PhysicsISC Class 12 · Electric Charges and Fields

Coulomb's Law, Quantisation of Charge and the Superposition Principle

Q: Does Coulomb's law hold for any charged bodies?

It holds exactly for point charges. For spheres with uniform charge, it also works using the distance between centres, because outside a uniform spherical charge the field is as if the charge were at the centre.

Q: Why is K always greater than 1?

A medium becomes polarised and its induced charges partly cancel the field of the original charges, so the force is weaker than in vacuum. Vacuum has K = 1 and air very nearly 1.

Q: Is the vector form with position vectors needed?

The ISC 2027 syllabus asks for the vector form of Coulomb's law, but says the position coordinates \vec{r}_1, \vec{r}_2 are not necessary.

Q: How is charge conserved in pair production?

A photon (no charge) creates an electron (−e) and a positron (+e), so the total charge stays zero. Conservation holds in every known process.

Q: How was this examined recently?

In 2024 a 2-mark question asked about the relative permittivity of water and whether a given charge was possible (quantisation).

Updated 15 October 2026 · Fact-checked

Coulomb's law: two point charges $q_1$ and $q_2$ a distance r apart in vacuum exert equal and opposite forces $F = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r^2}$ along the line joining them. In a medium of relative permittivity K the force is K times smaller. With several charges, the net force on one is the vector sum of the separate forces.

Understand Coulomb's Law and the Superposition Principle

Charge. There are two kinds of charge; like charges repel and unlike charges attract. Rubbing transfers electrons: the body that gains electrons becomes negative and the one that loses them becomes positive, by the same amount.

Conservation: the total charge of an isolated system never changes; charge is only transferred.
Quantisation: every charge is a whole-number multiple of the electronic charge, $q = n e$ with $e = 1.6 \times 1 0^{- 19}$ C and n an integer. A body cannot carry $2.4 \times 1 0^{- 19}$ C (that is 1.5e). The ISC 2024 examiners found candidates unaware of this.

Coulomb's law. The force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance:
$F = k \frac{q _{1} q _{2}}{r ^{2}}, k = \frac{1}{4 π ε _{0}} = 9 \times 1 0^{9} N m^{2} C^{- 2},$
where $ε_{0} = 8.85 \times 1 0^{- 12} C^{2} N^{- 1} m^{- 2}$ is the permittivity of free space. One coulomb is the charge that, placed 1 m from an equal charge in vacuum, repels it with $9 \times 1 0^{9}$ N.

Vector form. If $\overset{r}{^}_{12}$ is the unit vector from $q_{1}$ to $q_{2}$ , the force on $q_{2}$ due to $q_{1}$ is
$F_{21} = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}} \overset{r}{^}_{12}, F_{12} = - F_{21} .$
With the signs of the charges included, a positive product gives repulsion (along $\overset{r}{^}_{12}$ ) and a negative product attraction. The forces obey Newton's third law and act along the line joining the charges (central forces).

Effect of a medium. In a medium of permittivity $ε = K ε_{0}$ , $F_{m e d i u m} = \frac{1}{4 π ε _{0} K} \frac{q _{1} q _{2}}{r ^{2}} = \frac{F _{v a c uu m}}{K}$ . K (also written $ε_{r}$ ) is the relative permittivity or dielectric constant: the ratio of the force (or field) between two charges in vacuum to that in the medium, at the same separation. For water K ≈ 81, so forces and fields between charges in water are $\frac{1}{81}$ of their values in vacuum, and a capacitor filled with water would have 81 times the capacitance. K is not refractive index or permeability; the 2024 examiners saw these confused.

Comparison with gravitation. Both are inverse-square, central forces acting at a distance. Differences: electric force can attract or repel, gravity only attracts; electric force depends on the medium, gravity does not; electric force is enormously stronger (between two protons about $1 0^{36}$ times).

Superposition principle. The force between two charges is not affected by other charges nearby. So the net force on a charge $q_{1}$ due to $q_{2}, q_{3}, \dots$ is the vector sum:
$F_{1} = F_{12} + F_{13} + F_{14} + \dots$
Find each force's magnitude from Coulomb's law, draw its direction (repulsion away, attraction towards), and add by components or the parallelogram rule.

