Formula of Magnetic Flux: Φ = BA cos θ, Weber and Examples

The formula of magnetic flux is $\Phi_B = BA\cos\theta$ : field strength B times area A times cos θ, where θ tells you how far the field leans away from the line drawn perpendicular to the area. It measures how much magnetic field threads through a loop, and its SI unit is the weber (Wb), equal to one tesla square metre.

Flux matters because a change in it is what drives electromagnetic induction. Move a magnet toward a coil, spin a coil in a generator, or switch a nearby current on, and the flux changes. Faraday's law then gives the induced emf, $\varepsilon = -N\frac{d\Phi}{dt}$ . The field B itself is covered on the magnetic flux density formula page, and induction topics sit in the electricity and magnetism section.

Quick facts

Formula
$\Phi_B = BA\cos\theta = \vec{B} \cdot \vec{A}$
SI unit
weber (Wb) = T m² = V s
Dimensional formula
[M¹ L² T⁻² I⁻¹]
Type
Scalar, can be positive, negative or zero
Flux linkage
$N\Phi$ for a coil of N turns
Largest value
$BA$ , when the field is perpendicular to the loop's plane
CGS unit
maxwell; 1 Wb = 10⁸ maxwell

What each part of the formula of magnetic flux means

Inputs to Φ = BA cos θ
Symbol	Quantity	SI unit
$\Phi_B$	Magnetic flux	Wb
$B$	Magnetic field strength (flux density)	T
$A$	Area enclosed by the loop	m²
$\theta$	Angle between $\vec{B}$ and the area vector (the normal)	degree or radian

The area vector points straight out of the loop's face. When the field runs along that normal (θ = 0°), cos θ = 1 and the flux is the full BA. When the field lies in the plane of the loop (θ = 90°), no field lines pass through and Φ = 0. For a field that varies across the surface, the general form is $\Phi_B = \int \vec{B} \cdot d\vec{A}$ .

Getting the angle right

Most lost marks on flux come from the angle. Questions often give the angle between the field and the plane of the coil. The formula needs the angle with the normal, which is 90° minus the given value.

Read whether the angle is measured from the plane or from the normal.
If from the plane, subtract it from 90°.
Convert the area to m² (1 cm² = 10⁻⁴ m²).
Multiply B, A and cos θ; multiply by N if flux linkage is asked.

The calculator above expects θ measured from the normal, in degrees, and can also solve back for B, A or the angle.

Worked magnetic flux examples

Example 1: a square loop

A square loop of side 10 cm sits in a uniform field of 0.40 T. The field makes 60° with the normal to the loop. Find the flux.

A = (0.10)² = 0.010 m².

$\Phi = (0.40)(0.010)\cos 60^\circ = 0.0040 \times 0.5 = 2.0 \times 10^{-3}\ \text{Wb}$

Example 2: a coil whose plane is tilted (exam style)

A rectangular coil, 20 cm by 15 cm, with 50 turns, lies in a 0.25 T field. The plane of the coil makes 30° with the field. Find the flux through one turn and the flux linkage.

Area: 0.20 × 0.15 = 0.030 m². Angle with the normal: 90° − 30° = 60°.

$\Phi = (0.25)(0.030)\cos 60^\circ = 3.75 \times 10^{-3}\ \text{Wb}$

Flux linkage: $N\Phi = 50 \times 3.75 \times 10^{-3} = 0.19\ \text{Wb}$ (0.1875 Wb before rounding).

Example 3: from flux to induced emf

The coil in Example 2 is pulled clear of the field in 0.050 s. Estimate the average emf it develops.

$\varepsilon = N\frac{\Delta\Phi}{\Delta t} = \frac{50 \times 3.75 \times 10^{-3}}{0.050} = 3.75\ \text{V} \approx 3.8\ \text{V}$

Weber and its dimensions

One weber is the flux of a one tesla field through one square metre at right angles. Since the tesla is kg/(A s²), the weber is kg m²/(A s²). Faraday's law gives another reading: 1 Wb = 1 V s, because a flux change of one weber each second induces one volt.

$[\Phi] = [B][A] = [M^1 T^{-2} I^{-1}][L^2] = [M^1 L^2 T^{-2} I^{-1}]$

The derivation is set out in more detail on the dimensional formula of magnetic flux page.

Flux, inductance and the wider chapter

Formulas that build on Φ
Idea	Relation
Faraday's law	$\varepsilon = -N\frac{d\Phi}{dt}$
Self inductance	$N\Phi = LI$
Mutual inductance	$N_2\Phi_2 = MI_1$
Rotating coil (generator)	$\Phi = NBA\cos\omega t$ , so $\varepsilon = NBA\omega\sin\omega t$

The minus sign in Faraday's law is Lenz's law: the induced current opposes the change in flux. Follow these on the self inductance, mutual inductance and electromotive force pages. The electric version of the same idea is on the electric flux formula page.

Common mistakes with magnetic flux

Plane angle used directly. Change it to the angle with the normal first.
Confusing B and Φ. B is flux per square metre, in tesla; Φ is total flux, in weber.
Forgetting N. Flux through one turn is Φ; the coil as a whole links NΦ.
Closed surfaces. Net magnetic flux through any closed surface is zero, because field lines always form closed loops.

A coil turned through 180° does not lose its flux: Φ goes from +BA to −BA, a change of 2BA.

Frequently asked questions

What is the formula of magnetic flux?

Magnetic flux is Φ = BA cos θ, with θ measured between B and the normal of the loop. In vector form it is Φ = B · A.

Which unit is magnetic flux measured in?

The weber, Wb. One weber equals one tesla square metre, and also one volt second.

Does magnetic flux have a direction?

No, Φ is a scalar. Taking B · A collapses two vectors into one signed number, so flux has a size and a sign only.

When is magnetic flux zero?

When the field lies in the plane of the loop, so θ with the normal is 90° and cos θ = 0. Net flux through any closed surface is also zero.

What is the difference between magnetic flux and magnetic flux density?

Flux density B is the field per unit area, measured in tesla. Flux Φ is the total field passing through a given area, measured in weber.