Source: Physics 212 Spring 2022, University of Illinois at Urbana-Champaign
Tags: Faraday's law, electromagnetic induction, magnetic flux, Lenz's law, induced EMF, induced current direction, changing magnetic field, PHYS 212, electromagnetism
Difficulty: Intermediate Prerequisites: Magnetic fields and forces (Ch. on magnetism), vector dot products, right-hand rule for magnetic force, basic calculus (derivatives).
Faraday's law is one of Maxwell's four equations and describes how a changing magnetic field creates an electric field (and therefore an EMF). This is the bridge between electricity and magnetism: without it, there are no generators, no transformers, and no wireless charging. You need to be comfortable with magnetic fields, the dot product, and surface integrals before tackling this material. If you skipped the chapter on magnetic flux, go back and read it first.
A changing magnetic flux through a loop induces an EMF (voltage) in that loop. The induced current flows in whichever direction opposes the change in flux, per Lenz's law. The magnitude of the EMF equals the rate of change of flux.
Magnetic flux (Φ_B)
The total amount of magnetic field passing through a surface. Formally: Φ_B = ∫ B⃗ · dA⃗, where dA⃗ is the area element with direction along the surface normal. For a uniform field through a flat loop: Φ_B = BA cos θ.
In simple terms, think of it as "how much magnetic field threads through your loop." More field, bigger loop, or better alignment all mean more flux.
Faraday's law of induction
The induced EMF in a loop equals the negative rate of change of magnetic flux through that loop: ε = −dΦ_B/dt. For N turns of wire: ε = −N dΦ_B/dt.
In simple terms, this means a changing flux creates a voltage. The faster the change, the bigger the voltage.
Lenz's law
The direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is the physical content of the minus sign in Faraday's law.
Think of it as nature's resistance to change. If the flux through your loop is increasing, the induced current creates its own magnetic field pointing the opposite way, trying to keep the flux from growing.
Electromotive force (EMF, ε)
Despite the name, EMF is not a force. It is a voltage (measured in volts) that drives current around a circuit. In induction problems, it is produced by the changing flux rather than by a battery.
In simple terms, EMF is the "push" that makes charges move around the loop.
Induced current
The current that flows in a conducting loop as a result of an induced EMF. Its direction is determined by Lenz's law, and its magnitude is I = ε / R, where R is the resistance of the loop.
Faraday's law says EMF depends on dΦ/dt. Since Φ = BA cos θ, the flux can change if any of three things change:
The magnetic field B changes with time. The loop sits still, but someone turns up or down the external field. This is the scenario in questions 1 and 2 of the problem set.
The area A of the loop changes. A sliding bar on rails makes the enclosed area grow or shrink. This is the motional-EMF scenario (covered in Part 2).
The angle θ between B⃗ and the area normal changes. A rotating loop in a constant field. This is how generators work.
If none of these is changing, dΦ/dt = 0 and there is no induced EMF.
Determine the direction of the external magnetic field through the loop.
Determine whether the flux through the loop is increasing, decreasing, or constant.
If increasing: the induced current creates a field opposing the external field (i.e. in the opposite direction inside the loop).
If decreasing: the induced current creates a field in the same direction as the external field, trying to maintain the flux.
Use the right-hand rule to convert the direction of the induced B field into a current direction.
If the flux is constant: no induced current.
A square wire loop sits in a uniform magnetic field B⃗ pointing out of the page (+z⃗ direction). The field magnitude B_z(t) varies according to a graph:
From t = 0 to roughly t = 2 s, B_z increases (positive slope).
From t = 2 s to t = 4 s, B_z is constant (zero slope).
From t = 4 s to t = 6 s, B_z decreases (negative slope).
At t = α = 3 s (constant region): The slope of B_z(t) is zero, so dΦ/dt = 0. There is no induced EMF and no induced current. Answer: (c).
At t = β = 5 s (decreasing region): The flux is decreasing (B_z is getting smaller while still positive and pointing out of the page). By Lenz's law, the induced current must try to maintain the flux, so it produces a field out of the page inside the loop. By the right-hand rule, this requires a counterclockwise current (as viewed from above). Answer: (b).
A conducting bar slides in the −x⃗ direction along a U-shaped wire, forming a closed rectangular loop. The magnetic field B⃗ = B₀(+z⃗) is constant and points out of the page.
The bar moves left, so the enclosed area of the loop is decreasing.
The field is constant and out of the page, so with decreasing area, the flux Φ = BA is decreasing.
By Lenz's law, the induced current opposes the decrease, meaning it tries to maintain flux out of the page.
Right-hand rule: current flows counterclockwise around the loop.
In the bar itself, this means current flows from bottom to top. Answer: (b).
