Faraday's Law: Both Field and Area Changing at Once
1 · Predict
A loop's area AND the magnetic field through it are both oscillating simultaneously. Can you still find the induced EMF by simply adding the two separate effects (changing B alone, changing A alone) together?
- No — you need the full product rule d(BA)/dt = A·dB/dt + B·dA/dt, which is more than just adding the two separate EMFs.
- Yes — you can just add the changing-flux-b and changing-flux-area EMF formulas together.
- You multiply the two separate EMF formulas together.
2 · Set Up
- Open the faraday-lenz preset and press Reset. Both B(t) and A(t) oscillate: B₀ = 0.45 T, A₀ = 0.018 m², at 0.5 Hz.
- Enable the induced-EMF readout.
- Read the induced EMF at each listed time.
3 · Collect Data
| Time t (s) | Induced EMF (mV) |
|---|---|
| 0.3 | |
| 0.6 | |
| 1 |
Plot induced EMF (y-axis) against time t (x-axis) for your three readings.
4 · Analyze
- For one trial, compute B(t) = B₀(1 + 0.5sin(ωt)), A(t) = A₀(1 + 0.35cos(ωt)), B'(t) = B₀·0.5ω·cos(ωt), A'(t) = −A₀·0.35ω·sin(ωt), then EMF = −(A·B' + B·A') using B₀ = 0.45 T, A₀ = 0.018 m², ω = 2π×0.5 rad/s. Compare to the table.
- Explain why the product rule d(BA)/dt = A·dB/dt + B·dA/dt is exactly the right tool here — flux is a PRODUCT of B and A, and calculus's product rule (not simple addition) tells you how a product's rate of change depends on both factors changing together.
5 · Extend
- Compare this experiment's EMF values to changing-flux-b's and changing-flux-area's at similar times. Explain why you can't simply predict this combined result by adding the other two experiments' EMF values at the same time, even though all three use the product rule underneath.
- A transformer's changing flux comes from a changing current in one coil (which changes B), not from any moving parts (A is fixed there). Explain why the faraday-lenz experiment's combined-change scenario is more general than what a stationary transformer actually needs.
The Physics Behind This Experiment
Faraday's Law (Product Rule)
When both B and A change with time, Faraday's law requires the full product rule: EMF = −d(BA)/dt = −(A·dB/dt + B·dA/dt). Each term captures one factor's contribution while the other is held at its instantaneous value.