Electromagnetic Induction — AP Physics 2 Study Guide

For: AP Physics 2 candidates sitting AP Physics 2.

Covers: magnetic flux calculation, Faraday’s Law of electromagnetic induction, Lenz’s Law for induced current direction, motional emf, and problem-solving techniques for all AP Physics 2 question types.

You should already know: Magnetic force on moving charged particles. Right-hand rules for magnetic field direction. Ohm’s Law for DC electric circuits.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Physics 2 style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Electromagnetic Induction?

Electromagnetic induction is the generation of an electric potential (called electromotive force, or emf) and resulting induced current in a conductor exposed to a changing magnetic environment. Unlike sources like batteries that produce emf from chemical energy, induction converts mechanical energy or changing magnetic energy into electrical energy, forming the physical basis for all modern generators, transformers, and wireless charging technology. Per the AP Physics 2 CED, this topic makes up 10-15% of the total exam score within Unit 5 (Magnetism and Electromagnetic Induction), and appears regularly in both multiple-choice (MCQ) and free-response (FRQ) sections, often combined with prior concepts of electric circuits and magnetic fields. Common notation conventions: we use ${\Phi }_B$ for magnetic flux, $\epsilon$ for induced emf, $B$ for magnetic field strength, $A$ for the enclosed area of a conducting loop, and $\theta$ for the angle between the magnetic field vector and the unit normal vector to the loop's area. Motional emf is a common synonym for induction from a moving conductor, but it follows the same core rules as any other induction scenario.

2. Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing through a given area, and it is the change in flux that drives electromagnetic induction, per Faraday's Law. For a flat surface of area $A$ in a uniform magnetic field, the formula for magnetic flux is: $${\Phi }_B=\vec{B}\cdot \vec{A}=BA\cos \theta$$ where $\theta$ is explicitly defined as the angle between the magnetic field vector $\vec{B}$ and the normal vector (perpendicular) to the surface. If the magnetic field is parallel to the surface, $\theta =90^{\circ }$, so $\cos \theta =0$, and flux is zero even for very large $B$. If $B$ is perpendicular to the surface, $\theta =0^{\circ }$, $\cos \theta =1$, so ${\Phi }_B=BA$, the maximum possible flux for given $B$ and $A$. Units of flux are webers (Wb), where $1\text{Wb}=1\text{T}\cdot {\text{m}}^2$. Three independent changes can alter magnetic flux through a loop to produce induction: a change in magnetic field strength $B$, a change in the area $A$ of the loop exposed to the field, or a change in orientation $\theta$ of the loop relative to the field.

Worked Example

A square conducting loop with side length 0.2 m is placed in a uniform 0.5 T magnetic field. The plane of the loop is tilted so that the angle between the plane of the loop and the magnetic field is 30°. What is the magnetic flux through the loop?

1. The flux formula uses the angle between $\vec{B}$ and the normal to the plane, not the angle between $\vec{B}$ and the plane itself. For the given 30° angle between B and the plane, the angle between B and the normal is $90^{\circ }-30^{\circ }=60^{\circ }$.
2. Calculate the area of the square: $A=(0.2\text{m}{)}^2=0.04{\text{m}}^2$.
3. Substitute into the flux formula: ${\Phi }_B=BA\cos \theta =(0.5\text{T})(0.04{\text{m}}^2)\cos (60^{\circ })$.
4. $\cos (60^{\circ })=0.5$, so ${\Phi }_B=0.5\cdot 0.04\cdot 0.5=0.01\text{Wb}$.

Exam tip: Always confirm whether the problem gives the angle relative to the plane or the normal of the loop — 70% of student flux calculation errors on the exam come from mixing these two angles up.

