Magnetic Flux, Induced EMF, Faraday's and Lenz's Law

AP Physics 2· AP Physics 2 CED — Magnetism and Electromagnetic Induction· 14 min read

1. Magnetic Flux★★☆☆☆⏱ 4 min

Magnetic flux ( $Φ_{B}$ ) is a scalar quantity that measures the total magnetic field passing through a defined surface area, analogous to electric flux from Gauss's Law. For induction problems, we almost always work with flat open surfaces bounded by a closed loop of wire.

📘 Definition

Magnetic Flux (uniform field)

$Φ_{B}$

For a uniform magnetic field, magnetic flux is calculated as the product of field strength, loop area, and the cosine of the angle between the field and the normal to the surface. The SI unit is the weber (Wb).

Example:

$1 Wb = 1 T \cdotp m^{2}$ ; flux is zero when the field is parallel to the surface, maximum when perpendicular.

\Phi_B = B A \cos\theta

Where $B$ is magnetic field magnitude, $A$ is surface area, and $θ$ is the angle between the magnetic field vector and the normal (perpendicular) vector to the surface. Flux can change for three reasons, all of which produce induced EMF: $B$ changes, $A$ changes, or $θ$ changes.

📐 Worked Example

A circular wire loop with radius 0.10 m is placed in a uniform 0.50 T magnetic field. The plane of the loop makes a 30° angle with the magnetic field vector. What is the magnetic flux through the loop?

1
Calculate the area of the loop:
2
$A = \pi r^2 = \pi (0.10\ \text{m})^2 = 0.01\pi \approx 0.0314\ \text{m}^2$
3
Recall that $θ$ is measured relative to the normal, not the plane of the loop. If the plane is 30° to $B$ , the normal is $9 0^{\circ} - 3 0^{\circ} = 6 0^{\circ}$ to $B$ , so $θ = 6 0^{\circ}$ .
4
Substitute into the flux formula:
5
$\Phi_B = BA\cos\theta = (0.50\ \text{T})(0.0314\ \text{m}^2)(\cos 60^\circ)$
6
Calculate the final value, since $cos 6 0^{\circ} = 0.5$ :
7
$\Phi_B = 7.9 \times 10^{-3}\ \text{Wb}$

Exam tip:

Always confirm you are using the angle between the magnetic field and the normal to the loop, not the plane of the loop. This is the most common mistake on introductory flux calculation MCQs.

2. Faraday's Law of Induction★★★☆☆⏱ 5 min

Faraday's Law of Induction formalizes the relationship between changing magnetic flux and induced EMF. It states that the induced electromotive force around a closed loop is equal to the negative rate of change of magnetic flux through the loop, multiplied by the number of turns in the coil.

\varepsilon = -N \frac{d\Phi_B}{dt}

For a finite change in flux over a time interval $Δ t$ , we use the average induced EMF form:

\varepsilon_{\text{avg}} = -N \frac{\Delta \Phi_B}{\Delta t}

Each turn of the coil experiences the same flux change, so EMF adds in series, hence multiplication by $N$ . The negative sign encodes direction information, which we handle separately with Lenz's Law, so we almost always just calculate the magnitude of EMF first. A common special case is motional EMF, which simplifies to $ε = B Lv$ when speed $v$ is perpendicular to $B$ and rod length $L$ .

📐 Worked Example

A coil with 200 turns of wire has a cross-sectional area of 0.0025 m². The plane of the coil is perpendicular to a uniform magnetic field that increases linearly from 0.10 T to 0.35 T in 0.50 seconds. What is the magnitude of the average induced EMF in the coil?

1
Confirm angle: plane perpendicular to $B$ means normal is parallel to $B$ , so $θ = 0$ , $cos θ = 1$ .
2
Calculate initial and final flux per turn:
3
$\Phi_{B,i} = B_i A = (0.10\ \text{T})(0.0025\ \text{m}^2) = 2.5 \times 10^{-4}\ \text{Wb} \\ \Phi_{B,f} = B_f A = (0.35\ \text{T})(0.0025\ \text{m}^2) = 8.75 \times 10^{-4}\ \text{Wb}$
4
Find the change in flux:
5
$\Delta \Phi_B = \Phi_{B,f} - \Phi_{B,i} = 6.25 \times 10^{-4}\ \text{Wb}$
6
Apply Faraday's Law for average EMF, taking only the magnitude:
7
$|\varepsilon_{\text{avg}}| = N \left|\frac{\Delta \Phi_B}{\Delta t}\right| = 200 \cdot \frac{6.25 \times 10^{-4}\ \text{Wb}}{0.50\ \text{s}} = 0.25\ \text{V}$

Exam tip:

Write $N$ explicitly into your Faraday's Law equation at the start of every problem, even if $N = 1$ . This eliminates the common mistake of forgetting to multiply by $N$ for multi-turn coils on FRQs.

