Graphs of Potential Difference in the Coil

AC Potential Difference in Coil

When a coil rotates in a magnetic field, an alternating potential difference (p.d.) is induced across its ends. This p.d. changes continuously in magnitude and direction, producing an alternating current (AC) if the coil is part of a complete circuit.

The graph of potential difference against time for this coil is a sinusoidal waveform. This means the p.d. rises smoothly from zero to a positive peak, falls back through zero to a negative peak, and returns to zero, repeating this cycle continuously.

Key features of this AC waveform:

• Alternating polarity: The p.d. changes direction, shown by the graph crossing the zero line and going positive then negative.
• Frequency: The number of complete cycles per second (measured in hertz, Hz) matches the frequency of the AC supply or the coil's rotation rate.
• Amplitude: The peak value of the p.d. depends on factors like magnetic field strength and coil speed.

For example, a coil rotating once every 0.02 seconds produces a frequency of 50 Hz, which is the standard mains frequency in the UK.

The sinusoidal shape reflects the coil’s changing angle relative to the magnetic field, as the induced p.d. depends on the rate of change of magnetic flux through the coil.

For instance, if a coil rotates at 50 revolutions per second, the induced p.d. completes 50 full sine wave cycles each second, matching the mains frequency.

Induced Potential Graphs

The graph of induced potential difference (emf) in a coil rotating in a magnetic field shows several important features:

• Peak potential difference: The highest points on the graph represent the maximum emf induced when the coil's plane is perpendicular to the magnetic field lines.
• Zero crossing points: The points where the graph crosses zero correspond to the coil being parallel to the magnetic field, so no magnetic flux is changing and no emf is induced.
• Effect of coil rotation speed: Increasing the coil’s rotation speed increases the frequency of the emf waveform and also increases the peak emf, because the magnetic flux changes more rapidly.

The faster the coil spins, the more cycles per second appear on the graph, and the taller the peaks become.

For example, doubling the rotation speed doubles the frequency and roughly doubles the peak emf.

Transformer Coil Graphs

Transformers have two coils: the primary coil connected to the input voltage and the secondary coil where the output voltage is induced.

Graphs of potential difference against time for the primary and secondary coils show:

• Same frequency: Both coils have sinusoidal waveforms with the same frequency, since the magnetic flux changes at the same rate.
• Phase relationship: The primary and secondary voltages are in phase, meaning their peaks and zero crossings occur at the same times.
• Effect of turns ratio: The amplitude (peak voltage) of the secondary coil depends on the ratio of the number of turns in the secondary coil to the primary coil.

The transformer equation relates the voltages and turns:

$$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$

where $V_s$ and $V_p$ are the secondary and primary voltages, and $N_s$ and $N_p$ are the number of turns on the secondary and primary coils.

For example, if the secondary coil has twice as many turns as the primary, the secondary voltage will be twice the primary voltage, and the graph of the secondary voltage will have peaks twice as high.

Graph Interpretation Skills

To interpret graphs of potential difference in coils, you should be able to:

• Read amplitude and frequency: The height of the peaks shows the maximum voltage; the number of cycles per second shows the frequency.
• Relate graph shape to coil motion: The sinusoidal shape matches the coil’s smooth rotation in the magnetic field.
• Identify induced emf patterns: Zero crossings correspond to coil positions where no emf is induced; peak voltages occur when the coil cuts magnetic flux lines most effectively.

Understanding these graphs helps explain how AC generators and transformers work.

For example, if a graph shows a sinusoidal voltage with a frequency of 50 Hz and peak voltage of 230 V, this matches the UK mains supply voltage and frequency.