Equilibrium on a line. For two like charges, a third charge feels zero net force at a point between them, nearer the smaller charge. For unlike charges, the null point is outside, beyond the smaller charge. At that point $\frac{q _{1}}{x ^{2}} = \frac{q _{2}}{( d \pm x ) ^{2}}$ .

Continuous distributions. Charge spread along a line, over a surface or through a volume is described by $λ = \frac{d q}{d l}$ (C m⁻¹), $σ = \frac{d q}{d A}$ (C m⁻²) and $ρ = \frac{d q}{d V}$ (C m⁻³). Each small element $d q$ acts as a point charge, and superposition (integration) gives the total effect.

Coulomb's Law and the Superposition Principle: formulas and key facts

Quantisation: $q = n e, e = 1.6 \times 1 0^{- 19} C$; n is an integer.
Coulomb's law: $F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}}$; $\frac{1}{4 π ε _{0}} = 9 \times 1 0^{9}$ N m² C⁻².
Vector form: $F_{21} = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}} \overset{r}{^}_{12} = - F_{12}$; Force on 2 due to 1.
In a medium: $F_{m} = \frac{F _{0}}{K}, ε = K ε_{0}$; K = 81 for water.
Superposition: $F_{1} = \sum_{j} F_{1 j}$; Vector sum.
Charge densities: $λ = \frac{d q}{d l}, σ = \frac{d q}{d A}, ρ = \frac{d q}{d V}$; For continuous distributions.

How to solve Coulomb's Law and the Superposition Principle questions

Most problems are a Coulomb calculation followed by vector addition.

1Convert charges to coulombs (μC = $1 0^{- 6}$ , nC = $1 0^{- 9}$ ) and distances to metres.
2For each pair acting on the chosen charge, find the magnitude $k \frac{∣ q _{1} q _{2} ∣}{r ^{2}}$ .
3Mark directions on a diagram: like charges repel, unlike attract.
4Add the forces as vectors: along a line with signs, in a plane by components or the parallelogram rule.
5For a medium, divide the vacuum result by K.
6For null points or equilibrium, equate magnitudes and choose the region (between or outside) by the signs.

Coulomb's Law and the Superposition Principle in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Effect of the medium

Suppose a sodium ion and a chloride ion (charges +e and −e) are $2.8 \times 1 0^{- 10}$ m apart, first in vacuum and then in water (K = 81).

Compare the attractive forces. Why does salt dissolve easily in water?

Try it first, then show the solutionSolution

Vacuum: $F = \frac{9 \times 1 0 ^{9} \times ( 1.6 \times 1 0 ^{- 19} ) ^{2}}{( 2.8 \times 1 0 ^{- 10} ) ^{2}} \approx 2.9 \times 1 0^{- 9}$ N.
Water: $\frac{2.9 \times 1 0 ^{- 9}}{81} \approx 3.6 \times 1 0^{- 11}$ N.

Answer: About $2.9 \times 1 0^{- 9}$ N in vacuum but only $3.6 \times 1 0^{- 11}$ N in water; the much weaker attraction lets the ions separate.

Memory hook: Water divides electric forces by 81.

Quantisation of charge

Imagine a dust particle in a photocopier that picks up a charge of $- 8.0 \times 1 0^{- 18}$ C.

How many extra electrons does it carry? Could it carry $- 8.8 \times 1 0^{- 19}$ C instead?

Try it first, then show the solutionSolution

$n = \frac{8.0 \times 1 0 ^{- 18}}{1.6 \times 1 0 ^{- 19}} = 50$ electrons.
$\frac{8.8 \times 1 0 ^{- 19}}{1.6 \times 1 0 ^{- 19}} = 5.5$ , not a whole number, so that charge is impossible.

Answer: 50 extra electrons; $- 8.8 \times 1 0^{- 19}$ C cannot occur.

Memory hook: Divide by e: a whole number means possible.

Tricks to remember Coulomb's Law and the Superposition Principle

Short memory aids. Say each one aloud once, then use it on a question the same day.

