Electromagnetic Induction and Faraday’s Law
Choose an area normal, track changing flux, and connect induced current with external mechanical energy.
AP Physics 2 Topic 12.4 · College Board CED printed p. 87, 88 / PDF p. 94, 95 · See this topic in the course outline
A field can be strong and produce no induced voltage. Induction responds to changing flux. Choose a normal, track what changes, and let Lenz’s law tell you which current resists that change.
Before you start
- Area vectors and components
- Ohm’s law
- Work, power, and conservation of energy
Choose the area vector before calculating flux
Magnetic flux measures how much of B passes perpendicularly through an area. For a flat surface in a uniform field, , where θ is between B and the area vector, not the surface itself. The area vector has magnitude A and is perpendicular to the surface. For a closed surface it points outward at every patch. For an open loop choose either normal as positive and keep that choice throughout.
Flux is positive for a field component along the area vector, negative for a component opposite it, and zero when B lies in the plane. Units are webers: 1 Wb = 1 T·m². A single planar loop’s flux can be positive even though the net flux through a closed surface is zero because closed magnetic lines exit and reenter.
| Angle B to area vector | Flux (Wb) | Interpretation |
|---|---|---|
| 0° | +0.0080 | B through the positive face |
| 90° | 0 | B along the surface |
| 180° | −0.0080 | B through the negative face |
A changing flux induces emf
An emf is an energy transfer per unit charge around a circuit, measured in volts. Faraday’s law relates the average induced emf to changing flux: for one turn, or for N identical turns. Change B, the enclosed area, or the angle to change flux. A large constant flux produces no induced emf. A steeper flux-versus-time graph means a larger magnitude of emf; a flat interval means zero emf. The finite-change formula gives an interval average, and equals the instantaneous value when flux changes at a constant rate.
To use the minus sign, choose an area normal and take the positive circulation around the loop from the right-hand curl associated with that normal. A negative emf then drives circulation opposite that direction. Lenz’s law says the induced current’s field opposes the change in flux. If out-of-page flux increases, the current makes an into-page field (clockwise). If the same out-of-page flux decreases, the current makes an out-of-page field (counterclockwise). It can therefore reinforce an existing field while resisting its decrease. An emf can exist across an open circuit; sustained current additionally requires a closed conducting path.
A moving rod converts external work to electrical energy
Picture horizontal conducting rails joined on the left, with a vertical rod of length ℓ sliding right at speed v in uniform B into the page. As the rod moves, the enclosed area increases by . Applying Faraday to this changing area gives when rod, motion, and field are mutually perpendicular. Positive charges in the rod are driven upward by v × B, so its upper end is at higher potential.
For a closed circuit of total resistance R, conventional current goes up the rod and counterclockwise around the loop as seen by you, creating out-of-page B to oppose the increasing into-page flux. The external field exerts a leftward force on the rod. Holding v constant requires a rightward external force. Its power supplies the electrical heating. There is no free energy from Lenz’s law: the induced force resists the motion that changes flux. At the microscopic level the magnetic force still does no work on a point charge; the moving apparatus and electric fields transfer the external mechanical energy.
Read a flux graph and its sign
A single 0.0300 m² loop has its positive normal out of the page. A uniform out-of-page field rises linearly at a constant rate from 0.100 T at t = 0 to 0.500 T at t = 0.200 s, then stays at 0.500 T until t = 0.400 s. Total loop resistance is 2.00 Ω. Find the flux change during the ramp, graph signed flux and emf, and determine current direction in both intervals.
The field is parallel to the positive normal, so initial flux is +0.00300 Wb and final flux is +0.0150 Wb.
Flux rises linearly from +0.00300 Wb at 0 s to +0.0150 Wb at 0.200 s, then is constant through 0.400 s. Accessible flux-graph data t (s) Flux (Wb) 0 +0.00300 0.200 +0.0150 0.400 +0.0150 Why this step? Using the normal makes the flux sign unambiguous.
ΔΦ = +0.0120 Wb. ℰ = −0.0120/0.200 = −0.0600 V relative to positive counterclockwise circulation.
The first graph’s slope is +0.0600 Wb/s from 0 to 0.200 s. Negating that slope gives a constant −0.0600 V emf. In the hold interval the slope is zero, so emf is zero. At the ideal sharp transition the interval values are understood as left/right limits.
Emf is −0.0600 V from 0 to 0.200 s and 0 V from 0.200 to 0.400 s; positive circulation is counterclockwise. Accessible emf and current graph data Time interval (s) Emf (V) Current 0 to 0.200 −0.0600 0.0300 A clockwise 0.200 to 0.400 0 0 Why this step? Faraday gives a negative emf because positive flux is increasing.
