IB DP Physics: Electric charge, Coulomb's law, electric and magnetic fields (DP Physics)

This guide covers the standard level and higher level understandings of sub-topic D.2, Electric and magnetic fields, from the IB Physics guide (first assessment 2025). The higher level potential content in D.2 is taught in a separate unit and is not repeated here. Use the revision notes for recall and the practice questions to test yourself. Find weak spots first with the free 10-minute diagnostics.

Related pages: the course hub, the printable checklist and electric current, resistance and circuits.

What this unit covers

Understanding in D.2	What you must be able to do	Tier
Forces between charges	Give the direction of the force for like and unlike charges	SL and HL
Coulomb’s law	Use F = k q₁q₂/r² with k = 1/(4πε₀), for point charges	SL and HL
Conservation of charge	Apply it when charged bodies touch or react	SL and HL
Millikan’s experiment	Explain why it is evidence that charge is quantized	SL and HL
Charging	Describe friction, induction and contact, and the role of earthing	SL and HL
Electric field strength	Use E = F/q	SL and HL
Electric field lines	Sketch and interpret them; link line density to strength	SL and HL
Uniform field	Use E = V/d between parallel plates	SL and HL
Magnetic field lines	Sketch and interpret patterns for a bar magnet, straight wire, circular coil and air-core solenoid; find direction from the current	SL and HL

Forces between charges

There are two types of electric charge, called positive and negative. Like charges repel and unlike charges attract. The two bodies in a pair exert forces of equal size on each other, in opposite directions, as Newton’s third law requires.

Charge is measured in coulomb (C). Everyday charged objects carry nanocoulombs (10⁻⁹ C) to microcoulombs (10⁻⁶ C).

Coulomb’s law

For charged bodies small enough to treat as point charges, the magnitude of the force is

F = k q₁q₂ / r², with k = 1/(4πε₀)

Here r is the distance between the centres, and the force acts along the line joining them. In vacuum (and, to a good approximation, in air) k ≈ 8.99 × 10⁹ N m² C⁻², which corresponds to ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹. The guide expects you to meet a range of permittivity values, so if a question supplies the permittivity of another medium, use that value in place of ε₀.

Put charge magnitudes into the formula and decide attraction or repulsion from the signs. Doubling r divides F by 4.

Worked example 1: size and direction

A small sphere carries +3.6 nC and another carries −8.0 nC. Their centres are 2.5 cm apart. Find the force on each.

1. Convert: q₁ = 3.6 × 10⁻⁹ C, q₂ = 8.0 × 10⁻⁹ C, r = 0.025 m.
2. F = (8.99 × 10⁹ × 3.6 × 10⁻⁹ × 8.0 × 10⁻⁹) / 0.025²
3. F = 2.589 × 10⁻⁷ / 6.25 × 10⁻⁴ = 4.14 × 10⁻⁴ N

The charges have opposite signs, so each sphere is pulled towards the other with this force.

Worked example 2: scaling by proportion

Two point charges 6.0 cm apart exert forces of 18 mN on each other. What is the force when they are 2.0 cm apart?

The distance is divided by 3, so the force is multiplied by 3² = 9. F = 18 × 9 = 162 mN.

Electric and gravitational forces

Both laws are inverse square laws, but gravity only attracts. For two protons the ratio does not depend on r, because r² cancels:

F(electric)/F(gravity) = k e² / (G m²) = (8.99 × 10⁹ × (1.60 × 10⁻¹⁹)²) / (6.67 × 10⁻¹¹ × (1.67 × 10⁻²⁷)²) ≈ 1.24 × 10³⁶

Large bodies are almost neutral overall, so gravity dominates for them.

Conservation of charge

The total charge of an isolated system stays constant. Charge is neither created nor destroyed. It moves between bodies, and positive and negative charge can cancel.

Worked example 3: contact between conducting spheres

Two identical metal spheres carry +8.0 µC and −2.0 µC. They touch and are then separated. Find the final charge on each and the force when their centres are 0.30 m apart.

1. Total charge = +8.0 + (−2.0) = +6.0 µC.
2. Identical conductors share the total equally, so each has +3.0 µC.
3. F = (8.99 × 10⁹ × (3.0 × 10⁻⁶)²) / 0.30² = 0.899 N, repulsive.

Millikan’s experiment and quantization

Millikan observed tiny charged oil drops between two horizontal metal plates. By adjusting the voltage he could hold a drop stationary, so the upward electric force equalled the weight:

qE = mg, so q = mg/E, with E = V/d

Repeating this for many drops, he found that every measured charge was a whole-number multiple of one smallest value, the elementary charge e = 1.60 × 10⁻¹⁹ C. This is the evidence that charge is quantized: it comes in discrete units and not as a continuous range of values.

Worked example 4: a balanced drop

An oil drop of mass 1.96 × 10⁻¹⁵ kg is held stationary between plates 1.5 cm apart with 600 V across them. Find its charge in terms of e.

1. E = V/d = 600 / 0.015 = 4.0 × 10⁴ V m⁻¹
2. q = mg/E = (1.96 × 10⁻¹⁵ × 9.81) / 4.0 × 10⁴ = 4.81 × 10⁻¹⁹ C
3. q/e = 4.81 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ ≈ 3.0

The drop carries three elementary charges.

Worked example 5: interpreting data

Four drops have charges (in 10⁻¹⁹ C) of 3.2, 8.0, 4.8 and 11.2. Divide each by 1.6: you get 2, 5, 3 and 7. All are whole numbers, so the data are consistent with a smallest unit of 1.6 × 10⁻¹⁹ C. The largest value that divides every result into a whole number gives the elementary charge.