3. Faraday's Law of Induction

Faraday's Law is the core quantitative rule that relates changing magnetic flux to the magnitude of the induced emf. For a single conducting loop, the magnitude of the induced emf equals the magnitude of the rate of change of magnetic flux over time: $$|\epsilon |=\left|\frac{\Delta {\Phi }_B}{\Delta t}\right|$$ For a coil with $N$ identical turns of wire, the total emf is $N$ times the emf of a single turn, since each turn adds its emf in series: $$|\epsilon |=N\left|\frac{\Delta {\Phi }_B}{\Delta t}\right|$$ Faraday's Law gives the magnitude of induced emf, but not the direction of the resulting induced current. If the conductor forms a closed loop with total resistance $R$, the magnitude of the induced current is $I=|\epsilon |/R$, directly from Ohm's Law. AP Physics 2 problems most commonly test Faraday's Law for three scenarios: uniform $B$ changing at a constant rate, loop area changing at a constant rate, or a loop rotating at constant angular velocity. All can be solved by calculating the total change in flux over the given time interval.

Worked Example

A 50-turn circular coil has radius 0.1 m. The magnetic field through the coil is perpendicular to the plane of the coil, and increases at a constant rate from 0.1 T to 0.6 T over 0.25 s. The total resistance of the coil is 2.0 Ω. What is the magnitude of the induced current in the coil?

1. Calculate the area of the coil: $A=\pi r^2=\pi (0.1\text{m}{)}^2\approx 0.0314{\text{m}}^2$. $\theta =0^{\circ }$, so $\cos \theta =1$.
2. Calculate the change in flux per turn: $\Delta {\Phi }_B=A\Delta B=0.0314{\text{m}}^2(0.6\text{T}-0.1\text{T})=0.0157\text{Wb}$.
3. Apply Faraday's Law for $N$ turns to find induced emf: $|\epsilon |=N|\Delta {\Phi }_B/\Delta t|=50\cdot (0.0157\text{Wb}/0.25\text{s})\approx 3.14\text{V}$.
4. Use Ohm's Law to find current: $I=\epsilon /R=3.14\text{V}/2.0\Omega \approx 1.6\text{A}$ (2 sig figs).

Exam tip: If the problem describes a multi-turn coil, always scan for the number of turns $N$ and explicitly write it into your calculation — forgetting the $N$ multiplier is the second most common error on AP induction problems.

4. Lenz's Law for Direction of Induced Current

Lenz's Law is the rule that predicts the direction of induced current produced by changing flux. The law states: The induced current will flow in a direction such that the magnetic field produced by the induced current opposes the change in the original magnetic flux that created it. A common student misstatement is that the induced field "opposes the original magnetic field" — this is incorrect; it opposes the change in flux, not the field itself. To apply Lenz's Law correctly, follow this three-step process: (1) Determine the direction of the original magnetic field through the loop, and whether flux is increasing or decreasing. (2) The induced magnetic field will point opposite to the change: if original flux is increasing, induced $B$ points opposite original $B$; if original flux is decreasing, induced $B$ points in the same direction as original $B$. (3) Use the right-hand rule for current-carrying loops to find the direction of induced current that produces the required induced $B$.

Worked Example

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

1. The original magnetic field from the bar magnet points outward from the north pole, so through the loop, original $B$ points toward the viewer (on the side the magnet is approaching from). The magnet is moving toward the loop, so flux through the loop is increasing.
2. By Lenz's Law, the induced magnetic field must oppose the increase in flux, so induced $B$ points opposite to the original $B$, which is away from the viewer (toward the bar magnet).
3. Use the right-hand rule for current loops: point your right thumb in the direction of the induced $B$ (away from the viewer), and your fingers curl in the direction of current. This gives a clockwise current direction when viewed from the side of the approaching magnet.

Exam tip: Always start with "is flux increasing or decreasing?" before predicting direction — this one question eliminates 90% of direction errors.