The current magnitude is |ℰ|/R = 0.0300 A; it is clockwise and creates into-page B.
Why this step? Ohm’s law sets magnitude and Lenz’s law plus the loop right-hand rule sets direction.
A field held at 0.500 T afterward has zero induced emf in this stationary loop.
Why this step? Flux rate, not flux magnitude, causes induction.
Answer and check
During the linear ramp, ΔΦ = +0.0120 Wb, signed emf = −0.0600 V, and current = 0.0300 A clockwise. During the hold interval the flux is +0.0150 Wb but emf and current are zero. The flux graph has a rising straight segment then a horizontal segment; the emf graph has a negative constant segment then a zero segment.
Account for the energy of a sliding rod
A 0.200 m rod moves right at 3.00 m/s on closed rails in a 0.400 T into-page field. Total resistance is 2.00 Ω; ignore friction. Find emf, current, resisting force, and required mechanical power.
Use the perpendicular sliding-rod model. |ℰ| = Bℓv = (0.400)(0.200)(3.00) = 0.240 V.
Why this step? Motion changes area at ℓv and therefore changes flux.
I = 0.240/2.00 = 0.120 A, counterclockwise (up the rod).
Why this step? The induced field must oppose the increasing into-page flux.
F_B = IℓB = (0.120)(0.200)(0.400) = 0.00960 N leftward. The external force is equally large rightward at constant speed.
Why this step? The wire-force rule gives the resisting force; zero acceleration requires balance.
P_ext = Fv = (0.00960)(3.00) = 0.0288 W; I²R = (0.120)²(2.00) = 0.0288 W.
Why this step? The independent power calculation checks energy conservation and units.
Answer and check
0.240 V, 0.120 A counterclockwise, 0.00960 N resisting force, and 0.0288 W external power.
Make the model respond
Focused Physics 2 model
Flux change and induced emf
Predict first. If the flux decreases, which direction of induced field opposes that change?
Φ = BA cos θ per turn; flux linkage = NΦ; average ε = −N(Φf − Φi)/Δt
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
Flux increases along the chosen positive normal. The induced field would point opposite that normal if a current can flow; εavg = -0.2 V.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Turns | 100 | ||
| Area per turn | 0.01 | m² | |
| Initial B | 0.02 | T | |
| Final B | 0.06 | T | |
| Time interval | 0.2 | s | |
| Initial flux per turn | 2.000 × 10^−4 | Wb | |
| Final flux per turn | 6.000 × 10^−4 | Wb | |
| Change in flux per turn | 4.000 × 10^−4 | Wb | |
| Initial NΦ | 0.02 | Wb·turn | |
| Final NΦ | 0.06 | Wb·turn | |
| Average emf | -0.2 | V | |
| Induced field normal direction if current can flow | −normal |
B is signed along a fixed field axis, and θ is measured from the positive area normal to the positive field-axis direction; a negative B reverses the field along that axis. A field parallel to the plane contributes no flux. The minus sign is Lenz’s law: the induced response opposes the change in flux, rather than necessarily opposing the original field. A positive emf is defined around the positive circulation direction given by the right-hand rule about the positive area normal. Emf alone does not determine current; a closed conducting circuit and its resistance would also be needed.
Assumptions and limits. N identical fixed-area turns in a spatially uniform external field, with fixed area-normal and field-axis orientation during Δt. B may change magnitude or reverse sign along the fixed field axis. The displayed emf is an average over a finite specified change; an instantaneous value would require the time dependence. No resistance, induced current magnitude, self-inductance, or mechanical dynamics is assumed.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
B makes 30° with a loop’s plane. Which angle belongs in BA cos θ? Explain without calculating.
Worked answer and reason
θ = 60°, because the area vector is perpendicular to the plane.
Flux uses the field component along the surface normal.
Question 2
An into-page field through a stationary closed loop is decreasing. Which way does the induced current circulate?
Worked answer and reason
Clockwise; it creates an into-page field to oppose the decrease.
Lenz’s law opposes the change, so the induced field can point with the original field.
Question 3
Flux is a constant +0.80 Wb for one second, then falls linearly to +0.20 Wb over two seconds. Compare the two interval emfs for one turn.
Worked answer and reason
First interval: zero. Second: −(0.20−0.80)/2 = +0.30 V in the chosen positive circulation.
Average emf is the negative slope of the signed flux-versus-time graph.