Like repels, unlike pulls
Draw the arrow first, then the number.
Use it when: Every superposition problem.
Nine, nine, eight-eight-five
$k = 9 \times 1 0^{9}$ ; $ε_{0} = 8.85 \times 1 0^{- 12}$ .
Use it when: Constants.
Null point near the small one
Like charges: between, nearer the smaller. Unlike charges: outside, beyond the smaller.
Use it when: Where is the force or field zero?
K divides
A medium divides force and field by K and multiplies capacitance by K.
Use it when: Dielectric constant questions.

Coulomb's Law and the Superposition Principle: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Charges in water · 1 mark · MCQBoard level

The relative permittivity of water is 81. Compared with vacuum, the force between two fixed charges at the same separation in water is:
(a) 81 times
(b) 9 times
(c) $\frac{1}{9}$
(d) $\frac{1}{81}$

Try it first, then show the solutionSolution

$F_{m} = \frac{F _{0}}{K}$ .

Answer: (d) $\frac{1}{81}$ .

Example 2: Coulomb force in vector form · 2 marks · Short answerBoard level

A charge of +2 nC is at the origin and −5 nC at the point (3 cm, 4 cm). Find the magnitude of the force between them and the force on the −5 nC charge in vector form.

Try it first, then show the solutionSolution

r = 5 cm: $F = \frac{9 \times 1 0 ^{9} \times 2 \times 1 0 ^{- 9} \times 5 \times 1 0 ^{- 9}}{( 0.05 ) ^{2}} = 3.6 \times 1 0^{- 5}$ N (attractive). (1)
The force on −5 nC points towards the origin, along $- \overset{r}{^} = - (0.6 \hat{i} + 0.8 \hat{j})$ : $F = - (2.16 \hat{i} + 2.88 \hat{j}) \times 1 0^{- 5}$ N. (1)

Answer: $3.6 \times 1 0^{- 5}$ N; $F = (- 2.16 \hat{i} - 2.88 \hat{j}) \times 1 0^{- 5}$ N.

Example 3: Charges on a square · 3 marks · Short answerHarder board level

Charges of +1 μC are placed at three corners A, B and D of a square ABCD of side 10 cm in air. Find the force on a +1 μC charge placed at the fourth corner C.

Try it first, then show the solutionSolution

B and D are 10 cm from C: each force $\frac{9 \times 1 0 ^{9} \times 1 0 ^{- 12}}{0.01} = 0.9$ N, along BC and DC, at right angles. Resultant $0.9 2 \approx 1.27$ N along AC (away from A). (1)
A is $102$ cm away: $F_{A} = \frac{9 \times 1 0 ^{- 3}}{0.02} = 0.45$ N, also along AC. (1)
Total $1.27 + 0.45 \approx 1.72$ N, along the diagonal AC, away from A. (1)

Answer: About 1.72 N along the diagonal, directed away from A.

Example 4: Null point for unlike charges · 3 marks · Short answerHarder board level

A charge of +1 μC is at the origin and −4 μC is at x = 30 cm. Where on the x-axis will a test charge feel zero force?

Try it first, then show the solutionSolution

Unlike charges: the point must lie outside, on the side of the smaller charge (x < 0). Let it be d to the left of the origin. (1)
$\frac{1}{d ^{2}} = \frac{4}{( d + 0.3 ) ^{2}}$ , so $d + 0.3 = 2 d$ . (1)
$d = 0.3$ m: the point is at x = −30 cm. (1)

Answer: At x = −30 cm (30 cm from the +1 μC charge, on the side away from −4 μC).

Example 5: Quantisation and conservation · 3 marks · Short answerHarder board level

A glass rod rubbed with silk acquires $+ 4.8 \times 1 0^{- 10}$ C. (a) How many electrons were transferred, and in which direction? (b) What is the charge on the silk? (c) Can a body ever carry $5.6 \times 1 0^{- 19}$ C?

