Magnetic Flux Density Formula: B = Φ/A, Tesla, Examples

The magnetic flux density formula is $B = \frac{\Phi}{A}$ : divide the magnetic flux Φ by the area A it threads, with the area turned square on to the field. The answer, in webers per square metre or tesla, says how concentrated the field lines are.

Flux density is the same quantity that most chapters simply call the magnetic field B. Some books define it from flux, others from the force on a current, $B = \frac{F}{IL}$ . Both are on this page with examples, so you can use whichever matches the data a question hands you. The shorter search, flux density formula, means the same thing.

Quick facts

From flux
$B = \frac{\Phi}{A}$
From force on a wire
$B = \frac{F}{IL\sin\theta}$
Tilted area
$\Phi = BA\cos\theta$
SI unit
tesla (T) = Wb/m² = N/(A m)
Dimensions
[M¹ L⁰ T⁻² I⁻¹]
Older unit
1 T = 10⁴ gauss
Type
Vector

Reading the magnetic flux density formula

Letters used for flux density
Symbol	Quantity	Unit
$B$	Flux density, the magnetic field strength	tesla
$\Phi$	Total flux crossing the surface	weber
$A$	Area facing the field squarely	m²
$\theta$	Angle between B and the normal to the area	degree or radian

If the surface is tilted, only its projection catches the field. The flux becomes $\Phi = BA\cos\theta$ , so the density is $B = \frac{\Phi}{A\cos\theta}$ . When the field lies along the surface, θ = 90°, and no flux passes at all. The magnetic flux formula page covers Φ in more depth.

Defining B by force on a current

A straight wire of length L carrying current I across a field feels a sideways push $F = BIL\sin\theta$ . Rearranged, $B = \frac{F}{IL\sin\theta}$ . One tesla is the field that pushes a 1 m wire carrying 1 A at right angles with a force of 1 N. The magnetic force formula page uses this relation the other way round.

Both definitions give the same B. The flux definition suits coils and induction; the force definition suits wires and motors.

Solved problems on flux density

Problem 1: flux through a square loop

Exactly 4.5 × 10⁻³ Wb threads a loop of area 0.015 m² held square to the field. Find B.

$B = \frac{4.5\times10^{-3}}{0.015} = 0.30\ \text{T}$

Problem 2: a tilted coil

A coil of area 0.040 m² has its normal at 60° to a uniform field. The flux through it is 1.6 × 10⁻³ Wb. Find the flux density.

$B = \frac{\Phi}{A\cos 60^\circ} = \frac{1.6\times10^{-3}}{0.040\times0.50} = 0.080\ \text{T}$

Problem 3: from a force with length in cm

A 40 cm wire carrying 2.5 A at right angles to a field feels 0.030 N. Find B.

L = 0.40 m, so IL = 1.0 A m.

$B = \frac{0.030}{1.0} = 0.030\ \text{T} = 30\ \text{mT}$

Sizes of B you should recognise

Typical magnetic flux densities
Source	Rough value
Earth's field at the surface	about 25 to 65 μT
Fridge magnet	a few mT
School bar magnet near a pole	about 0.01 to 0.1 T
Strong lab electromagnet	around 1 to 2 T
Hospital scanner magnet	about 1.5 to 3 T

These give a sense of scale. If your answer for a classroom magnet comes out at hundreds of tesla, check the units again. Older books and some exam questions quote gauss: divide by 10⁴ to get tesla.

Unit and dimension of the tesla

From $B = \frac{F}{IL}$ , the tesla is N/(A m). The newton is kg m s⁻², so

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

Length cancels completely. The same result comes from webers per square metre, because the weber is V s. Since flux density and magnetic field share one symbol and one unit, the magnetic field dimensions page covers both.

Where students go wrong

Using sin θ in the flux formula. Flux uses cos θ, measured from the normal.
Area left in square centimetres. Multiply by 10⁻⁴ to get square metres.
Confusing flux (Wb) with flux density (T).
Forgetting that 1 mT = 10⁻³ T and 1 μT = 10⁻⁶ T.

For fields made by currents, see the magnetic field formula page and the electricity hub for a chapter overview.

Frequently asked questions

What is the flux density formula?

B = Φ / A, magnetic flux divided by the perpendicular area. With a wire in the field you can also use B = F / (IL sin θ).

Which unit is magnetic flux density measured in?

The tesla (T), equal to one weber per square metre or one newton per ampere metre.

Is magnetic flux density the same as magnetic field?

In most school and college courses, yes. B is called both the magnetic field and the magnetic flux density. A separate quantity H, the magnetising field, appears in more advanced work.

How do you convert gauss to tesla?

Divide by 10⁴. So 500 gauss equals 0.05 T.

Is flux density a vector?

Yes. B has a direction, shown by field lines running from north to south outside a magnet. Flux Φ itself is a scalar.