Flux linkage is cosine, EMF is sine — and they're a quarter cycle apart

Electromagnetic Induction — AQA A-Level Physics

For a coil rotating at constant angular velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The rate of change of angular displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m).. The angle swept per unit time for an object moving in a circle. Measured in rad s⁻¹. in a uniform field, the flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns). and EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). vary sinusoidally but are 90° (a quarter of a cycle) out of phase.
Flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns).: NΦ = BAN cos(ωt) — a cosine curve.
Induced EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V).: ε = BANω sin(ωt) — a sine curve.
The EMFElectromotive force. The energy transferred per unit charge by a source in driving charge around a complete circuit. Measured in volts (V). is the gradient (rate of change) of the flux linkageThe product of magnetic flux and the number of turns of a coil. Measured in weberThe SI unit of magnetic flux. One weber is the flux through an area of 1 m² when the magnetic flux density is 1 T perpendicular to the area.-turns (Wb turns). graph. That's Faraday's lawThe magnitude of the induced EMF is proportional to the rate of change of magnetic flux linkage. in graphical form.

Key points on the graphs

When NΦ is at its maximum or minimum value, the flux linkage is momentarily not changing. The gradient of the NΦ graph is zero. So the EMF is zero at these instants.
When NΦ passes through zero, the flux linkage is changing most rapidly. The gradient of the NΦ graph is steepest. So the EMF is at its maximum (positive or negative) at these instants.
The coil is face-on to the field ($N\Phi = max)$ when $\varepsilon = 0. The coil$ is edge-on to the field ($N\Phi = 0)$ when $\varepsilon = max$.
If you can sketch the NΦ graph, you can deduce the ε graph by asking: where is the gradient steepest? Where is it zero? That gives you the shape of the EMF graph.

NΦ and ε vs time graphs

Two aligned graphs sharing the same time axis. Top graph: NΦ = BAN cos(ωt), a cosine wave starting at maximum. Bottom graph: ε = BANω sin(ωt), a sine wave starting at zero. Dashed vertical lines show that ε = 0 when NΦ is at max/min, and ε is at max/min when NΦ = 0.

Examiner Tips and Tricks

A very common exam question gives you a flux linkage-time graph and asks you to sketch or describe the EMF-time graph.
Remember: the EMF graph is the gradient of the flux linkage graph.
Where the NΦ curve is flat (turning points), $\varepsilon = 0$.
Where NΦ is steepest (crossing the axis), ε is at its peak.

Common Mistake MEDIUM

Students often: Thinking the EMF is maximum when the flux linkage is maximum.
Instead: It's the opposite. Maximum flux linkage means zero rate of change, so zero EMF. Maximum EMF occurs when flux linkage is zero (changing fastest).