3. Lenz's Law★★★☆☆⏱ 3 min

Lenz's Law gives the direction of induced EMF and induced current in a closed conducting loop. It states: The induced current flows in a direction that creates a magnetic field that opposes the change in magnetic flux that produced the induction. A critical point: Lenz's Law opposes the change in flux, not the flux itself — this is the most common point of confusion for students.

Find the direction of the original magnetic field through the loop.
Determine if the total flux through the loop is increasing or decreasing.
If flux is increasing, the induced magnetic field points opposite the original field; if flux is decreasing, the induced magnetic field points in the same direction as the original field.
Use the right-hand rule for current loops to find the direction of induced current from the direction of the induced magnetic field.

📐 Worked Example

A north pole of a bar magnet is moving toward a stationary circular conducting loop along the loop's central axis. What is the direction of the induced current in the loop, as viewed from the side where the magnet is approaching?

1
Original magnetic field direction: Magnetic field lines exit the north pole, so through the loop, original $B$ points toward the viewer (viewed from the magnet's side).
2
Change in flux: As the magnet approaches, the strength of $B$ through the loop increases, so flux is increasing.
3
Induced magnetic field direction: Lenz's Law requires induced $B$ to oppose the increase, so it points opposite original $B$ : away from the viewer, through the loop toward the back side.
4
Right-hand rule: Curl your right hand's fingers in the direction of current; your thumb points in the direction of induced $B$ . With thumb pointing away from you, fingers curl clockwise, so induced current is clockwise as viewed from the magnet side.

Exam tip:

Always explicitly note whether flux is increasing or decreasing before finding the direction of induced $B$ . This step prevents the common mistake of always making induced $B$ opposite the original field.

4. Common AP Problem Types★★★★☆⏱ 5 min

✓ Calculator OK

📐 Worked Example

A square loop of wire with side length 0.2 m is moved at constant speed from a region of zero magnetic field into a uniform 0.4 T magnetic field, where the field is perpendicular to the plane of the loop. Which of the following correctly describes the magnitude of induced EMF as a function of distance moved, from when the leading edge enters the field to when the entire loop is fully inside the field?

A) EMF increases linearly from 0 to maximum when the loop is fully inside B) EMF is constant and non-zero from entry to full immersion, then drops to zero C) EMF is constant and non-zero the entire time, including when fully inside D) EMF decreases linearly from maximum to zero when the loop is fully inside

1
As the loop enters the field, the area inside the field increases at a constant rate because speed is constant:
2
$\frac{dA}{dt} = L \frac{dx}{dt} = Lv = \text{constant}$
3
The rate of change of flux $d Φ_{B} / d t = B d A / d t$ is therefore constant, so EMF magnitude is constant while the loop enters.
4
Once the entire loop is inside, flux is constant, so $d Φ_{B} / d t = 0$ and EMF drops to zero. This matches option B.

📐 Worked Example

A 0.20 m long conducting rod slides at constant speed 3.0 m/s along two parallel conducting rails connected to a 6.0 Ω resistor, forming a closed rectangular loop. The entire setup is in a uniform 1.5 T magnetic field perpendicular to the plane of the loop, pointing into the page. The resistance of the rod and rails can be neglected.

(a) Calculate the magnitude of the induced EMF in the loop. (b) Find the magnitude of the induced current in the loop and the direction of current through the rod. (c) Calculate the force required to keep the rod moving at constant speed to the right, and explain why a force is needed.

1
(a) $B$ , $L$ , and $v$ are mutually perpendicular, so apply the motional EMF formula:
2
$|\varepsilon| = BLv = (1.5\ \text{T})(0.20\ \text{m})(3.0\ \text{m/s}) = 0.90\ \text{V}$
3
(b) Use Ohm's Law to find current magnitude:
4
$I = \frac{\varepsilon}{R} = \frac{0.90\ \text{V}}{6.0\ \Omega} = 0.15\ \text{A}$
5
For direction: Flux into the page is increasing as the rod moves right, so induced $B$ must point out of the page to oppose the increase. A counterclockwise current produces out-of-page $B$ inside the loop, so current flows upward through the sliding rod.
6
(c) Calculate the magnetic force on the current-carrying rod:
7
$F = ILB = (0.15\ \text{A})(0.20\ \text{m})(1.5\ \text{T}) = 0.045\ \text{N}$
8
By Lenz's Law, this force opposes the motion of the rod, pulling it to the left. For constant speed, net force must be zero, so an external force of 0.045 N pointing to the right is required to counteract the magnetic drag force.

