Induced emf and induced current
An emf is induced whenever flux changes, even in an open coil. A current flows only in a closed circuit, and its size is I = ε/R.
Induced emf and induced current
A changing flux always produces an induced emf, whether or not the circuit is closed. An emf is like a push: it is the work done per unit charge. A current flows only if the circuit is closed. If the total resistance of the closed circuit is R, the induced current is I = ε/R = (N/R) dΦ/dt in magnitude. So a coil with open ends has an emf across its ends but no current. The induced current lasts only while the flux changes. Extra reading, not in your textbook now: the total charge that flows, q = NΔΦ/R, does not depend on how fast the flux changes.
Finding the induced emf and current, step by step
- 1Find the flux through one turn at the start and at the end: Φ = BA cos θ.
- 2Find the change in flux ΔΦ and the time Δt it takes.
- 3Average emf (Faraday's law): ε = N ΔΦ/Δt, where N is the number of turns.
- 4If the circuit is closed, the induced current is I = ε/R, where R is the total resistance.
- 5Find the direction of the current with Lenz's law: it opposes the change in flux.
- 6★So, more turns, a faster change of flux or a smaller resistance gives a larger induced current.
★ key step
Reading a graph of flux against time
- The induced emf equals the slope of the Φ to t graph (with a minus sign, for one turn).
- A straight line rising steadily gives a constant emf.
- A flat (horizontal) part means the flux is constant, so the emf is zero.
- A steeper part gives a larger emf.
- When the slope changes from rising to falling, the emf reverses direction.
Worked example: emf, current and charge
A coil of 100 turns has a resistance of 5 Ω. The flux through each turn changes steadily from 3 × 10⁻³ Wb to 1 × 10⁻³ Wb in 0.02 s. Find (a) the induced emf, (b) the induced current and (c) the charge that flows.
- Given: N = 100, R = 5 Ω, ΔΦ = 3 × 10⁻³ − 1 × 10⁻³ = 2 × 10⁻³ Wb, Δt = 0.02 s.
- (a) ε = N ΔΦ/Δt = 100 × 2 × 10⁻³ Wb ÷ 0.02 s = 0.2 ÷ 0.02 = 10 V.
- (b) I = ε/R = 10 V ÷ 5 Ω = 2 A.
- (c) The flux changes steadily, so the current is steady. Charge q = IΔt = 2 A × 0.02 s = 0.04 C.
(a) 10 V; (b) 2 A; (c) 0.04 C
Worked example: from a flux graph
The flux through a single loop of resistance 2 Ω rises uniformly from 0 to 0.06 Wb in the first 0.3 s. It stays constant from 0.3 s to 0.5 s, then falls uniformly to zero by 0.6 s. Find the induced current in each interval.
- 0 to 0.3 s: slope = 0.06 Wb ÷ 0.3 s = 0.2 V, so ε = 0.2 V and I = 0.2 V ÷ 2 Ω = 0.1 A.
- 0.3 s to 0.5 s: the flux is constant, so the slope is 0. Then ε = 0 and I = 0.
- 0.5 s to 0.6 s: slope = 0.06 Wb ÷ 0.1 s = 0.6 V (falling), so ε = 0.6 V and I = 0.6 V ÷ 2 Ω = 0.3 A.
- In the last interval the current flows in the opposite direction to the first, because the flux now decreases.
0.1 A, then zero, then 0.3 A in the opposite direction
On a bicycle dynamo lamp, pedalling faster turns the magnet faster, so the flux changes faster and the lamp glows brighter. Stop pedalling and the lamp goes out: no change in flux, no emf.
2 marks: “A coil whose ends are not joined lies in a changing magnetic field. Is an emf induced in it? Does a current flow? Give a reason for each.” Yes, an emf is induced, because the flux through the coil changes (Faraday's law). No current flows, because there is no closed path.
3 marks: “The flux through each turn of a 200-turn coil of resistance 8 Ω falls steadily from 6 mWb to 2 mWb in 0.04 s. Find the induced emf, the current and the charge that flows.” ΔΦ = 4 × 10⁻³ Wb; ε = 200 × 4 × 10⁻³ ÷ 0.04 = 20 V; I = 20 V ÷ 8 Ω = 2.5 A; the current is steady, so q = IΔt = 2.5 A × 0.04 s = 0.1 C.
1 marks: “The flux through a coil stays constant at 0.5 Wb. What is the induced emf?” Zero, because the flux is not changing.
emf needs change; current needs a closed loop. Then the current is I = ε/R.