Question 4
At fixed B, ℓ, and R, a sliding rod’s speed doubles. How do emf, current, resisting force, and heating power change?
Worked answer and reason
Emf, current, and resisting force each double. I²R and external power each quadruple.
Use ℰ = Bℓv, I = ℰ/R, F = IℓB, and power rather than assuming every quantity scales identically.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- Lenz’s law always makes a field opposite the existing field.
- It opposes the change. A decreasing field is supported by a field in the same direction.
- The angle in BA cos θ is measured from the surface.
- Measure from its perpendicular area vector.
- A generator gets electrical energy directly from a static magnetic field.
- An external agent supplies work against the induced resisting force.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 7 · flux and emf graphs
- Problem 8 · verbal and mathematical
- Problem 11 · numerical table and experimental comparison
Scientific calculator workspace
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Scientific calculator
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For the teacher
- Have students choose one normal and associated positive circulation before assigning algebraic signs.
- Distinguish an induced emf from a current; a closed path is needed for sustained current.
- Use piecewise linear flux graphs so AP algebra gives the interval emfs.
Source objectives and scope
- 12.4.A — Describe the induced electric potential difference resulting from a change in magnetic flux.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 CED, Unit 12 (supplied pinned source) — Official Unit 12 LO/EK scope and quantitative moving-charge angle boundary; raw 12.1.B4 normalized as 12.1.B.4
- OpenStax College Physics 2e: magnetic field lines — Closed-loop field convention and compass alignment
- OpenStax College Physics 2e: force on moving charge — Magnetic force magnitude and right-hand rule
- OpenStax College Physics 2e: Hall effect — Transverse force balance and Hall polarity
- OpenStax College Physics 2e: magnetic fields produced by currents — Long-wire field model and current-loop direction
- OpenStax College Physics 2e: force on a current-carrying conductor — Wire force and in-field length
- OpenStax College Physics 2e: Faraday and Lenz laws — Flux change and Lenz direction
- OpenStax College Physics 2e: motional emf — Sliding rod and mechanical-to-electrical energy accounting
Glossary and deeper reading · 12 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Potential difference
Potential difference is the change in electric potential between two locations. It represents electric potential-energy change per unit charge and is measured in volts. For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
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Electric current
Electric current is the net rate of charge flow through a specified cross section. It is measured in amperes, or coulombs per second, with direction defined by positive-charge motion. For example, the same steady current passes through each element of a simple unbranched series loop.
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Vector
A vector is a quantity with a magnitude and a direction, represented by an arrow or by components along chosen axes. Vectors add by adding corresponding components rather than simply adding their magnitudes. For example, an eastward displacement followed by a northward displacement produces a resultant pointing northeast. The zero vector has zero magnitude and no unique direction.
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Vector component
A vector component describes the part of a vector along a chosen axis. Its signed value changes when the axes change, even though the physical vector remains the same. For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.
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Ohm’s law
Ohm’s law describes an element whose current is proportional to its potential difference under the stated conditions. Such an element has constant resistance over the modeled range, giving a straight-line current–voltage relationship through the origin. For example, a fixed-temperature ideal resistor follows this model, whereas a warming filament bulb need not.
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Mechanical work
Mechanical work is energy transferred by a force through displacement of its point of application. For a constant force on a particle, it depends on the force component along the particle’s displacement, and it can be positive, negative, or zero. For example, a magnetic force perpendicular to a particle’s displacement does zero work on that particle.
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Power
Power is the rate at which energy is transferred or converted. It is measured in watts, with one watt equal to one joule per second. For example, two motors can lift the same load through the same height but have different average powers if they take different times.
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Magnitude
Magnitude is the nonnegative size of a vector quantity. It does not specify direction, and it differs from a signed component. For example, a westward velocity can have a negative x component while its speed, the velocity magnitude, stays positive.
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Electromotive force
Electromotive force is energy supplied per unit charge by a source or induction process. Despite its name it is measured in volts, rather than newtons, and can exist even when no current flows. For example, an ideal battery supplies an emf while an induced emf can be produced by changing magnetic flux.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Electrical resistance
Electrical resistance relates the potential difference across an element to the current through it. It is measured in ohms and depends on the element’s material, geometry, and conditions. For example, a longer uniform wire has greater resistance when cross-sectional area and material remain fixed.
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Conventional current
Conventional current points in the direction that positive charge would move through a circuit. In a metal wire, electrons usually drift in the opposite direction. For example, a current arrow pointing right in a metal wire does not mean the conduction electrons drift right.