📐 Worked Example

A small portable AC generator for camping uses a 500-turn circular coil with area 0.015 m² rotating at 60 Hz (60 rotations per second) in a uniform 0.10 T permanent magnet field. What is the maximum induced EMF produced by this generator, and is it sufficient to power a portable device that requires a maximum input voltage of 120 V?

1
For a rotating coil, flux as a function of time is $Φ_{B} (t) = B A cos (ω t)$ , where angular frequency $ω = 2 π f$ :
2
$\omega = 2\pi (60\ \text{Hz}) = 120\pi\ \text{rad/s}$
3
Apply Faraday's Law to find EMF magnitude:
4
$|\varepsilon| = N \left|\frac{d\Phi_B}{dt}\right| = NBA\omega |\sin(\omega t)|$
5
Maximum EMF occurs when $∣ sin (ω t) ∣ = 1$ , so:
6
$\varepsilon_{\text{max}} = NBA\omega = 500(0.10\ \text{T})(0.015\ \text{m}^2)(120\pi) \approx 283\ \text{V}$
7
This maximum voltage is higher than the 120 V requirement, so the generator can produce enough voltage to power the device.

5. Common Pitfalls

Wrong move:

Uses the angle between the magnetic field and the plane of the loop for $θ$ in the flux formula.

Why:

Problems often give the angle between the plane and $B$ directly, leading students to confuse the definition of $θ$ .

Correct move:

Always draw the normal vector to the loop, then measure $θ$ between $B$ and this normal before plugging into the flux formula.

Wrong move:

Forgets to multiply induced EMF by the number of turns $N$ for a multi-turn coil.

Why:

Students treat a coil the same as a single loop, forgetting each turn adds EMF in series.

Correct move:

Always write $N$ explicitly in your Faraday's Law equation at the start of the problem.

Wrong move:

Claims induced magnetic field always opposes the original magnetic field, regardless of flux change direction.

Why:

Students misremember Lenz's Law as 'opposes the magnetic field' instead of 'opposes the change in flux'.

Correct move:

After finding original $B$ direction, always first note if flux is increasing or decreasing, then set induced $B$ direction: opposite for increasing, same for decreasing.

Wrong move:

Uses the original flux value instead of the change in flux when calculating induced EMF.

Why:

Students confuse flux with change in flux, especially for rotation problems that take a loop from maximum to zero flux.

Correct move:

Always calculate $Δ Φ_{B} = Φ_{f ina l} - Φ_{ini t ia l}$ , never just use final or initial flux alone.

Wrong move:

For motional EMF, uses full speed $v$ even when velocity is not perpendicular to $B$ and the rod.

Why:

Students memorize $ε = B Lv$ without remembering the restriction that $v$ must be the perpendicular component.

Correct move:

Derive motional EMF from Faraday's Law directly if velocity is at an angle, to avoid component errors.

6. Quick Reference Cheatsheet

Category	Formula	Notes
Magnetic Flux (uniform $B$ )	$Φ_{B} = B A cos θ$	$θ$ = angle between $B$ and normal to surface; unit: Wb ( $1 Wb = 1 T \cdotp m^{2}$ )
Faraday's Law (instantaneous EMF)	$ε = - N \frac{d Φ _{B}}{d t}$	$N$ = number of turns; negative sign indicates direction per Lenz
Faraday's Law (average EMF)	$ε_{avg} = - N \frac{Δ Φ _{B}}{Δ t}$	Use for finite time interval flux changes
Motional EMF (perpendicular $B, L, v$ )	$ε = B Lv$	Only valid when all three are mutually perpendicular
Lenz's Law (increasing flux)	Induced $B$ = opposite original $B$	Opposes the increase in flux
Lenz's Law (decreasing flux)	Induced $B$ = same as original $B$	Opposes the decrease in flux
Maximum EMF (rotating AC generator)	$ε_{max} = N B A ω$	$ω = 2 π f$ , $f$ = rotation frequency in Hz
Magnetic drag force (motional rod)	$F = I L B$	Opposes motion of the rod per Lenz's Law

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · MCQ
Flux angle calculation
2022 · FRQ
Motional EMF problem

What's Next

This topic is the foundational prerequisite for all remaining electromagnetic induction concepts in AP Physics 2 Unit 5. Next you will apply Faraday’s and Lenz’s Law to analyze the behavior of mutual inductance, transformers, and alternating current (AC) circuits, which make up a large share of Unit 5 exam questions. Without mastering flux calculation, Lenz’s direction rules, and Faraday’s magnitude calculation, you will not be able to solve transformer voltage/current problems or energy conservation questions for induction systems, which regularly appear on FRQs. This topic also sets up the study of Maxwell’s equations, where changing magnetic fields induce changing electric fields to form electromagnetic waves.