The formula
- Faraday's law for an N-turn coil
ε = -N·ΔΦ / Δt- Instantaneous form
ε = -N·dΦ/dt- Magnetic flux through a flat loop in a uniform field
Φ = BA·cos θ
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
ε | Induced electromotive force | V |
N | Number of turns in the coil | dimensionless |
Φ | Magnetic flux through one turn | Wb (1 Wb = 1 T·m²) |
Δt | Time over which the flux changes | s |
B | Magnetic field strength | T |
A | Area of the loop | m² |
θ | Angle between the field and the normal to the loop | degrees or radians |
When it applies
- Anything that changes the flux works: a changing field strength, a changing loop area, or a loop rotating in a steady field.
- A steady field through a stationary loop induces nothing at all. Only the rate of change matters.
- Use the magnitude to size the emf, then apply Lenz's law separately to get the direction of the induced current.
Worked example
Problem. A 200-turn coil of area 0.010 m² sits with its plane perpendicular to a magnetic field. The field rises steadily from 0.20 T to 0.60 T over 0.50 s. What emf is induced?
- The field is perpendicular to the plane of the coil, so it is parallel to the normal, θ = 0 and Φ = BA.
- Flux change per turn: ΔΦ = A·ΔB = (0.010 m²)(0.60 - 0.20 T) = 4.0 × 10⁻³ Wb.
- Faraday's law, taking magnitudes: |ε| = N·ΔΦ / Δt = (200)(4.0 × 10⁻³) / 0.50 s.
- |ε| = 0.80 / 0.50 = 1.6 V.
- By Lenz's law the induced current circulates in whichever sense makes its own magnetic field point opposite to the growing applied field inside the coil.
Answer. 1.6 V while the field is changing, dropping to zero the moment the field steadies.
Common mistakes
- Leaving out the number of turns. A 200-turn coil gives 200 times the emf of a single loop.
- Measuring θ from the plane of the loop instead of from its normal, which swaps the sine and the cosine.
- Thinking a strong field alone induces an emf. Only a changing flux does.
- Reading the minus sign as a negative voltage. It encodes direction, not a sign on the magnitude.
Related formulas
- Coulomb's law:
F = k·|q₁q₂| / r² - Kirchhoff's laws:
ΣI_in = ΣI_out - Ohm's law:
V = IR