Try it first, then show the solutionSolution

(a) $n = \frac{4.8 \times 1 0 ^{- 10}}{1.6 \times 1 0 ^{- 19}} = 3 \times 1 0^{9}$ electrons, from the glass to the silk. (1)
(b) By conservation, the silk carries $- 4.8 \times 1 0^{- 10}$ C. (1)
(c) $\frac{5.6}{1.6} = 3.5$ , not an integer, so no. (1)

Answer: (a) $3 \times 1 0^{9}$ , glass to silk; (b) $- 4.8 \times 1 0^{- 10}$ C; (c) no.

Example 6: Three charges in equilibrium · 4 marks · Case-basedHOTS / competency

Suppose charges of +2 μC and +8 μC are fixed 30 cm apart.
(i) Find the force between them. (1 mark)
(ii) Where should a third charge Q be placed on the line joining them so that Q feels no net force? (1 mark)
(iii) Find Q so that all three charges are in equilibrium. (2 marks)

Try it first, then show the solutionSolution

(i) $F = \frac{9 \times 1 0 ^{9} \times 2 \times 1 0 ^{- 6} \times 8 \times 1 0 ^{- 6}}{0.09} = 1.6$ N, repulsive.
(ii) Between them, x from +2 μC: $\frac{2}{x ^{2}} = \frac{8}{( 0.3 - x ) ^{2}}$ , so $0.3 - x = 2 x$ and x = 10 cm.
(iii) For +2 μC to be in equilibrium, the pull of Q must balance the push of +8 μC: $\frac{k ∣ Q ∣ ( 2 μ C )}{0. 1 ^{2}} = 1.6$ N, so $∣ Q ∣ = \frac{1.6 \times 0.01}{9 \times 1 0 ^{9} \times 2 \times 1 0 ^{- 6}} = 8.9 \times 1 0^{- 7}$ C.
Q must attract, so $Q = - \frac{8}{9} μ C \approx - 0.89 μ$ C (check: on +8 μC, $\frac{k ( 8/9 ) ( 8 )}{0. 2 ^{2}} \times 1 0^{- 12} = 1.6$ N, balancing).

Answer: (i) 1.6 N; (ii) 10 cm from +2 μC; (iii) $Q = - \frac{8}{9} μ$ C.

Where students lose marks in Coulomb's Law and the Superposition Principle

The slips examiners see most often, and what to write instead.

Mistake: Not knowing quantisation
The ISC 2024 examiners found candidates who said a non-integral multiple of e was possible, or gave no reason.
Do this: Divide by $1.6 \times 1 0^{- 19}$ C; if the result is not an integer, the charge is impossible.
Usually costs: 1 mark.
Mistake: Confusing permittivity with permeability or refractive index
Also in 2024: relative permittivity was linked to refractive index or permeability, with non-standard notation.
Do this: Learn the definition: $K = \frac{ε}{ε _{0}} = \frac{F _{v a c uu m}}{F _{m e d i u m}}$ .
Usually costs: 1 mark.
Mistake: Adding forces as scalars
Forces in different directions are added as numbers.
Wrong working
$0.9 + 0.9 + 0.45 = 2.25$ N for the square.
Right working
$0.9 2 + 0.45 \approx 1.72$ N.
Do this: Draw every force and add by components.
Usually costs: 1 to 2 marks.
Mistake: Wrong region for a null point
Looking between unlike charges, where both forces act the same way.
Do this: Like charges: between. Unlike charges: outside, near the smaller one.
Usually costs: 1 mark.

Exam tips

Write $F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}}$ before substituting; it earns the formula mark.
State "attractive" or "repulsive" with every force.
Keep three significant figures; e, k and ε₀ are given in the paper.
In vector form, say which force acts on which charge.