A square wire loop rotates around the y-axis in a constant field B⃗ = B₀(+x⃗). The angle θ (between the loop normal and B⃗) is increasing at the moment shown.
Φ = BA cos θ. Since θ is increasing from a value where cos θ > 0, the flux is decreasing (cos θ is getting smaller).
By Lenz's law, the induced current opposes the decrease, trying to keep flux positive.
Working out the geometry (the normal is rotating away from +x⃗), the current flows from α to β. Answer: (a).
Quantity | Formula | Units |
|---|---|---|
Magnetic flux | Φ_B = ∫ B⃗ · dA⃗ = BA cos θ (uniform B, flat surface) | T·m² (Weber, Wb) |
Faraday's law | ε = −dΦ_B / dt | V (Volts) |
Faraday's law (N turns) | ε = −N dΦ_B / dt | V |
Induced current | I = ε / R | A (Amps) |
Faraday's law is why generators produce electricity: a coil rotating in a magnetic field experiences a continuously changing flux, which produces an alternating EMF. Every coal, gas, nuclear, wind, and hydroelectric power station relies on this principle. Wireless phone chargers work on the same idea: a changing current in the charging pad creates a changing magnetic field, which induces an EMF in a coil inside the phone.
Students often think a strong magnetic field alone is enough to induce a current. It is not. The flux must be changing. A loop sitting in a strong but constant field has zero induced EMF.
Students sometimes confuse the direction of the induced current with the direction of the external magnetic field. Lenz's law says the induced current's own field opposes the change in flux, not the flux itself.
A common error is forgetting that Φ = BA cos θ. If B is perpendicular to the area normal (θ = 90°), the flux is zero regardless of how strong the field is.
Students occasionally apply Lenz's law backwards, creating a current that reinforces the change rather than opposing it. If that were the case, you would get runaway energy from nothing, which violates conservation of energy.
⚠️ "No change, no EMF" is the single most tested idea. If B, A, and θ are all constant, the answer is zero.
⚠️ Lenz's law direction problems appear on nearly every exam. Practise the step-by-step method above until it is automatic.
⚠️ Be careful with signs. The minus sign in Faraday's law encodes Lenz's law. If you define a positive normal direction and a positive current direction (via the right-hand rule), the signs take care of themselves.
⚠️ Know the three ways flux can change (B, A, θ). Exam questions often test whether you can identify which one is changing.
True or false: A loop in a strong, constant, uniform magnetic field has an induced EMF.
Fill in the blank: The induced EMF in a loop is proportional to the ______ of change of magnetic flux.
True or false: If the magnetic flux through a loop is increasing, the induced current creates a magnetic field in the same direction as the external field.
Fill in the blank: The unit of magnetic flux is the ______.
True or false: Doubling the rate at which the magnetic field changes will double the induced EMF.
Answers: 1. False. 2. Rate. 3. False (it opposes the change). 4. Weber (Wb), or equivalently T·m². 5. True.
Q: A circular loop sits in a uniform magnetic field that points to the right and is increasing in magnitude. In what direction does the induced current flow (as viewed from the right)?
A: Clockwise (as viewed from the right). The flux is increasing to the right, so the induced current opposes this by creating a field to the left inside the loop, which requires clockwise current by the right-hand rule.
Q: A square loop is being pulled out of a region of uniform magnetic field pointing into the page. Does the induced current flow clockwise or counterclockwise?
A: Counterclockwise. The flux into the page is decreasing (less area in the field region), so the induced current tries to maintain flux into the page. By the right-hand rule, this is clockwise. Wait, let us be careful: the induced B must point into the page to oppose the decrease, and by the right-hand rule (curl fingers in the direction of current, thumb points in direction of B), that requires clockwise current. So the answer is clockwise.
Q: A loop of wire is in a region where B is constant in magnitude but the loop is being rotated so that its normal goes from parallel to B to perpendicular to B. Is there an induced EMF during this rotation?
A: Yes. The angle θ is changing, so Φ = BA cos θ is changing, and there is a nonzero dΦ/dt.
Q: At the instant when a rotating loop's normal is exactly perpendicular to B (θ = 90°), is the flux a maximum, minimum, or zero? Is the EMF a maximum, minimum, or zero?
A: The flux is zero (cos 90° = 0). The EMF is at its maximum magnitude, because dΦ/dt is largest when Φ passes through zero (the cosine function changes fastest at its zero crossings).
This connects directly to Maxwell's equations: Faraday's law is one of the four. It also sets the stage for inductance (self and mutual), where the changing current in one circuit induces an EMF in itself or a nearby circuit. In AC circuit analysis, the voltage across an inductor comes from Faraday's law applied to the inductor's own changing flux.
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