Magnetic Flux

Q: What is magnetic flux?

Magnetic flux is Φ = ∫ B·dA, the total field passing through a surface. For uniform B and flat surface, Φ = BA cos θ where θ is the angle between B and surface normal.

Q: What is the SI unit of magnetic flux?

The weber (Wb), where 1 Wb = 1 T·m² = 1 V·s. In CGS the unit is the maxwell; 1 Wb = 10⁸ Mx.

Q: What is Gauss's law for magnetism?

∮B·dA = 0 for any closed surface. Total magnetic flux out of any closed surface is always zero because field lines are closed loops. Equivalently, isolated magnetic monopoles do not exist.

Q: Why is the total magnetic flux through any closed surface zero?

Because magnetic field lines form closed loops with no start or end, every line entering a closed surface must also exit. There are no monopole sources or sinks of B.

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

Flux Φ is total through a surface (weber). Flux density B is flux per unit area (tesla = Wb/m²). Related by Φ = ∫B·dA.

Q: What is flux linkage?

Ψ = NΦ, total flux linked with an N-turn coil. Appears in Faraday's law for multi-turn coils: ε = -dΨ/dt.

Q: How is magnetic flux related to induced EMF?

Faraday's law: ε = -dΦ/dt (single loop) or -N dΦ/dt (N-turn coil). Any change in flux through a circuit induces an EMF; by Lenz's law it opposes the change.

Q: Does the shape of a surface spanning a loop affect the flux through it?

No. Because ∇·B = 0, all surfaces sharing the same boundary loop enclose the same flux.

Q: Can magnetic flux be negative?

Yes, if B has a component opposite to the chosen surface normal. The sign becomes physically meaningful in Faraday's law, determining the direction of induced current.

PhysicsMagnetic Effects of Current and MagnetismFor JEE aspirants

Magnetic flux $Φ_{B}$ through a surface measures how many magnetic field lines thread that surface. Formally, $Φ_{B} = \int B \cdot d A$ , and for a uniform field and flat surface, $Φ_{B} = B A cos θ$ . Its SI unit is the weber (Wb), where $1 Wb = 1 T\cdotm^{2}$ . A crucial law is Gauss's law for magnetism: the net flux through any closed surface is always zero ( $\oint B \cdot d A = 0$ ), which is the mathematical statement that isolated magnetic poles (monopoles) do not exist. Magnetic flux is the essential bridge from magnetostatics to electromagnetic induction (Faraday's law $ε = - d Φ_{B} / d t$ ), the very next unit in JEE Main and NEET Physics.

Key Formulas - Quick Reference

Flux through a surface: $Φ_{B} = \int_{S} B \cdot d A$
Uniform field, flat surface: $Φ_{B} = B \cdot A = B A cos θ$ , where $θ$ is the angle between $B$ and the surface normal
SI unit: weber (Wb); $1 Wb = 1 T\cdotm^{2} = 1 V\cdots$
CGS unit: maxwell (Mx); $1 Wb = 1 0^{8} Mx$
Flux linkage: $N Φ_{B}$ (for a coil of $N$ turns)
Gauss's law for magnetism: $\oint_{S} B \cdot d A = 0$ (always, for any closed surface)
Faraday's law (used in EMI): induced EMF $ε = - \frac{d Φ _{B}}{d t}$

1. Definition of Magnetic Flux

Magnetic flux through a surface is the "amount of $B$ " passing through that surface, weighted by the projection onto the surface's normal:

Φ_{B} = \int_{S} B \cdot d A

where $d A$ is an element of surface area with direction along the local outward (or chosen) normal. For a uniform field over a flat surface:

Φ_{B} = B \cdot A = B A cos θ

Here $θ$ is the angle between $B$ and the surface normal $\overset{n}{^}$ . Special cases:

$θ = 0°$ ( $B ∥ \overset{n}{^}$ , i.e., $B$ perpendicular to the surface): $Φ_{B} = B A$ (maximum).
$θ = 90°$ ( $B$ lies in the surface, parallel to it): $Φ_{B} = 0$ (no field lines cross).
$θ = 180°$ ( $B$ opposite to $\overset{n}{^}$ ): $Φ_{B} = - B A$ (sign of flux depends on chosen normal direction).

Figure 1: Only the part of the surface that faces the field counts. Sliding each corner along

B

gives the dashed projected area

A cos θ

, so

Φ_{B} = B \cdot A = B A cos θ

. Tilt the surface until it lies along the field and the shadow shrinks to nothing.

Figure 2: Exam questions phrase this both ways. If a problem gives the angle with the plane, that is

α

, and you must use

θ = 90° - α

Φ = B A cos θ

- equivalently

Φ = B A sin α

. Misreading this one angle is the single most common flux mistake.

