(i) e.m.f. = − rate of change of magnetic flux linkage; ( ) E t N T T z =-

The idea

This statement gives the quantitative form of Faraday's law: the induced e.m.f. E equals the negative rate of change of magnetic flux linkage, E = −N ΔΦ/Δt, where N is the number of turns, ΔΦ is the change in flux through one turn in webers and Δt is the time interval in seconds. The negative sign expresses Lenz's law. For a 200-turn coil in which flux through each turn falls from 4.0 × 10⁻³ Wb to zero in 0.050 s, ΔΦ/Δt = −8.0 × 10⁻² Wb s⁻¹, so E = −200 × (−8.0 × 10⁻²) = 16 V. The unit check is Wb s⁻¹ = V.

What earns the marks

• The induced e.m.f. is given by E = −N ΔΦ/Δt, the negative rate of change of magnetic flux linkage.
• N is the number of turns, ΔΦ is the change in flux through one turn in webers and Δt is the time interval in seconds.
• The unit of ΔΦ/Δt is Wb s⁻¹, which is equivalent to the volt.
• The negative sign represents Lenz's law: the induced e.m.f. opposes the change in flux linkage.
• Worked check: N = 200, ΔΦ = −4.0 × 10⁻³ Wb, Δt = 0.050 s gives E = −200 × (−4.0 × 10⁻³ ÷ 0.050) = 16 V.

Common mistakes and how to put them right

• Using the total flux linkage change without dividing by the time interval.

  Put it right: e.m.f. is a rate of change, so divide Δ(NΦ) by Δt.

• Forgetting to multiply the flux change per turn by the number of turns N.

  Put it right: Use the change in flux linkage NΔΦ.

• Dropping the negative sign when the direction of the induced e.m.f. is required.

  Put it right: Keep the sign to represent Lenz's law.

Exam tips

• Write E = −N ΔΦ/Δt and identify N, ΔΦ and Δt before substituting numerical values.
• Check units: flux in webers, time in seconds, e.m.f. in volts.
• Use the sign of ΔΦ to decide the direction of the induced e.m.f. and state that it opposes the change.