Magnetic flux density and magnetic flux

Magnetic flux density, B, measures how strong a magnetic field is. It is defined by the force on a current-carrying wire at right angles to the field: B = F ÷ (Il), so the familiar equation is F = BIl. Its unit is the tesla, T. A field of 1 T pushes with a force of 1 N on each metre of wire carrying 1 A. Magnetic flux, Φ, is the flux density multiplied by the area it passes through at right angles: Φ = BA, measured in webers, Wb.

What is magnetic flux density?

Picture a magnetic field as field lines. Where the lines are packed close together, the field is strong; where they spread out, it is weak. Magnetic flux density is a measure of how closely packed the lines are: the amount of flux through each square metre. That is where the name comes from, and why B is sometimes simply called the magnetic field strength.

You can see this in the Field Lines Simulator: place a magnet and watch the lines crowd together near the poles, where the flux density is greatest, and spread apart further away.

The magnetic flux density formula

B is defined using the motor effect. A wire carrying current I, with length l inside a field at right angles to it, feels a force F:

AQA's A-level specification links the definition of the tesla to F = BIl, for a field perpendicular to the current, and its GCSE specification gives the same equation for Higher tier, with B in tesla. The direction of the force comes from Fleming's left-hand rule.

The tesla and the magnetic flux density symbol

The symbol is B and the unit is the tesla, T. Rearranging B = F ÷ (Il) shows that 1 T = 1 N A⁻¹ m⁻¹. One tesla is a strong field. The European Commission's scientific committees put the Earth's magnetic field at its surface at about 50 µT, which is 50 × 10⁻⁶ T, and that is why field strengths are often quoted in millitesla (mT) or microtesla (µT).

When the wire is at an angle

Only the part of the field at right angles to the wire pushes on it. If the wire makes an angle θ with the field, the force is:

At θ = 90° this is the full BIl. At θ = 0, when the wire runs along the field lines, there is no force at all.

Force on a moving charge: F = BQv

A current is moving charge, so a single charged particle moving through a field also feels a force:

where Q is the charge in coulombs and v the speed in m/s, with the velocity perpendicular to the field. Because this force always acts sideways to the motion, it changes the particle's direction but not its speed, and the particle moves in a circle. AQA's A-level specification uses this for devices such as the cyclotron.

Magnetic flux

Magnetic flux, Φ (the Greek letter phi), is the total amount of field passing through an area. When the field is at right angles to the area:

Its unit is the weber, Wb, where 1 Wb = 1 T m². So magnetic flux density really is flux per unit area: B = Φ ÷ A, which is why the tesla can also be written Wb m⁻².

Flux at an angle and flux linkage

If the field is not at right angles to the area, only the component of B along the normal to the area counts. With θ the angle between the field and the normal:

For a coil of N turns, each turn links the flux, so the flux linkage is NΦ = BAN cos θ. AQA's data booklet gives both Φ = BA and NΦ = BAN cos θ.

Changing flux and induced emf

When the flux linkage through a coil changes, an emf is induced. Faraday's law, from AQA's booklet, gives its size:

So moving a magnet faster, using a stronger magnet or adding more turns all give a bigger emf. This is how generators and transformers work.

Magnetic flux density worked examples

Example 1: finding B

A 5.0 cm length of wire carries 2.0 A at right angles to a magnetic field and feels a force of 0.012 N. Find the magnetic flux density.

Convert first: 5.0 cm = 0.050 m.

Example 2: finding the force

A wire 8.0 cm long carries 3.0 A at right angles to a 0.25 T field. Find the force on it.

If the same wire were at 30° to the field instead, the force would be 0.060 × sin 30° = 0.030 N.

Example 3: a charged particle

A proton, charge 1.60 × 10⁻¹⁹ C, moves at 2.0 × 10⁶ m/s at right angles to a 0.50 T field. Find the force on it.

Example 4: flux and flux linkage

A coil of 150 turns has an area of 5.0 cm² and sits at right angles to a 0.40 T field. Find the flux through one turn and the flux linkage. What happens if the coil turns so its normal is at 60° to the field?

Convert the area: 5.0 cm² = 5.0 × 10⁻⁴ m².

The flux linkage is NΦ = 150 × 2.0 × 10⁻⁴ = 0.030 Wb. At 60°, multiply by cos 60° = 0.5, giving 0.015 Wb.

Example 5: induced emf

The coil in Example 4 is pulled out of the field, so its flux linkage falls from 0.030 Wb to zero in 0.12 s. Find the average induced emf.

Example 6: the top-pan balance practical

AQA's A-level required practical investigates how the force on a wire varies with flux density, current and length, using a top-pan balance. A magnet sits on the balance and a fixed wire runs between its poles. When 3.0 A flows through 5.0 cm of wire in the field, the balance reading changes by 2.45 g. Find B.

The change in reading is the force between the wire and the magnet: F = 0.00245 kg × 9.81 N/kg = 0.0240 N.

Plotting the force against the current for several currents gives a straight line through the origin with gradient Bl, which is more reliable than a single reading. The Practical Data Lab helps you fit the line and find its gradient.

Try it

Field Lines Simulator

Place charges and magnets and see their electric and magnetic fields

Open the simulatorGo to its page

Magnetic flux density vs magnetic flux

Common mistakes

• Length in cm. Convert to metres: 5.0 cm is 0.050 m.
• Area in cm². 1 cm² = 1 × 10⁻⁴ m², not 10⁻² m².
• mT and µT. 30 mT is 0.030 T; 50 µT is 5.0 × 10⁻⁵ T.
• The angle in Φ = BA cos θ. θ is measured between the field and the normal to the coil. If you are given the angle between the field and the plane of the coil, use sin instead.
• Mixing up B and Φ. B is per square metre; Φ is the total through an area.

Which equations are given?

AQA's GCSE Physics equations sheet for June 2026 gives F = BIl for Higher tier: force on a conductor (at right angles to a magnetic field) carrying a current = magnetic flux density × current × length. AQA's A-level data booklet gives F = BIl, F = BQv, Φ = BA, NΦ = BAN cos θ and ε = N ΔΦ ÷ Δt under magnetic fields. In AQA's A-level, magnetic fields are taught in the second year. This was checked in November 2026; other boards publish their own sheets, so check yours.

Practise fields in a short timed paper

A-level Physics: Year 2 and GCSE Physics Higher each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so magnetic fields turn up regularly but not in every paper.

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