three standard applications.

Electric charge and its properties

Charge is the property that makes a body exert and feel electric forces. There are two kinds, called positive and negative by convention (a convention due to Franklin). Like charges repel and unlike charges attract. The SI unit is the coulomb (C), which is the charge that flows in 1 s through a wire carrying 1 A.

• Additivity: charge is a scalar, so the total charge of a system is the algebraic sum, e.g. +1, +2, −3 and +5 units add to +5.
• Conservation: charge can be transferred but not created or destroyed in an isolated system. Even when particles are created (pair production), the net charge before and after is the same.
• Quantisation: any charge is an integer multiple of the basic charge, q = ne, with e = 1.6 × 10⁻¹⁹ C. At the large scale (a few μC is already about 10¹³ electrons) quantisation can be ignored and charge treated as continuous.

When a glass rod is rubbed with silk, electrons move from glass to silk; the rod ends up positive and the silk equally negative. Mass changes by a negligible amount, since electrons are very light.

Conductors, insulators and induction

Metals, the human body and the earth are conductors: they have free electrons that move through the material. Glass, plastic and nylon are insulators, so charge put on them stays where it is placed. Semiconductors lie in between.

In charging by induction a charged rod is brought near (not touching) a conductor. Opposite charge collects on the near side. The conductor is then earthed so that the like charge escapes, the earth connection is removed, and only then is the rod taken away. The conductor is left with a charge opposite to that of the rod. The same idea explains why a charged comb attracts small bits of paper: it induces charges in the paper, and the nearer unlike charge is attracted more than the farther like charge is repelled.

Coulomb's law

The force between two point charges at rest is along the line joining them, proportional to the product of the charges and inversely proportional to the square of the distance.

In vector form, the force on q₂ due to q₁ is F₂₁ = k q₁q₂ r₂₁/r₂₁³, where r₂₁ points from q₁ to q₂. The law obeys Newton's third law, F₁₂ = −F₂₁. In a medium of dielectric constant K, the force becomes F/K.

The electric force between a proton and an electron is about 10³⁹ times their gravitational attraction, which is why gravity plays no part in atomic structure.

Superposition

For many charges, the force on any one is the vector sum of the forces due to each of the others taken one at a time. The presence of other charges does not change the force between any pair. Most mistakes here come from adding magnitudes instead of vectors, so always draw the force arrows first.

Worked example: Charges +3 μC and −2 μC are placed 30 cm apart in air. Find the force between them.
Solution: F = 9 × 10⁹ × (3 × 10⁻⁶)(2 × 10⁻⁶)/(0.30)² = (5.4 × 10⁻²)/0.09 = 0.6 N. The charges are unlike, so the force is attractive, 0.6 N on each charge, directed towards the other.

Electric field

The electric field at a point is the force per unit positive test charge placed there, E = F/q₀ (strictly the limit as q₀ → 0, so that the test charge does not disturb the source). Its unit is N C⁻¹, which is the same as V m⁻¹.

The field of a group of charges is again the vector sum of individual fields. The field is useful because it separates the problem into two parts: the source sets up a field, and any charge placed there feels F = qE. For time-varying situations the field turns out to be a physical entity with its own energy and momentum, and it carries disturbances at a finite speed.

Electric field lines

A field line is a curve whose tangent at any point gives the direction of the field there. The density of lines (lines per unit area normal to them) shows the field strength. For a point charge, the number of lines through a sphere does not depend on its radius, which is another way of seeing the 1/r² law.

• Lines start on positive charges and end on negative charges; for a single charge they start or end at infinity.
• In a charge-free region they are continuous curves without breaks.
• Two field lines never cross, since the field has a single direction at every point.
• Electrostatic field lines do not form closed loops, because the field is conservative.

Electric flux

Flux through a small area element is ΔΦ = E · ΔS = EΔS cos θ, where ΔS is along the normal to the area. For a closed surface the normal is taken outward, so lines going out count positive and lines coming in count negative. Flux is a scalar with unit N m² C⁻¹. It is proportional to the number of field lines crossing the surface.

Electric dipole

Two equal and opposite charges q and −q separated by a distance 2a form a dipole. The dipole moment is p = q × 2a, directed from −q to +q, unit C m. The total charge is zero, but the field is not, because the two charges are at different places. At large distances the dipole field falls as 1/r³, faster than the 1/r² of a single charge.

So at the same large distance, the axial field is twice the equatorial field. Molecules such as H₂O have a permanent dipole moment because the centres of positive and negative charge do not coincide; CO₂ and CH₄ do not, but they acquire an induced moment in an external field.

Dipole in a uniform external field

The forces qE and −qE on the two ends are equal and opposite, so the net force is zero. They do not act along the same line, so they form a couple:

The torque tries to align p with E. It is maximum (pE) at θ = 90° and zero at 0° and 180°. In a non-uniform field there is also a net force; a dipole aligned with the field moves towards the region of stronger field. This is how a charged comb pulls neutral paper, whose induced dipoles are aligned with the comb's field.

Continuous charge distributions

For charge spread over a line, surface or volume we use densities: linear λ (C m⁻¹), surface σ (C m⁻²) and volume ρ (C m⁻³). The field is found by dividing the body into small elements, writing the field of each as that of a point charge (for example ΔE = kρΔV/r²), and adding vectorially. Direct integration is often messy, and Gauss's law gives the answer much faster when the distribution is symmetric.

Gauss's law

The total flux through any closed surface equals the net charge enclosed divided by ε₀. Points to remember:

• The result does not depend on the shape or size of the surface.
• Charges outside the surface contribute zero net flux, but the field E in the integral is the total field due to all charges, inside and outside.
• The surface (the Gaussian surface) must not pass through a discrete charge, although it may pass through a continuous distribution.
• Gauss's law is true for any field that falls as 1/r²; it is equivalent to Coulomb's law.

For the wire, the flat ends of the cylinder have no flux because the field is parallel to them, so only the curved surface counts: E · 2πrl = λl/ε₀. Always explain your choice of surface in board answers: symmetry tells you the direction of E and that its magnitude is constant over the part of the surface that carries flux.

Worked example: A point charge of 1.0 μC sits at the centre of a cube. Find the flux through the whole cube and through one face.
Solution: Total flux = q/ε₀ = (1.0 × 10⁻⁶)/(8.854 × 10⁻¹²) ≈ 1.13 × 10⁵ N m² C⁻¹. By symmetry each of the six faces gets one-sixth, about 1.88 × 10⁴ N m² C⁻¹. The size of the cube does not matter.

Common mistakes: (1) Adding forces or fields as numbers when they point in different directions. (2) Taking the direction of p from + to −; it is from −q to +q. (3) Forgetting that the equatorial field of a dipole is opposite to p. (4) Thinking zero flux through a surface means zero field on it; a dipole inside a closed surface gives zero flux but the field is clearly not zero. (5) Using E = σ/ε₀ for a sheet; that is the field just outside a charged conductor, while an isolated thin sheet gives σ/2ε₀.

JEE and NEET focus

• Coulomb's law with superposition, including equilibrium of a third charge placed on the line joining two charges.
• Field of a dipole on the axis and the equatorial line, and the 2 : 1 ratio at large distance.
• Torque on a dipole in a uniform field and the condition for zero net force.
• Gauss's law problems, especially flux through a cube face or a hemisphere, and the fields of a wire, sheet and shell.
• Properties of field lines and quantisation of charge.

Practice questions