Difficulty: Intermediate–Advanced | Prerequisites: Lorentz force, magnetic flux, basic calculus (derivatives of trig functions).
Everything so far has dealt with static situations: steady currents in fixed wires, or charges moving through constant fields. Now things start to change with time. When a conductor moves through a magnetic field, or when a magnetic field itself changes, an EMF (voltage) is induced. Faraday's Law is the universal statement of this principle, and Lenz's Law tells you the direction of the induced current. This is the physics behind electric generators, transformers, and wireless charging. It also closes the conceptual loop: not only do currents create magnetic fields (Ampere's Law), but changing magnetic fields create electric fields and drive currents (Faraday's Law).
Moving a conductor through a magnetic field generates a voltage (EMF = vBL). More generally, Faraday's Law says any change in magnetic flux through a loop induces an EMF: ε = −dΦ_B/dt. The minus sign is Lenz's Law, which says the induced current opposes the change in flux that caused it.
Motional EMF
The voltage induced in a conductor moving through a magnetic field. For a straight conductor of length L moving at velocity v perpendicular to a uniform field B: EMF = vBL. Think of it as the Lorentz force doing the work of a battery, pushing charges along the conductor.
Magnetic flux (Φ_B)
Φ_B = ∫B · dA = BAcosθ (for a uniform field). It measures how much magnetic field "passes through" a surface. Units: Weber (Wb) = T·m².
Faraday's Law
ε = −dΦ_B/dt. The induced EMF in a loop equals the negative rate of change of magnetic flux through the loop. This is one of Maxwell's equations and is completely general.
Lenz's Law
The direction of the induced current is such that its own magnetic field opposes the change in flux that produced it. This is the physical meaning of the minus sign in Faraday's Law. In simple terms: nature resists change in magnetic flux.
Induced EMF
A voltage that appears in a conductor or loop without any battery present, driven entirely by changing magnetic flux.
A straight conducting bar of length L moves with velocity v through a uniform magnetic field B (field perpendicular to both L and v):
The free charges inside the conductor experience a Lorentz force: F = qv × B.
This force pushes positive charges toward one end of the bar and negative charges toward the other.
A charge separation builds up, creating an electric field E inside the conductor that opposes further charge motion.
Equilibrium: the electric force (qE) balances the magnetic force (qvB).
qvB = qE → E = vB
The potential difference across the bar (the motional EMF):
V = EL = vBL
If the bar slides on rails connected to a resistor R, forming a closed circuit:
The motional EMF drives a current: I = V/R = vBL/R
This current in the bar, sitting in the B field, experiences a force: F = ILB = vB²L²/R
This force opposes the motion (it acts as a braking force). You must do work to keep the bar moving.
Power dissipated in the resistor: P = Fv = v²B²L²/R = I²R
The mechanical energy you put in is converted to electrical energy and then to heat in the resistor.
The motional EMF result is a special case. Faraday's Law is the general principle:
ε = −dΦ_B/dt
Where magnetic flux is:
Φ_B = ∫B · dA = BAcosθ (for a uniform field through a flat loop)
The flux can change because:
B changes in magnitude.
The area A changes (e.g. a loop expanding or contracting).
The angle θ between B and the area vector changes (e.g. a loop rotating).
Any combination of the above.
The EMF can also be expressed as:
ε = ∮ E · dl = −dΦ_B/dt
This tells you that a changing magnetic flux produces an electric field, even in empty space (no wire needed). The wire just gives the charges somewhere to flow.
A circular loop of radius r rotates with angular velocity ω in a uniform magnetic field B:
The angle between B and the area vector changes with time: θ = ωt
Flux: Φ_B = BAcosθ = BAcos(ωt) = Bπr²cos(ωt)
EMF: ε = −dΦ_B/dt = BAωsin(ωt) = Bπr²ωsin(ωt)
Maximum EMF occurs when ωt = 90° (when the loop is parallel to B, flux is changing fastest): ε_max = BAω = Bπr²ω
Minimum EMF (zero) occurs when ωt = 0° (when the loop is perpendicular to B, flux is momentarily not changing): ε = 0
This is exactly how an AC generator works.
The minus sign in Faraday's Law is not just mathematical bookkeeping. It encodes a physical principle:
If the flux through a loop is increasing, the induced current flows in a direction that creates a magnetic field opposing the increase (i.e. the induced B points opposite to the external B inside the loop).
If the flux is decreasing, the induced current creates a field that tries to maintain the flux (induced B points in the same direction as the external B).
Lenz's Law is a consequence of conservation of energy. If the induced current reinforced the change, you would get a runaway process, violating energy conservation.
How to apply Lenz's Law:
Determine whether the flux through the loop is increasing or decreasing.
The induced current must create a magnetic field that opposes that change.
Use the right-hand rule to find the current direction that produces the opposing field.