Figure 3: Rotating a loop in a steady field changes the flux smoothly as

cos θ

. Notice the flux is zero, not maximum, when the field lies in the plane of the loop. A coil spun steadily traces this curve, and its rate of change is exactly what Faraday's law turns into an alternating EMF.

2. Units of Magnetic Flux

System	Unit	Equivalence
SI	weber (Wb)	$1 Wb = 1 T\cdotm^{2} = 1 V\cdots = 1 J/A$
CGS	maxwell (Mx)	$1 Wb = 1 0^{8} Mx$

Because $B$ is defined as flux per unit area, the SI unit of magnetic field, the tesla, is $1 T = 1 Wb/m^{2}$ . Historically, $B$ is even called "magnetic flux density".

3. Flux Linkage

For a coil of $N$ turns, each turn is threaded by the same flux $Φ_{B}$ . The total "linked flux" is:

Ψ = N Φ_{B}

Flux linkage $Ψ$ is what appears in Faraday's law when the coil has more than one turn:

$ε = - \frac{d Ψ}{d t} = - N \frac{d Φ _{B}}{d t}$

This " $N$ factor" is why transformer secondaries and multi-turn induction coils generate much larger EMFs than single-loop devices.

4. Gauss's Law for Magnetism

The magnetic flux through any closed surface is always zero:

\oint_{S} B \cdot d A = 0

Figure 4: Field lines leave the N pole, curve round to the S pole, and continue inside the magnet from S back to N. Because they never begin or end anywhere, any closed surface you draw is pierced equally in both directions and

\oint B \cdot d A = 0

. Contrast this with electric field lines, which start and stop on charges.

Physical meaning

Because magnetic field lines always form closed loops (they have no beginning or end), whatever flux enters a closed surface must also leave it. Equivalently: isolated magnetic poles do not exist. You cannot cut a magnet to obtain a pure north or south pole - each piece is a complete dipole with both poles.

Contrast with the electrostatic case: $\oint E \cdot d A = Q_{enc} / ε_{0}$ - Gauss's law for electricity is non-zero because isolated electric charges (monopoles) do exist. Gauss's law for magnetism has zero on the right side because magnetic monopoles have never been observed.

Solved Example 1

A rectangular coil of area

200 cm^{2}

is placed in a uniform magnetic field of

0.4 T

. Find the magnetic flux when (i) the coil is perpendicular to the field, (ii) the plane of the coil makes

30°

with the field, (iii) the coil is parallel to the field.

Solution:

Area $A = 200 \times 1 0^{- 4} = 0.02 m^{2}$ ; $B A = 0.4 \times 0.02 = 8 \times 1 0^{- 3} Wb$ .

(i) Coil perpendicular to field means $B$ is along the normal, $θ = 0°$ : $Φ = B A = 8 \times 1 0^{- 3} Wb = 8 mWb$ .

(ii) If the plane makes $30°$ with $B$ , then the normal makes $60°$ with $B$ : $Φ = B A cos 60° = 4 mWb$ .

(iii) Coil parallel to field: $B$ lies in the plane, $θ = 90°$ : $Φ = 0$ .

Solved Example 2

A cube of side

10 cm

sits in a uniform magnetic field of

0.5 T

directed along the

+ x

-axis. Find the net magnetic flux through the cube.

Solution:

By Gauss's law for magnetism, the net flux through any closed surface is zero. No calculation needed.

Check: Flux entering the left face: $B A = 0.5 \times 0.01 = 5 \times 1 0^{- 3} Wb$ . Flux leaving the right face: same, $5 \times 1 0^{- 3} Wb$ . Flux through the top, bottom, front, back faces: zero (field lies in those planes). Net: $- 5 \times 1 0^{- 3} + 5 \times 1 0^{- 3} = 0$ . ✓

Figure 5: Worked Example 2 needs no arithmetic. Whatever enters the left face leaves the right face, and the four remaining faces contain the field so contribute nothing. This holds for any closed surface in any magnetic field, not just this symmetric case.

Solved Example 3

A circular coil of

50

turns and radius

0.05 m

carries no current, but sits with its normal at

60°

to a uniform field

B = 0.2 T

. Find the flux and flux linkage.

Solution:

$A = π (0.05)^{2} = 7.85 \times 1 0^{- 3} m^{2}$ .

$Φ_{B} = B A cos 60° = 0.2 \times 7.85 \times 1 0^{- 3} \times 0.5 = 7.85 \times 1 0^{- 4} Wb \approx 0.785 mWb$ .

Flux linkage $Ψ = N Φ_{B} = 50 \times 7.85 \times 1 0^{- 4} = 3.93 \times 1 0^{- 2} Wb-turns \approx 39.3 mWb-turns$ .