Charging bodies

In solids only electrons move: a body becomes negative by gaining them and positive by losing them.

• Friction. Rubbing two insulators together transfers electrons from one to the other. One becomes positive and the other negative, with equal and opposite charges, so the total is conserved.
• Contact. A charged body touches a conductor and shares charge with it, leaving charge of the same sign.
• Induction. A charged body is brought near, but does not touch, an isolated conductor. The free electrons redistribute. To leave a net charge, you connect the conductor to the ground (earth), then remove the connection, then remove the charged body.

Earthing (grounding) means connecting to the Earth, which can supply or accept electrons in huge numbers.

Worked example 6: positively charging a sphere by induction

You have a negatively charged rod and a neutral metal sphere on an insulating stand.

1. Bring the rod near the sphere without touching. Electrons in the sphere are repelled to the far side. The near side becomes positive.
2. Touch the far side with a wire to earth. Electrons flow from the sphere to the earth.
3. Remove the earth connection while the rod is still there.
4. Take the rod away. The remaining positive charge spreads over the sphere.

Removing the rod before the earth connection lets the electrons redistribute and leaves the sphere neutral, so the order matters.

Electric field strength and field lines

Electric field strength at a point is defined as the force each coulomb of a small positive test charge would feel there:

E = F/q

Its unit is N C⁻¹ (equivalent to V m⁻¹). It is a vector. A positive charge is pushed along E and a negative charge against it.

For a point charge Q, combining E = F/q with Coulomb’s law gives E = kQ/r², pointing outwards for positive Q and inwards for negative Q.

Worked example 7: field from a point charge and from a pair

(a) A test charge of +1.8 nC feels 7.2 µN. E = F/q = 7.2 × 10⁻⁶ / 1.8 × 10⁻⁹ = 4.0 × 10³ N C⁻¹.

(b) For Q = +7.5 nC at r = 0.30 m: E = (8.99 × 10⁹ × 7.5 × 10⁻⁹) / 0.30² = 7.49 × 10² N C⁻¹, pointing outwards.

(c) Charges of +3.0 nC and −3.0 nC are 0.50 m apart. At the midpoint each is 0.25 m away and gives E = (8.99 × 10⁹ × 3.0 × 10⁻⁹) / 0.25² = 4.32 × 10² N C⁻¹. Both fields point from the positive charge towards the negative one, so they add: E = 8.63 × 10² N C⁻¹.

Rules for field lines

• Lines leave positive charges and arrive at negative ones (or run to or from infinity).
• The arrow gives the direction of the force on a positive test charge.
• Lines never cross: a crossing would give one point two field directions.
• Where lines are closer together, the field is stronger.
• Lines meet the surface of a conductor at right angles.

Patterns to sketch: a single point charge (radial, evenly spaced lines); two point charges (lines curve from positive to negative; two equal like charges push their lines apart, with zero field midway between them); a charged conducting sphere.

For a charged conducting sphere, the charge sits on the surface. The field inside is zero, and outside it is radial, like the field of a point charge at the centre.

Uniform field between parallel plates

For two oppositely charged parallel plates with potential difference V across a separation d, the field between them is uniform:

E = V/d

Lines are straight, parallel and evenly spaced, from the positive plate to the negative plate. Near the edges they bulge outwards, so the field is weaker and not uniform there.

Worked example 8: force on an electron

Plates 5.0 mm apart have 1.35 kV across them. Find E, the force on an electron, and its acceleration.

1. E = 1350 / 5.0 × 10⁻³ = 2.7 × 10⁵ V m⁻¹
2. F = eE = 1.60 × 10⁻¹⁹ × 2.7 × 10⁵ = 4.3 × 10⁻¹⁴ N, towards the positive plate
3. a = F/m = 4.32 × 10⁻¹⁴ / 9.11 × 10⁻³¹ = 4.7 × 10¹⁶ m s⁻²

The weight of an electron is negligible by comparison.

Magnetic field lines

A north pole placed in a magnetic field is pushed along the field lines. They form closed loops, running outside a magnet from north to south. Closer lines mean a stronger field. Directions below use conventional current.

Source	Pattern
Bar magnet	Loops out of the north pole and into the south pole; strongest, with lines closest, near the poles
Straight wire	Concentric circles around the wire; lines spread out further away, so the field weakens
Circular coil	Circles around each part of the wire; nearly straight through the centre of the coil
Air-core solenoid	Straight, parallel and close together inside, like a bar magnet outside

Direction from the current

Use the right-hand grip rule for a straight wire: point your thumb along the conventional current, and your fingers then wrap the way the field circulates.

For a coil or solenoid, curl your fingers along the current round the loops. Your thumb points along the field inside, towards the north end.

Worked example 9: a vertical wire carries current upwards. Seen from above, the field circles anticlockwise. At a point east of the wire, the field points north. For a solenoid, if the current looks anticlockwise from one end, that end is the north pole.

For a current into the page, the circles are clockwise.

Common errors

• Using r from the surface rather than the centre in Coulomb’s law.
• Not converting nC, µC or cm to SI units.
• Adding field vectors without checking their directions.
• Field lines that intersect or lack arrows.
• Using electron flow, not conventional current, in the grip rule.

Official syllabus

This page follows the IB Physics guide, first assessment 2025, published by the International Baccalaureate Organization, sub-topic D.2.