5. Motional Emf

Motional emf is the emf induced in a moving conductor in a magnetic field, a special case of Faraday's Law where flux changes because the area of the conducting loop changes. For a conducting rod of length $L$ moving with speed $v$ perpendicular to a uniform magnetic field $B$, the motional emf across the ends of the rod is: $$\epsilon =BLv$$ This formula can be derived directly from Faraday's Law: if the rod slides along parallel conducting rails to form a closed loop, the area of the loop increases by $\Delta A=Lv\Delta t$ in time $\Delta t$, so $\Delta {\Phi }_B=B\Delta A=BLv\Delta t$, and $\epsilon =\Delta {\Phi }_B/\Delta t=BLv$. It can also be derived from the magnetic force on charges in the moving rod: $F=qvB$, which separates charge until the electric force balances the magnetic force, giving $E=vB$ and $\epsilon =EL=BLv$. For a closed loop with total resistance $R$, the induced current is $I=\epsilon /R=BLv/R$.

Worked Example

A 0.5 m long conducting bar slides at constant speed 4 m/s along two parallel conducting rails connected by a 10 Ω resistor at one end. The entire system is in a uniform 0.2 T magnetic field perpendicular to the plane of the loop. What is the magnitude and direction of the current through the resistor, if the bar slides away from the resistor?

1. Calculate motional emf: $\epsilon =BLv=(0.2\text{T})(0.5\text{m})(4\text{m/s})=0.4\text{V}$.
2. Use Ohm's Law for current: $I=\epsilon /R=0.4\text{V}/10\Omega =0.04\text{A}$.
3. For direction: flux into the plane of the loop increases as the bar slides away from the resistor. By Lenz's Law, induced B points out of the plane, so current flows counterclockwise around the loop, which means it flows from top to bottom through the resistor.

Exam tip: Motional emf only exists for velocity components perpendicular to both B and the length of the rod. If the rod moves parallel to its own length, the induced emf is zero.

6. Common Pitfalls (and how to avoid them)

• Wrong move: Using the angle between the magnetic field and the plane of the loop directly as $\theta$ in the flux formula. Why: Students forget $\theta$ is defined relative to the normal vector, and it feels natural to use the angle given in the diagram directly. Correct move: Always subtract the given angle from 90° if the problem gives the angle relative to the plane of the loop before plugging into the flux formula.
• Wrong move: Forgetting to multiply induced emf by the number of turns $N$ for a multi-turn coil. Why: Students remember Faraday's Law for a single loop and stop there, assuming the problem asks for flux per turn. Correct move: Scan any problem mentioning a coil for the number of turns, write $N$ at the top of your working, and always multiply by $N$ when calculating total emf.
• Wrong move: Stating that Lenz's Law requires the induced magnetic field to oppose the original magnetic field. Why: The common phrasing "opposes the change" is often misremembered as opposing the field, leading to wrong direction when flux is decreasing. Correct move: Always first confirm if flux is increasing or decreasing; only oppose the original field when flux is increasing.
• Wrong move: Calculating motional emf as $BLv$ when the rod moves parallel to its length. Why: Students memorize $BLv$ and forget the perpendicular requirement for all three quantities. Correct move: If velocity is parallel to the rod's length, no flux change occurs, so induced emf is zero.
• Wrong move: Assuming any moving conductor in a magnetic field has an induced current. Why: Students confuse induced emf with induced current. Correct move: Check if the conductor is a closed loop; an isolated moving rod has induced emf (potential difference) but no induced current.

7. Practice Questions (AP Physics 2 Style)

Question 1 (Multiple Choice)

A student moves a bar magnet at constant speed completely through a circular conducting loop, north pole first, entering from the left side and exiting out the right side. Which of the following correctly describes the induced current in the loop as viewed from the left (entry) side? A: The current is always clockwise the entire time the magnet passes through the loop B: The current is clockwise when the magnet enters, and counterclockwise when the magnet exits C: The current is counterclockwise when the magnet enters, and clockwise when the magnet exits D: The current is always zero because the speed of the magnet is constant

Worked Solution: Apply Lenz's Law step by step. When the magnet enters, the north pole approaches the left side, so flux pointing to the right (toward the exit side) increases through the loop. The induced field must oppose this increase, so induced B points to the left. Using the right-hand rule with thumb pointing left, the current curls clockwise when viewed from the left. When the magnet exits the right side, the north pole moves away from the left side, so flux pointing right is decreasing. Induced B points right to oppose the decrease, so thumb points right, and current curls counterclockwise when viewed from the left. Constant speed does not eliminate flux change. The correct answer is B.

