AP Physics 2 Unit 4 Notes: Magnetism and Electromagnetic Induction (Algebra-Based)

Magnetic Fields and Magnetic Forces

What a magnetic field is (and how it’s different from an electric field)

A magnetic field is a vector field that describes how magnetic forces act in space. In AP Physics 2, you’ll usually represent the magnetic field with the symbol \vec{B}, and its unit is the tesla (T).

Magnetic fields matter because they explain:

  • How permanent magnets interact (attraction/repulsion)
  • How electric currents experience forces (which is how motors work)
  • How moving charges follow curved paths (important in mass spectrometers and particle accelerators)

A crucial conceptual difference from electric fields is this:

  • Electric fields act on charges whether the charges are moving or not.
  • Magnetic fields exert forces on moving charges (and on currents, which are moving charges in bulk).

So if a charge is sitting still in a magnetic field, the magnetic force on it is zero.

Magnetic field lines and what they mean

You’ll often see magnetic fields drawn using field lines. The rules are similar to electric field lines, but with one key difference.

  • The direction of \vec{B} at a point is tangent to the field line.
  • A denser cluster of lines means a stronger magnetic field.
  • Magnetic field lines form closed loops—they do not start or end the way electric field lines do.

That “closed loop” behavior connects to a deep fact: isolated magnetic “charges” (magnetic monopoles) are not part of the standard AP Physics 2 model. Magnets come as north and south poles together; you can’t isolate one pole by cutting a magnet.

Force on a moving point charge in a magnetic field

If a charge q moves with velocity \vec{v} through a magnetic field \vec{B}, it experiences a magnetic force

\vec{F}_B = q\,\vec{v} \times \vec{B}

For AP Physics 2, you typically use the magnitude form

F_B = |q|vB\sin\theta

where \theta is the angle between \vec{v} and \vec{B}.

How it works (step-by-step):

  1. The force depends on motion: if v = 0 then F_B = 0.
  2. The force depends on direction: if the particle moves parallel to the field, \theta = 0 and F_B = 0.
  3. The force is maximum when motion is perpendicular to the field, \theta = 90^\circ.
  4. The force is perpendicular to both \vec{v} and \vec{B}. That perpendicular nature is why magnetic forces often cause circular motion instead of speeding something up.

Direction and the right-hand rule:
Because the force comes from a cross product, direction is not “along” the field lines. For a positive charge, you can use this procedure:

  • Point your fingers in the direction of \vec{v}
  • Curl them toward \vec{B}
  • Your thumb points in the direction of \vec{F}_B

For a negative charge, the force points opposite that right-hand-rule result.

A common misconception is to forget the sign of the charge and always use the right-hand rule directly.

Magnetic force does no work on a point charge

Because magnetic force is always perpendicular to velocity (when it’s nonzero), it changes the direction of motion but not the speed. That means magnetic fields do no work on a point charge:

  • Kinetic energy stays constant (if only magnetic forces act)
  • Speed stays constant
  • Path bends (changes direction)

This idea is often tested conceptually: a charge can spiral or curve, but a pure magnetic field alone cannot increase its speed.

Circular motion of a charged particle in a uniform magnetic field

If a charged particle moves perpendicular to a uniform magnetic field, it undergoes uniform circular motion because the magnetic force provides the centripetal force.

Set magnetic force equal to centripetal force:

|q|vB = \frac{mv^2}{r}

Solve for the radius:

r = \frac{mv}{|q|B}

The period of circular motion is

T = \frac{2\pi m}{|q|B}

Notice something surprising: T does not depend on speed v. Faster particles make bigger circles, but they still take the same time per revolution (in the ideal model).

Example: radius of curvature

A proton with mass m = 1.67\times 10^{-27}\ \text{kg} and charge q = 1.60\times 10^{-19}\ \text{C} moves at v = 3.0\times 10^6\ \text{m/s} perpendicular to a uniform field B = 0.50\ \text{T}. Find r.

Use

r = \frac{mv}{|q|B}

Compute:

r = \frac{(1.67\times 10^{-27})(3.0\times 10^6)}{(1.60\times 10^{-19})(0.50)}

Numerator:

1.67\times 10^{-27}\times 3.0\times 10^6 = 5.01\times 10^{-21}

Denominator:

1.60\times 10^{-19}\times 0.50 = 8.0\times 10^{-20}

So

r \approx \frac{5.01\times 10^{-21}}{8.0\times 10^{-20}} = 6.3\times 10^{-2}\ \text{m}

So the radius is about 0.063\ \text{m}.