Quantity | Formula |
|---|---|
Motional EMF (straight bar) | ε = vBL |
Current from motional EMF | I = vBL/R |
Braking force on the bar | F = ILB = v B²L²/R |
Power dissipated | P = Fv = v²B²L²/R = I²R |
Magnetic flux | Φ_B = ∫B · dA = BAcosθ |
Faraday's Law | ε = −dΦ_B/dt |
Rotating loop EMF | ε = BAωsin(ωt) |
Maximum EMF (rotating loop) | ε_max = BAω |
Electric generators are literally Faraday's Law in a box: a coil rotates in a magnetic field, and the changing flux induces an AC voltage. Transformers use a changing current in one coil to create a changing flux that induces a voltage in a second coil. Induction cooktops use a rapidly changing magnetic field to induce currents (and therefore heat) directly in the pot. Electromagnetic braking on roller coasters and trains uses the braking force from motional EMF to slow the vehicle without friction.
Students often think a magnetic field alone is enough to induce an EMF. It is not. The flux must be changing. A loop sitting in a constant, uniform magnetic field has constant flux and zero induced EMF, regardless of how strong the field is.
Lenz's Law does not say the induced field cancels the external field. It says the induced field opposes the change. If B is increasing, the induced field fights the increase but does not eliminate B entirely.
The negative sign in Faraday's Law is Lenz's Law, not a sign error. Students who drop it and then try to figure out the current direction separately often get confused. Keep the sign and it does the work for you.
For a rotating loop, ε is maximum when the flux is changing fastest (loop parallel to B), not when the flux is at a maximum (loop perpendicular to B). The EMF depends on the rate of change of flux, not the flux itself.
⚠️ Expect at least one Faraday's Law problem, either a bar on rails or a rotating loop. Know both setups.
⚠️ Lenz's Law direction questions are a staple. Given a changing field and a loop, determine the direction of the induced current. Practise several of these.
⚠️ The distinction between when ε is maximum vs. when Φ is maximum in a rotating loop is a favourite conceptual question.
⚠️ Energy and power in the sliding-bar circuit (mechanical energy → electrical energy → heat) is a common quantitative problem.
1. Fill in the blank: The motional EMF of a 0.5 m bar moving at 2 m/s through a 0.3 T field is ε = ______.
A: ε = vBL = 2 × 0.3 × 0.5 = 0.3 V.
2. True or false: A loop in a constant, uniform magnetic field has a constant induced EMF.
A: False. If the flux is not changing, the induced EMF is zero.
3. True or false: Lenz's Law states that the induced current always flows clockwise.
A: False. The direction depends on whether the flux is increasing or decreasing and the orientation of the loop.
4. Fill in the blank: Faraday's Law states ε = ______.
A: −dΦ_B/dt.
5. True or false: In a rotating loop, the EMF is maximum when the magnetic flux through the loop is maximum.
A: False. EMF is maximum when dΦ/dt is maximum, which is when the flux is at zero (changing fastest).
Q: A conducting bar of length 0.4 m slides at 3 m/s on frictionless rails in a 0.5 T field. The rails are connected by a 6 Ω resistor. Find the current and the force required to maintain constant velocity.
A: ε = vBL = 3 × 0.5 × 0.4 = 0.6 V. I = ε/R = 0.6/6 = 0.1 A. Force = ILB = 0.1 × 0.4 × 0.5 = 0.02 N.
Q: A circular loop of radius 5 cm is in a magnetic field that increases uniformly from 0 to 0.8 T in 0.2 s. What is the magnitude of the induced EMF?
A: Φ_i = 0. Φ_f = BA = 0.8 × π(0.05)² = 6.28 × 10⁻³ Wb. ε = ΔΦ/Δt = 6.28 × 10⁻³ / 0.2 = 0.0314 V ≈ 31.4 mV.
Q: A bar magnet is pushed toward a loop of wire. Using Lenz's Law, determine the direction of the induced current (clockwise or counterclockwise as seen from the magnet's approach side).
A: As the magnet approaches, the flux through the loop increases. By Lenz's Law, the induced current must create a field that opposes the increase, so it creates a field pointing back toward the magnet (repelling it). Using the right-hand rule, the induced current flows counterclockwise as seen from the magnet's side.
Q: A loop rotates at 60 revolutions per second in a 1.2 T field. The loop area is 0.02 m². What is the peak EMF?
A: ω = 2π × 60 = 377 rad/s. ε_max = BAω = 1.2 × 0.02 × 377 = 9.05 V.
Motional EMF ties directly back to the Lorentz force from the magnetic force topic. Faraday's Law, together with Ampere's Law, forms two of the four Maxwell's equations, which unify all of electromagnetism. The rotating-loop result is the foundation for understanding AC circuits and transformers. The braking force from motional EMF connects to energy dissipation in RC circuits (both convert one form of energy to heat in a resistor).
motional EMF, Faraday's law, Lenz's law, magnetic flux, induced EMF, electromagnetic induction, changing magnetic flux, rotating loop, AC generator, sliding bar on rails, magnetic braking, eddy currents, rate of change of flux, Weber, PHY 212, university physics, electricity and magnetism