Question 2 (Free Response)

A rectangular loop of wire with mass $m$, resistance $R$, vertical length $L$, and horizontal width $w$ is dropped vertically from rest into a region of uniform horizontal magnetic field $B$ perpendicular to the plane of the loop, pointing into the page. The top edge of the loop has not entered the field when the bottom edge is already inside. (a) Derive an expression for the magnitude of the induced current in the loop at the instant when it has a downward speed $v$. (b) Derive an expression for the magnitude and direction of the net magnetic force on the loop at this instant. (c) Explain why the loop will eventually reach a constant terminal speed, and derive an expression for terminal speed in terms of given quantities.

Worked Solution: (a) The bottom edge of the loop cuts through magnetic field lines, producing motional emf. $\epsilon =BLv$, so by Ohm's Law: $$I=\frac{\epsilon }{R}=\frac{BLv}{R}$$ (b) The magnetic force on the current-carrying bottom edge is $F=ILB$. Substitute $I$ from (a): $$F=\left(\frac{BLv}{R}\right)LB=\frac{B^2{L}^{2}v}{R}$$ By Lenz's Law, force opposes the downward motion of the loop, so the net magnetic force points upward. (c) As the loop accelerates downward due to gravity, speed $v$ increases, so the upward magnetic force increases proportionally. When the upward magnetic force equals the downward gravitational force, net force is zero, acceleration is zero, and speed becomes constant (terminal speed). Set $F=mg$: $$\frac{B^2{L}^2{v}_t}{R}=mg⟹v_t=\frac{mgR}{B^2{L}^2}$$

Question 3 (Application / Real-World Style)

A small hand-cranked phone charger has a 150-turn coil of area $0.002{\text{m}}^2$ that you rotate by hand at 2 revolutions per second in a 0.15 T permanent magnetic field. What is the average magnitude of the induced emf when the coil rotates 90 degrees from perpendicular to parallel relative to the magnetic field? Interpret your result in context.

Worked Solution: At 2 revolutions per second, the time for one full revolution is $T=1/2=0.5\text{s}$, so the time to rotate 90° (1/4 revolution) is $\Delta t=0.5/4=0.125\text{s}$. The initial flux per turn is ${\Phi }_i=BA=(0.15\text{T})(0.002{\text{m}}^2)=0.0003\text{Wb}$, and final flux is ${\Phi }_f=0$, so $|\Delta {\Phi }_B|=0.0003\text{Wb}$. Apply Faraday's Law: $$|\epsilon |=N\left|\frac{\Delta {\Phi }_B}{\Delta t}\right|=150\cdot \frac{0.0003\text{Wb}}{0.125\text{s}}=0.36\text{V}$$ This average output is enough to trickle-charge a modern USB phone, which typically accepts input voltages as low as 0.5 V, so cranking faster than 2 revolutions per second would produce enough voltage to charge the device.

8. Quick Reference Cheatsheet

9. What's Next

Electromagnetic induction is the foundational prerequisite for the remaining topics in Unit 5 of AP Physics 2: alternating current circuits and transformers, which rely entirely on Faraday’s Law of induction to function. Without mastering flux calculation, Faraday’s Law, and Lenz’s Law, you will not be able to solve for voltage and current ratios in transformers, or understand how AC power generation and bulk transmission works. Beyond Unit 5, induction is also the core physical principle behind electromagnetic waves, which rely on mutually inducing changing electric and magnetic fields to propagate through space, covered later in the course. The reasoning skills you built here for relating changing fields to induced emf are directly transferable to those more advanced topics.

AC Circuits and Transformers Electromagnetic Waves Magnetic Forces and Fields Review