Magnetic flux

CIE A-Level Physics· 6 min read

1. Definition of Magnetic Flux★★☆☆☆⏱ 15 min

📘 Definition

Magnetic Flux

\Phi

The product of the component of magnetic flux density perpendicular to an area and the magnitude of the area itself. It measures how many magnetic field lines pass through the area.

Example:

A flat coil aligned parallel to a uniform B field has zero flux, as no field lines pass through the coil.

\Phi = BA \cos\theta

In this formula, $θ$ is the angle between the magnetic field vector and the normal (perpendicular) to the plane of the area, not the angle between the field and the plane itself. This is the most commonly tested convention in exams.

📐 Worked Example

A rectangular coil of area $2.0 \times 1 0^{- 4}$ m² is placed in a uniform magnetic field of flux density 0.15 T. The normal to the coil makes an angle of 30° with the magnetic field. Calculate the magnetic flux through the coil.

1
1. Recall the flux formula:
$\Phi = BA \cos\theta$
2
1. Substitute the given values:
$\Phi = (0.15)(2.0 \times 10^{-4})(\cos 30^\circ)$
3
1. Calculate the final value, $cos 3 0^{\circ} \approx 0.866$ :
$\Phi = 2.6 \times 10^{-5} \text{ Wb}$

Exam tip:

If the question gives the angle between the field and the plane of the coil, subtract this angle from 90° to get $θ$ for the formula.

2. Magnetic Flux Linkage★★☆☆☆⏱ 15 min

📘 Definition

Magnetic Flux Linkage

N\Phi

The total flux linked with a coil of N identical turns, equal to the product of the number of turns and the magnetic flux through one turn of the coil.

Example:

A 100-turn coil with $1 \times 1 0^{- 5}$ Wb flux per turn has a total flux linkage of $1 \times 1 0^{- 3}$ Wb turns.

Flux linkage is the quantity that directly appears in Faraday's Law of electromagnetic induction. It accounts for the fact that each turn of the coil cuts magnetic field lines independently, so more turns produce a larger induced emf for the same rate of change of flux per turn.

📐 Worked Example

A 50-turn circular coil of radius 1.5 cm is placed perpendicular to a uniform magnetic field of 0.20 T. Calculate the total flux linkage of the coil.

1
1. Calculate the area of the coil:
$A = \pi r^2 = \pi (0.015)^2 = 7.07 \times 10^{-4} \text{ m}^2$
2
1. Calculate flux per turn, $θ = 0^{\circ}$ so $cos 0^{\circ} = 1$ :
$\Phi = BA = (0.20)(7.07 \times 10^{-4}) = 1.41 \times 10^{-4} \text{ Wb}$
3
1. Multiply by number of turns for flux linkage:
$N\Phi = 50 \times 1.41 \times 10^{-4} = 7.1 \times 10^{-3} \text{ Wb turns}$

Exam tip:

Always write the unit of flux linkage as Wb turns, even though some mark schemes accept Wb. This avoids losing unnecessary marks.

3. Change in Magnetic Flux★★★☆☆⏱ 20 min

Most exam questions on magnetic flux ask for the change in flux or flux linkage when a coil rotates, moves, or the magnetic field strength changes. Change in flux is calculated as $ΔΦ = Φ_{f ina l} - Φ_{ini t ia l}$ , and change in flux linkage is $Δ (N Φ) = N ΔΦ$ .

📐 Worked Example

A flat 100-turn coil is initially placed with its plane parallel to a uniform 0.10 T magnetic field. It is rotated 90° so its plane is now perpendicular to the field. The coil area is $4.0 \times 1 0^{- 3}$ m². Calculate the change in flux linkage.

1
1. Find initial flux per turn: when plane is parallel, normal is perpendicular to B, so $θ_{ini t ia l} = 9 0^{\circ}$ :
$\Phi_{initial} = BA\cos 90^\circ = 0$
2
1. Find final flux per turn: when plane is perpendicular, normal is parallel to B, so $θ_{f ina l} = 0^{\circ}$ :
$\Phi_{final} = BA\cos 0^\circ = (0.10)(4.0 \times 10^{-3}) = 4.0 \times 10^{-4} \text{ Wb}$
3
1. Calculate change in flux linkage:
$\Delta(N\Phi) = N(\Phi_{final} - \Phi_{initial}) = 100(4.0 \times 10^{-4} - 0) = 4.0 \times 10^{-2} \text{ Wb turns}$

✓ Quick check

Test your understanding of angle conventions:

The plane of a coil makes an angle of 20° with a uniform magnetic field. What is $θ$ (for the flux formula)?
- 20°
- 70°
- 90°
- 0°
Reveal answer
1 —
Correct: $θ$ is the angle between the normal (perpendicular to the plane) and the field, so $9 0^{\circ} - 2 0^{\circ} = 7 0^{\circ}$ .

4. Common Pitfalls

Wrong move:

Taking $θ$ as the angle between the plane of the coil and the magnetic field

Why:

The flux formula is defined using the angle between the magnetic field and the normal to the plane, not the plane itself

Correct move:

If given the angle to the plane, subtract it from 90° to get $θ$ before substituting into $Φ = B A cos θ$

Wrong move:

Forgetting to multiply by the number of turns when calculating flux linkage

Why:

Flux linkage describes total flux across all turns of a coil, not just flux through one turn

Correct move:

Always multiply flux per turn by $N$ when asked for flux linkage for Faraday's law calculations

Wrong move:

Getting a change of zero when a coil flips 180°

Why:

Flips reverse the sign of flux: flux changes from $+ B A$ to $- B A$ , not from $B A$ to $B A$

Correct move:

Calculate $ΔΦ = - B A - B A = - 2 B A$ , so magnitude of change is $2 B A$ for a 180° flip

Wrong move:

Claiming flux is maximum when the plane of the coil is parallel to the magnetic field

Why:

Maximum flux occurs when the maximum number of field lines pass through the coil area

Correct move:

Flux is maximum when the plane is perpendicular to the magnetic field, and zero when it is parallel

5. Quick Reference Cheatsheet

Quantity	Symbol	Formula/Rule	Unit
Magnetic flux	$Φ$	$B A cos θ$ , $θ$ = angle to normal	Weber (Wb)
Magnetic flux linkage	$N Φ$	$N \times Φ$ (per turn)	Weber turns (Wb turns)
Change in flux	$ΔΦ$	$Φ_{f ina l} - Φ_{ini t ia l}$	Weber (Wb)
Change in flux linkage	$Δ (N Φ)$	$N \times ΔΦ$	Weber turns (Wb turns)
Maximum flux	$Φ_{ma x}$	$B A$ (normal parallel to B)	Weber (Wb)
Zero flux	$Φ = 0$	Normal perpendicular to B	Weber (Wb)

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 1
Flux calculation with angle
2022 · 2
Change in flux linkage for rotation
2021 · 4
Flux for moving coil in B field

What's Next

Magnetic flux is the foundational quantity for the entire topic of electromagnetic induction, which contributes 10-15% of total marks across CIE A-Level Physics papers 2 and 4. Mastery of flux conventions and calculations is essential to avoid losing marks on Faraday's law and induced emf problems, which are frequent high-mark questions. Next, you will build on this knowledge to learn how changing flux produces induced emf, and apply Faraday's and Lenz's laws to solve a wide range of exam problems. This concept also underpins later topics including alternating current, transformers, and electromagnetic generators.