Unit 13 · Topic 13.1

13.1 Magnetic Flux

Magnetic flux measures how much magnetic field passes through a surface. It's the quantity whose change drives everything else in this unit, so you need to calculate it confidently for uniform fields, tilted surfaces and fields that vary across a surface.

Key terms

magnetic flux
area vector
dot product
weber
surface integral

The area vector

To describe which way a flat surface faces, use an area vector $\vec{A}$ . Its size is the surface's area, and its direction is perpendicular (normal) to the surface. A loop lying flat on a table has an area vector pointing straight up or straight down.

For a closed surface, the area vector always points outward. For a loop in a circuit, you choose one of the two directions, and that choice sets the sign convention for the flux and for the emf in 13.2.

Flux in a uniform field

Magnetic flux is the dot product of the field and the area vector: $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$ , where θ is the angle between $\vec{B}$ and the area vector, which is the normal to the surface.

Its unit is the weber: 1 Wb = 1 T·m². Flux is a scalar, but it has a sign. It's positive when the field goes through the surface in the same direction as $\vec{A}$ and negative when it goes the other way.

A good way to picture flux is the number of field lines passing through the surface. Field lines that skim along the surface don't pass through it at all.

Orientation of the surface	θ (field to normal)	Flux
Field passes straight through the face	0°	BA (maximum)
Surface tilted	θ	BA cos θ
Field runs along the surface	90°	0
Surface flipped over	180°	−BA

Flux when the field varies

If B changes across the surface, split the surface into small strips where B is nearly constant, find each strip's flux, and add: $\Phi_B = \displaystyle\int \vec{B} \cdot d\vec{A}$ .

The classic case is a rectangular loop next to a long straight wire, in the same plane. The wire's field, $B = \dfrac{\mu_0 I}{2\pi r}$ , changes with distance r but is the same along any strip parallel to the wire. A strip of length $\ell$ and width dr has area $\ell\,dr$ , so $\Phi_B = \displaystyle\int_a^b \frac{\mu_0 I}{2\pi r}\ell\,dr = \frac{\mu_0 I\ell}{2\pi}\ln\frac{b}{a}$ , where a and b are the distances from the wire to the loop's near and far sides.

Closed surfaces

Through any closed surface, the net magnetic flux is zero, because field lines that go in must come out (Gauss's law for magnetism, 12.1). Flux questions in this unit are almost always about an open surface bounded by a loop of wire.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
A tilted loop (classic trap)
A rectangular loop measures 0.20 m by 0.30 m. A uniform 0.40 T magnetic field makes a 30° angle with the plane of the loop. Find the magnetic flux through the loop.
Show the solutionHide the solution
1. Step 1: The angle in $\Phi_B = BA\cos\theta$ is between the field and the normal to the loop. The field makes 30° with the plane, so it makes 90° − 30° = 60° with the normal.
2. Step 2: Area: (0.20)(0.30) = 0.060 m². Flux: $\Phi_B = (0.40)(0.060)\cos 60^\circ = 0.012\text{ Wb}$ .
3. Step 3: The trap is using 30° directly, which gives 0.021 Wb. Always check whether an angle is measured from the plane or from the normal.
Answer: 0.012 Wb
Example 2Calculator allowed
Flux from a long straight wire
A long straight wire carries 20 A. A rectangular loop lies in the same plane, with its 0.50 m sides parallel to the wire. Its near side is 0.10 m from the wire and its far side is 0.30 m away. Find the magnetic flux through the loop.
Show the solutionHide the solution
1. Step 1: B varies with distance from the wire, so integrate over strips parallel to the wire, each of area $(0.50)\,dr$ .
2. Step 2: $\Phi_B = \displaystyle\int_{0.10}^{0.30} \frac{\mu_0 I}{2\pi r}(0.50)\,dr = \frac{\mu_0 I(0.50)}{2\pi}\ln\frac{0.30}{0.10}$ .
3. Step 3: $\Phi_B = \dfrac{(4\pi \times 10^{-7})(20)(0.50)}{2\pi}\ln 3 = (2.0 \times 10^{-6})(1.0986) \approx 2.2 \times 10^{-6}\text{ Wb}$ .
4. Step 4: Using the field at the middle of the loop times the area would give a different, wrong answer, because B isn't linear in r.
Answer: About 2.2 × 10⁻⁶ Wb

Common mistakes

Measuring θ from the surface instead of from its normal.
Multiplying B by the area when B varies across the loop. Set up an integral over strips where B is constant.
Thinking flux is zero just because the field is weak. It's zero when no field passes through the surface, such as when the field runs parallel to the loop's plane.
Forgetting that flux has a sign set by your chosen area vector.

On the exam

Expect a loop near a long wire and a request to derive the flux as an integral. Show the strip, its area, the field at the strip and the limits.
Graph questions ask for flux against time as a loop moves into, through and out of a field region. Flux rises while the loop enters, stays constant while it's fully inside a uniform field, and falls as it leaves.

Connected topics

Videos

Topic 13.1 - Magnetic Flux
Lessons With LondotWatch on YouTube (opens in a new tab)
Magnetic Flux
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Dan Fullerton (APlusPhysics)Watch on YouTube (opens in a new tab)
Magnetic Flux | Physics with Professor Matt Anderson | M24-01
Physics with Professor Matt AndersonWatch on YouTube (opens in a new tab)
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The Organic Chemistry TutorWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 13.1 Magnetic Flux. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A square loop with sides 20 cm long is in a uniform 0.50 T magnetic field. The line perpendicular to the loop’s plane makes an angle of 60° with the field. What is the magnetic flux through the loop?

Question 2 of 4Calculator allowed

A flat loop of area A is in a uniform magnetic field B. The plane of the loop makes a 30° angle with the field lines. What is the magnitude of the flux through the loop?

Question 3 of 4Calculator allowed

A rectangular loop lies in the xy-plane from x = 0 to x = L, with width w along y. A magnetic field perpendicular to the plane has magnitude $B = kx$ , where k is a constant. What is the magnetic flux through the loop?

Question 4 of 4Calculator allowed

A long straight wire carries 20 A. A rectangular loop in the same plane has its 30 cm sides parallel to the wire; its near side is 5.0 cm from the wire and its far side is 15 cm from the wire. What is the magnetic flux through the loop?

0 of 4 answered