What Magnetic Flux Means
Faraday's insight, NCERT records, lay in finding a single mathematical relation to tie together a long series of induction experiments. That relation rests on one preliminary idea: magnetic flux, written $\Phi_B$. Flux is defined in exactly the same way as electric flux was in the electrostatics chapter, and it answers a physical question — how much of the magnetic field passes through a given surface.
A useful picture is to think of the magnetic field as a set of field lines. The flux through a loop is then proportional to the number of field lines that pierce the area bounded by that loop. A surface held face-on to the field is threaded by many lines and carries large flux; the same surface turned edge-on is pierced by none and carries zero flux. Nothing about the field itself has changed — only its geometric relationship to the surface.
The Defining Equation
For a plane surface of area $A$ placed in a uniform magnetic field $\mathbf{B}$, NCERT §6.3 defines the magnetic flux as a dot product:
$$\Phi_B = \mathbf{B}\cdot\mathbf{A} = BA\cos\theta$$
Here $\theta$ is the angle between $\mathbf{B}$ and the area vector $\mathbf{A}$. The dot product compresses three factors — field strength, area, and orientation — into a single number. When the field is not uniform, or the surface is curved, the area is divided into small elements $d\mathbf{A}_i$ and the contributions are summed, $\Phi_B = \sum_{\text{all}} \mathbf{B}_i \cdot d\mathbf{A}_i$, but for NEET the planar uniform-field form is what is used in practice.
The Angle and the Area Vector
The area of a surface is treated as a vector. Its magnitude is the numerical area, and its direction is along the normal to the surface — perpendicular to the plane of the loop. This is the source of nearly every flux error: $\theta$ is measured from the field to that normal, never from the field to the surface itself.
Two limiting cases anchor the formula. When the field is perpendicular to the plane of the loop, it lies along the area vector, so $\theta = 0^\circ$, $\cos\theta = 1$, and the flux is maximum, $\Phi_B = BA$. When the field lies in the plane of the loop, it is perpendicular to the area vector, so $\theta = 90^\circ$, $\cos\theta = 0$, and the flux is zero. The intermediate orientation gives an in-between value scaled by $\cos\theta$.
θ is measured from the normal, not the surface
A question may state that the magnetic field "makes an angle of 30° with the plane of the loop." Students often substitute $\cos 30^\circ$ straight into $BA\cos\theta$. That is wrong. The 30° is measured to the plane, so the angle to the area vector (the normal) is its complement, $90^\circ - 30^\circ = 60^\circ$, and the correct factor is $\cos 60^\circ$.
If a problem gives the angle to the plane, use the complement. If it gives the angle to the normal or to the area vector, use it directly. Flux is maximum when B is along A, and zero when B lies in the plane.
Units and Related Quantities
The SI unit of magnetic flux is the weber (Wb). Because flux is field times area, one weber equals one tesla metre squared (T m²). Magnetic flux is a scalar quantity, even though it is built from two vectors, because the dot product of two vectors returns a scalar. The table below collects the three quantities in the defining equation with their symbols and units.
| Quantity | Symbol | SI Unit | Nature |
|---|---|---|---|
| Magnetic flux | $\Phi_B$ | weber (Wb) = T m² | Scalar |
| Magnetic field | $B$ | tesla (T) | Vector |
| Area (vector) | $\mathbf{A}$ | metre² (m²) | Vector (along normal) |
| Angle | $\theta$ | degree / radian | Field-to-normal angle |
Flux Through a Coil of N Turns
Most practical coils have many turns, not one. For a closely wound coil of $N$ turns, the same magnetic flux is linked with every turn. The total quantity of interest is then the flux linkage, $N\Phi_B$:
$$N\Phi_B = N\,BA\cos\theta$$
This product, not the single-turn flux, is what enters Faraday's law for a coil. When the flux through each turn changes, every turn contributes to the induced emf, so the total emf is multiplied by $N$. This is precisely why coils are wound with many turns — it amplifies the induced emf without changing the underlying field.
Once flux is defined, its time rate of change gives the induced emf. See Faraday's Law of Induction for how $-N\,d\Phi_B/dt$ produces the emf.
Three Ways to Change Flux
Since $\Phi_B = BA\cos\theta$ contains three factors, NCERT notes that flux can be varied by changing any one or more of $B$, $A$, or $\theta$. Each route produces an induced emf, and NEET problems are built around all three.
| Factor changed | How it is done | Typical setup |
|---|---|---|
| Field $B$ | Move a magnet toward or away from the coil, or switch a neighbouring coil's current on and off | Bar magnet plunged into a coil |
| Area $A$ | Shrink or stretch the loop, changing the area it encloses in the field | Sliding conducting rod on rails |
| Angle $\theta$ | Rotate the coil in the field so the orientation of the normal changes | Coil spinning in a uniform field |
The rotating-coil case is the basis of the AC generator, where $\theta = \omega t$ makes the flux vary as $\cos\omega t$. The sliding-rod case is the basis of motional emf. Both reduce to the same principle: it is the change in flux, by whatever means, that induces the emf.
Worked Examples
A coil is placed so that the angle made by its area vector with the magnetic field is 45°, and the flux is then reduced to zero in 0.70 s. If the initial flux is $0.1\sqrt{2}\times 10^{-2}$ Wb, the calculation begins from the flux value.
The initial flux follows directly from $\Phi = BA\cos\theta$ with $\theta = 45^\circ$. Because the angle quoted is already the angle between the area vector and B, it is substituted without modification. NCERT then uses this initial flux, the final flux of zero, and the 0.70 s interval to find the induced emf. The point for flux specifically: the 45° is used directly only because it was given relative to the area vector.
A circular coil of radius 10 cm with 500 turns sits with its plane perpendicular to the horizontal component of the earth's field, $B = 3.0\times10^{-5}$ T. It is rotated about a vertical diameter through 180°. Find the initial and final flux through one turn.
With the plane perpendicular to the field, the area vector is along the field, so $\theta = 0^\circ$: $$\Phi_{\text{initial}} = BA\cos 0^\circ = 3.0\times10^{-5}\times(\pi\times10^{-2})\times 1 = 3\pi\times10^{-7}\ \text{Wb}.$$ After a 180° rotation the area vector reverses, so $\theta = 180^\circ$ and $\cos 180^\circ = -1$: $$\Phi_{\text{final}} = 3.0\times10^{-5}\times(\pi\times10^{-2})\times(-1) = -3\pi\times10^{-7}\ \text{Wb}.$$ The flux does not merely vanish — it changes sign, so the total change in flux per turn has magnitude $6\pi\times10^{-7}$ Wb. This is the quantity Faraday's law then multiplies by $N$ and divides by the time.
Magnetic Flux at a Glance
- Magnetic flux $\Phi_B = \mathbf{B}\cdot\mathbf{A} = BA\cos\theta$, where $\theta$ is the angle between the field and the area vector (the normal).
- Flux is a scalar; its SI unit is the weber (Wb) = T m².
- Flux is maximum ($BA$) when B is along the area vector ($\theta = 0^\circ$) and zero when B lies in the plane of the loop ($\theta = 90^\circ$).
- For $N$ turns the flux linkage is $N\Phi_B$; this is what Faraday's law acts on.
- Flux changes by varying $B$, $A$, or $\theta$ — and any such change induces an emf.
- If a problem gives the angle to the plane, take its complement before using $\cos\theta$.