Lenz's Law — AP Physics C: E&M Study Guide

For: AP Physics C: E&M candidates sitting AP Physics C: E&M.

Covers: This chapter covers the definition of Lenz's Law, step-by-step techniques to find direction of induced current, induced emf, and induced magnetic fields, with applications to sliding conducting rails and moving loops in magnetic fields.

You should already know: Faraday's Law of electromagnetic induction; right-hand rule for magnetic fields from current loops; magnetic flux calculation.

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 C: E&M 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 Lenz's Law?

Lenz's Law is a fundamental rule of electromagnetic induction that predicts the direction of the induced current and induced emf generated by a changing magnetic flux. It is often described as the law of inertia for electromagnetism, because it states that the induced effect always opposes the change that created it. According to the AP Physics C: E&M Course and Exam Description (CED), Lenz's Law is a core concept within Unit 5: Electromagnetism, which accounts for 16-24% of the total exam score. It appears on both multiple-choice (MCQ) and free-response (FRQ) sections: it is required for full credit in nearly all induction FRQ problems, and is a common source of conceptual MCQ questions testing for misconceptions. The formal statement of Lenz's Law is: The induced current flows in a direction such that the induced magnetic field created by the current opposes the change in magnetic flux that produced the induction. A standard notation convention defines positive flux as pointing in the direction of the loop's area normal vector (set by the right-hand rule), with negative induced emf corresponding to a direction that opposes the change in flux.

2. The Core 4-Step Method for Applying Lenz's Law

The most common student mistake when starting with Lenz's Law is confusing "opposing change" with "opposing the original magnetic field". Lenz's Law does not say the induced field always points opposite the original field: it opposes the change in flux, so the induced field can point in the same direction as the original field if the original flux is decreasing. The consistent 4-step method to get the correct direction every time is:

1. Identify the direction of the original external magnetic field $\vec{B}$ through the conducting loop.
2. Determine whether the total magnetic flux ${\Phi }_B=\vec{B}\cdot \vec{A}$ through the loop is increasing or decreasing.
3. Assign the direction of the induced magnetic field ${\vec{B}}_{ind}$: if flux is increasing, ${\vec{B}}_{ind}$ points opposite to $\vec{B}$; if flux is decreasing, ${\vec{B}}_{ind}$ points in the same direction as $\vec{B}$.
4. Use the right-hand rule for current loops: point your right thumb in the direction of ${\vec{B}}_{ind}$, and your fingers curl in the direction of the induced current.

Worked Example

A circular loop of wire lies flat on a horizontal table. The external magnetic field through the loop points straight up, out of the table, and is steadily decreasing in magnitude. What is the direction of the induced current, as viewed from above the table?

1. Original $\vec{B}$ points up, out of the table through the loop.
2. Flux ${\Phi }_B$ is decreasing, because $|\vec{B}|$ decreases and the loop area is constant.
3. For decreasing flux, ${\vec{B}}_{ind}$ points in the same direction as the original $\vec{B}$, so ${\vec{B}}_{ind}$ is also up out of the table.
4. Point the right thumb up (direction of ${\vec{B}}_{ind}$): the fingers curl counterclockwise. The induced current is counterclockwise when viewed from above.

Exam tip: Always specify your reference frame for direction (e.g., "as viewed from above the loop") on the AP exam, to avoid losing points for ambiguous answers.

3. Lenz's Law for Moving Conducting Loops

One of the most common exam problem types is a conducting loop moving through a region of magnetic field, entering or exiting a uniform B region. Flux changes here because the area of the loop inside the B region changes, so induction occurs. A useful shortcut from Lenz's Law applies directly to the net magnetic force on the loop: the net force always opposes the motion of the loop, because the motion causes the flux change, so the induced force resists that change. If the loop is entering the field, flux is increasing, so force pulls the loop out (opposing entry); if exiting, flux is decreasing, so force pulls the loop back in (opposing exit).

Worked Example

A square conducting loop moves to the right at constant speed $v$, entering a large region of uniform magnetic field pointing into the page. What is the direction of the induced current and the net magnetic force on the loop?