Practice questions on Coulomb's Law and the Superposition Principle

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
If both charges are doubled and their separation is halved, the Coulomb force becomes:
(a) 4 times
(b) 8 times
(c) 16 times
(d) unchanged
Hint
$F \propto \frac{q _{1} q _{2}}{r ^{2}}$ .
Answer
(c).
Q2 · 1 mark · Very short answerBoard level
The relative permittivity of glycerine is 42. Two charges repel with 2.1 N in vacuum. What force do they exert at the same separation in glycerine, and what does "relative permittivity 42" mean? (A variation on ISC 2024.)
Hint
Compare force or field with vacuum.
Answer
$\frac{2.1}{42} = 0.05$ N. The permittivity of glycerine is 42 times that of vacuum, so the force (or field) between two charges in it is $\frac{1}{42}$ of its value in vacuum at the same separation.
Q3 · 2 marks · Short answerHarder board level
Two charges exert a force F in air. One charge is halved, the distance is halved and the pair is placed in a medium of K = 2. Find the new force.
Hint
Multiply the three factors.
Answer
$F \times \frac{1}{2} \times 4 \times \frac{1}{2} = F$ : unchanged.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The force between two charges is not changed by bringing a third charge near them.
Reason (R): By the superposition principle, each pair of charges interacts independently of the others.
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
Net force changes; the pair force does not.
Answer
(a).
Q5 · 1 mark · Very short answerHarder board level
How many electrons must be removed from a neutral metal sphere to give it a charge of $+ 3.2 μ$ C?
Hint
$n = \frac{q}{e}$ .
Answer
$2 \times 1 0^{13}$ .
Q6 · 3 marks · Short answerHOTS / competency
Two point charges with a total of $+ 10 μ$ C repel each other with 5.4 N when 20 cm apart in air. Find the charges.
Hint
Find $q_{1} q_{2}$ , then solve a quadratic.
Answer
$q_{1} q_{2} = \frac{5.4 \times 0.04}{9 \times 1 0 ^{9}} = 2.4 \times 1 0^{- 11}$ ; with $q_{1} + q_{2} = 1 0^{- 5}$ : 6 μC and 4 μC.
Q7 · 4 marks · Case-basedHOTS / competency
Case study: Suppose two identical small balls, each of mass 1 g and carrying equal charge q, hang from the same point on silk threads 50 cm long and settle 6 cm apart. (g = 9.8 m s⁻²)
(i) Find q. (2 marks)
(ii) About how many electrons has each ball lost if q is positive? (1 mark)
(iii) If each charge leaked to half its value, what would the new separation be, using the small-angle approximation? (1 mark)
Hint
$tan θ = \frac{F}{m g}$ ; for small angles $r^{3} \propto q^{2}$ .
Answer
(i) $tan θ = \frac{0.03}{0.25 - 0.0009} \approx 0.0601$ ; $F = m g tan θ \approx 5.89 \times 1 0^{- 4}$ N; $q = r \frac{F}{k} \approx 1.5 \times 1 0^{- 8}$ C. (ii) $\approx 9.6 \times 1 0^{10}$ . (iii) $r = 6 \times (\frac{1}{4})^{1/3} \approx 3.8$ cm.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.State the quantisation of charge.Show answer
Any charge is an integral multiple of e: q = ne.
2.Value of $\frac{1}{4 π ε _{0}}$ ?Show answer
$9 \times 1 0^{9}$ N m² C⁻².
3.Define dielectric constant.Show answer
The ratio of the force between two charges in vacuum to that in the medium at the same separation (= $\frac{ε}{ε _{0}}$ ).
4.Force between 1 μC and 1 μC at 1 m in vacuum?Show answer
$9 \times 1 0^{- 3}$ N.
5.State the superposition principle.Show answer
The net force on a charge is the vector sum of the forces due to each other charge acting alone.

Coulomb's Law and the Superposition Principle: frequently asked questions

Does Coulomb's law hold for any charged bodies?

It holds exactly for point charges. For spheres with uniform charge, it also works using the distance between centres, because outside a uniform spherical charge the field is as if the charge were at the centre.

Why is K always greater than 1?

A medium becomes polarised and its induced charges partly cancel the field of the original charges, so the force is weaker than in vacuum. Vacuum has K = 1 and air very nearly 1.

Is the vector form with position vectors needed?

The ISC 2027 syllabus asks for the vector form of Coulomb's law, but says the position coordinates \vec{r}_1, \vec{r}_2 are not necessary.

How is charge conserved in pair production?

A photon (no charge) creates an electron (−e) and a positron (+e), so the total charge stays zero. Conservation holds in every known process.

How was this examined recently?

In 2024 a 2-mark question asked about the relative permittivity of water and whether a given charge was possible (quantisation).