5. Bridge to Electromagnetic Induction

Magnetic flux is the central quantity in the next unit, EMI. The key result there:

Faraday's law: $ε = - \frac{d Φ _{B}}{d t}$ (for a single loop) or $ε = - N \frac{d Φ _{B}}{d t}$ (for $N$ -turn coil).

An EMF is induced whenever $Φ_{B}$ through a circuit changes - by moving the coil, changing $B$ , rotating the coil, or deforming the loop. The minus sign is Lenz's law: the induced current opposes the change in flux.

Common Mistakes to Avoid

Watch out

Confusing the angle in $Φ = B A cos θ$ : $θ$ is the angle between $B$ and the surface normal, not between $B$ and the plane of the surface. If the problem says "coil's plane makes angle $α$ with the field", the angle to use is $θ = 90° - α$ .
Thinking flux depends on the shape of the surface for a fixed boundary loop: it does not. All surfaces spanning the same loop enclose the same flux (a consequence of $\nabla \cdot B = 0$ ).
Setting Gauss's law for magnetism to something other than zero: the right side is always zero, regardless of what magnets, currents, or fields exist inside.
Confusing $Φ_{B}$ (through a surface) with $Ψ = N Φ_{B}$ (flux linkage, for $N$ turns): only the linkage appears in Faraday's law for multi-turn coils.
Forgetting the sign of flux: it depends on the chosen direction of $\overset{n}{^}$ ; the sign of the induced EMF ties to this convention.
Using webers as an SI unit of field: weber is the unit of flux; $Wb/m^{2} = T$ is the unit of field.

Frequently Asked Questions

Q1. What is magnetic flux?

Magnetic flux through a surface is the total "amount of magnetic field" passing through it, defined as $Φ_{B} = \int B \cdot d A$ . For uniform $B$ and a flat surface, $Φ_{B} = B A cos θ$ , with $θ$ the angle between $B$ and the surface normal. It measures how many field lines thread the surface.

Q2. What is the SI unit of magnetic flux?

The weber (Wb), where $1 Wb = 1 T\cdotm^{2} = 1 V\cdots$ . In CGS the unit is the maxwell; $1 Wb = 1 0^{8} Mx$ .

Q3. What is Gauss's law for magnetism?

$\oint B \cdot d A = 0$ for any closed surface. The total magnetic flux out of a closed surface is always zero because magnetic field lines are closed loops - every line entering must leave. Equivalently, isolated magnetic charges (monopoles) do not exist.

Q4. Why is the total magnetic flux through any closed surface zero?

Because magnetic field lines form closed loops (they have no start or end), any line entering a closed surface must also exit. There are no "sources" (isolated north poles) or "sinks" (isolated south poles) of $B$ . This is why Gauss's law for magnetism has zero on the right, unlike Gauss's law for electricity.

Q5. What is the difference between magnetic flux and magnetic flux density?

Magnetic flux $Φ_{B}$ is the total flux through a surface, measured in webers. Magnetic flux density $B$ (also called magnetic induction, or just magnetic field) is flux per unit area, measured in $Wb/m^{2} = tesla$ . The two are related by $Φ_{B} = \int B \cdot d A$ .

Q6. What is flux linkage?

Flux linkage $Ψ = N Φ_{B}$ is the total flux linked with a coil of $N$ turns, since each turn is threaded by the same $Φ_{B}$ . It is the quantity that appears in Faraday's law for multi-turn coils: $ε = - d Ψ/ d t$ .

Q7. How is magnetic flux related to induced EMF?

Via Faraday's law: $ε = - d Φ_{B} / d t$ (single loop) or $ε = - N d Φ_{B} / d t$ ( $N$ -turn coil). A changing magnetic flux through a circuit induces an EMF whose direction (by Lenz's law) opposes the change. This is covered in the electromagnetic induction unit.

Q8. Does the shape of a surface spanning a loop affect the flux through it?

No. Because $\nabla \cdot B = 0$ (Gauss's law for magnetism), the flux through any two surfaces sharing the same boundary loop is the same. You can choose whatever surface is easiest to integrate over - flat, curved, hemispherical - and get the same answer.

Q9. Can magnetic flux be negative?

Yes, depending on the choice of surface normal. If $B$ has a component opposite to $\overset{n}{^}$ , the flux $B \cdot A$ is negative. The sign is a bookkeeping convention that becomes physically meaningful in Faraday's law, where it determines the direction of the induced current.

Previous year questions on Magnetic Flux

5 questions from past papers, each with a step-by-step solution.

Ready to master Magnetic Effects of Current and Magnetism?

Take a full mock test, practice concept-by-concept, and get an AI-powered rank prediction — all on Fundamenthol.

Practice questions →Take a mock test