1. Original $\vec{B}$ points into the page, and the area of the loop inside the B region increases as it moves right, so flux into the page is increasing.
2. To oppose the increasing flux, ${\vec{B}}_{ind}$ points out of the page.
3. Right-hand rule: thumb pointing out of the page gives a counterclockwise induced current when viewed from the front.
4. Only the leading edge of the loop (already inside the B field) carries current in a magnetic field, so net force acts here. Current in the leading edge flows upward, so $F=I\vec{L}\times \vec{B}$ gives force pointing to the left, opposite to the direction of motion, matching the Lenz's force shortcut.

Exam tip: If a loop moves entirely inside a uniform magnetic field, flux through the loop is constant, so there is no induced current and no net force — this is a common distractor in MCQs.

4. Lenz's Law for Sliding Conducting Rail Problems

Sliding conducting rail setups are the most common FRQ context for motional emf, and Lenz's Law is required to find the direction of current through the circuit's resistor and the force on the sliding bar. The setup forms a closed conducting loop with fixed rails connected by a resistor at one end, and a movable conducting bar that changes the area of the loop as it slides. Flux changes proportionally to the change in area, so Lenz's Law can be applied directly to the changing area to find current direction.

Worked Example

Two parallel horizontal rails separated by distance $d$ are connected at their left end by a resistor $R$. A uniform magnetic field $B$ points vertically upward, perpendicular to the plane of the rails. A conducting bar slides to the right along the rails, increasing the area of the loop. What is the direction of current through the resistor $R$, as viewed from above?

1. Original $\vec{B}$ points upward through the loop. As the bar moves right, area increases, so upward flux through the loop is increasing.
2. To oppose the increasing upward flux, ${\vec{B}}_{ind}$ points downward through the loop.
3. Right-hand rule: thumb pointing downward gives a clockwise current around the loop, viewed from above.
4. Tracing the current around the loop: a clockwise current flows from the top rail through the resistor to the bottom rail, so the current through $R$ is downward (top to bottom). This is confirmed by the force shortcut: force on the bar opposes motion to the right, so force is left, which matches the calculated current direction.

Exam tip: Always trace the current around the full loop when asked for direction through a specific component like the resistor — direction in the bar is opposite to direction in the resistor, so it is easy to mix up.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Stating that the induced magnetic field always points opposite to the original external magnetic field. Why: Students memorize the word "oppose" and apply it to the original field instead of the change in flux, which gives the wrong direction when flux is decreasing. Correct move: Always explicitly identify whether flux is increasing or decreasing first, then assign induced B direction based on the change, not the original field direction.
• Wrong move: Claiming there is induced current when a loop moves entirely inside a uniform magnetic field. Why: Students see a moving conductor in a magnetic field and automatically assume induction occurs, even when flux does not change. Correct move: Always check if $\frac{d{\Phi }_B}{dt}=0$ before solving for current; if flux is constant, $\epsilon =0$ and $I=0$.
• Wrong move: Giving a direction of "counterclockwise" or "clockwise" with no reference to viewing perspective. Why: Students assume the perspective is obvious, but AP exam graders require explicit direction to award full credit. Correct move: Always add a reference like "counterclockwise as viewed from above the loop".
• Wrong move: Assigning the induced force direction as aiding the motion of the conductor. Why: Students confuse the direction of the change, leading to a result that violates conservation of energy. Correct move: Use the conservation of energy shortcut: induced magnetic force always opposes the motion that causes the flux change, so direction is opposite to velocity.
• Wrong move: Stating the current direction through the resistor matches the current direction through the sliding bar in a rail problem. Why: It is easy to forget the current flows around a closed loop, so direction reverses in different components. Correct move: After finding the loop direction, trace the current step by step through the specific component the question asks for.
• Wrong move: Ignoring the negative sign in $\epsilon =-\frac{d{\Phi }_B}{dt}$ for all problems, because "the question only asks for magnitude". Why: Many FRQs require direction or sign for full credit, and missing the sign leads to wrong differential equations for circuits. Correct move: Always apply Lenz's Law to get the correct sign of emf when using Faraday's Law in differential form.

6. Practice Questions (AP Physics C: E&M Style)

Question 1 (Multiple Choice)

A bar magnet is held vertically above a horizontal circular loop of wire, with the north pole of the magnet pointing down toward the center of the loop. The bar magnet is dropped toward the loop. What is the direction of the induced current in the loop, as viewed from above the loop, and what is the correct reasoning? A) Clockwise, because the magnetic flux through the loop pointing down is increasing B) Clockwise, because the magnetic flux through the loop pointing up is increasing C) Counterclockwise, because the magnetic flux through the loop pointing down is increasing D) Counterclockwise, because the magnetic flux through the loop pointing up is decreasing

Worked Solution: Magnetic field lines exit the north pole of a bar magnet, so the field through the loop points downward when the north pole is above pointing down. As the magnet falls closer, the magnitude of B increases, so downward flux through the loop is increasing. Lenz's Law says induced B opposes the increase, so induced B points upward. The right-hand rule gives that thumb pointing upward (direction of induced B) corresponds to a counterclockwise current viewed from above. The correct answer is C.

Question 2 (Free Response)

A rectangular conducting loop has mass $m$, resistance $R$, vertical side length $a$ and horizontal side length $b$. The loop is dropped from rest into a region of uniform magnetic field $B$ pointing into the page, which fills all space below the loop's starting position. The top edge of the loop starts a height $h$ above the top boundary of the magnetic field region. Answer the following: (a) When the loop is entering the magnetic field region (only the bottom of the loop is inside B), what is the direction of the induced current? Justify your answer with Lenz's Law. (b) Derive an expression for the magnitude of the induced current when the loop is entering the field and has speed $v$. (c) What is the direction of the net magnetic force on the loop while it is entering the field? Justify your answer in terms of Lenz's Law.

Worked Solution: (a) The magnetic field points into the page, and the area of the loop inside the field increases as the loop falls, so flux into the page through the loop is increasing. By Lenz's Law, the induced magnetic field opposes this increase, so it points out of the page. The right-hand rule gives a counterclockwise induced current, viewed from the front. (b) The magnitude of the motional emf across the vertical side of the loop inside the field is $|\epsilon |=Bav$. By Ohm's law, the current magnitude is: $$I=\frac{|\epsilon |}{R}=\frac{Bav}{R}$$ (c) By Lenz's Law, the induced effect opposes the change that created it. The change in flux is caused by the loop falling downward into the field, so the net force opposes the motion. The net magnetic force is upward.

Question 3 (Application / Real-World Style)

Eddy current braking is a contactless braking system used in roller coasters that relies on Lenz's Law. A 0.15 m wide leading edge of a metal train car passes through a 0.80 T magnetic field perpendicular to the edge at a speed of $2.5\text{ m/s}$. The total resistance of the induced eddy current path is $0.020\Omega$. What is the magnitude of the force opposing the motion of the train car, and what effect does this force have?

Worked Solution: The magnitude of the induced emf is $|\epsilon |=\frac{d{\Phi }_B}{dt}=BLv$, where $L=0.15\text{ m}$ is the width of the leading edge. Substituting values: $$|\epsilon |=(0.80\text{ T})(0.15\text{ m})(2.5\text{ m/s})=0.30\text{ V}$$ The induced current is $I=\frac{|\epsilon |}{R}=\frac{0.30\text{ V}}{0.020\Omega }=15\text{ A}$. The magnitude of the opposing force is: $$F=ILB=(15\text{ A})(0.15\text{ m})(0.80\text{ T})=1.8\text{ N}$$ By Lenz's Law, this force points opposite to the direction of the train's motion, so it slows the train down without any friction or contact between moving parts.

7. Quick Reference Cheatsheet

8. What's Next

Mastering Lenz's Law is a prerequisite for all other induction topics in AP Physics C: E&M. Next you will combine Lenz's Law with Faraday's Law to solve problems involving induced electric fields, eddy currents, and inductors in RL circuits. Without correctly finding the direction of induced emf and current using Lenz's Law, you will not be able to set up the correct differential equations for RL circuits, which are a frequent FRQ topic on the exam. Lenz's Law is also the core principle behind many real-world technologies that appear on AP application questions, from generators to magnetic braking to transformers. The conservation of energy principle inherent in Lenz's Law underpins all electromagnetic energy conversion, a unifying theme across the entire unit of electromagnetism.

Faraday's Law of Induction Motional Emf Inductors and RL Circuits Maxwell's Equations