Force on a current-carrying wire

A current in a wire is many charges moving together. A straight segment of wire of length L carrying current I in a uniform magnetic field experiences a force

F = ILB\sin\theta

where \theta is the angle between the current direction (along the wire) and \vec{B}.

Direction: use a right-hand rule variant for \vec{F} = I\,\vec{L} \times \vec{B}:

  • Fingers along current direction (direction of conventional current)
  • Curl toward \vec{B}
  • Thumb gives force on the wire

This matters because it is the core principle behind electric motors: forces on different sides of a loop produce a torque.

Torque on a current loop and magnetic dipole moment

A rectangular loop in a magnetic field experiences a torque that tends to rotate it.

Define the magnetic dipole moment \vec{\mu} of a loop:

  • Magnitude: \mu = NIA
  • Direction: perpendicular to the plane of the loop (given by the right-hand rule)

Here, N is the number of turns and A is the area of one turn.

The maximum torque magnitude is

\tau = \mu B\sin\theta

where \theta is the angle between \vec{\mu} and \vec{B}.

Physical meaning: the loop behaves like a tiny bar magnet. The magnetic field tries to align \vec{\mu} with \vec{B}. This is also how a compass works: Earth’s magnetic field exerts torque on the compass needle.

Example: force vs torque confusion

A common error is to think “a loop in a uniform field feels a net force.” In a uniform magnetic field, opposite sides of the loop experience equal and opposite forces, so the net force is zero. The loop can still rotate because those forces form a couple and create a net torque.

Exam Focus
  • Typical question patterns:
    • Determine the direction of force on a moving charge or current-carrying wire using right-hand rules (often with a negative charge twist).
    • Relate circular motion radius or period of a charged particle to m, q, v, and B.
    • Explain conceptually why a magnetic field changes direction but not speed (work/energy reasoning).
  • Common mistakes:
    • Using the right-hand rule for a positive charge and forgetting to reverse direction for a negative charge.
    • Plugging into F = qvB without the \sin\theta factor when motion is not perpendicular to the field.
    • Claiming a magnetic field does work because the path is curved (curved path does not automatically mean work is done).

Sources of Magnetic Fields (Currents and Magnetic Materials)

The big idea: moving charges create magnetic fields

Electric charges produce electric fields. Moving charges (currents) produce magnetic fields. In AP Physics 2, the key magnetic-field sources you’re expected to know quantitatively are:

  • Long straight current-carrying wires
  • Circular loops of wire
  • Solenoids (coils)

These models let you predict field strength and direction in many practical devices—electromagnets, speakers, MRI machines, and motors.

The permeability constant \mu_0

Many magnetic field formulas include the constant \mu_0, called the permeability of free space:

\mu_0 = 4\pi\times 10^{-7}\ \text{T·m/A}

In matter (like iron cores), magnetic fields can be much stronger because the material’s magnetic domains align, but AP Physics 2 typically treats that through qualitative reasoning (or through “ideal transformer” assumptions) rather than heavy material science.

Magnetic field of a long straight wire

For a long straight wire carrying current I, the magnetic field at a distance r from the wire has magnitude

B = \frac{\mu_0 I}{2\pi r}

Direction: the field lines form circles around the wire. Use the right-hand grip rule:

  • Thumb points in the direction of conventional current
  • Fingers curl in the direction of the magnetic field lines

Why the direction matters: Many problems involve multiple wires or a wire near a loop. Direction determines whether fields add or cancel.

Example: field from a straight wire

A wire carries I = 8.0\ \text{A}. Find B at r = 0.040\ \text{m}.

B = \frac{\mu_0 I}{2\pi r}

B = \frac{(4\pi\times 10^{-7})(8.0)}{2\pi(0.040)}

Cancel \pi and simplify:

B = \frac{(32\times 10^{-7})}{0.080}

B = 4.0\times 10^{-5}\ \text{T}

So B = 40\ \mu\text{T} (about comparable to Earth’s field magnitude).

Magnetic field at the center of a circular current loop

For a single circular loop of radius R carrying current I, the field at the center is

B = \frac{\mu_0 I}{2R}

For N turns (a coil), multiply by N:

B = \frac{\mu_0 N I}{2R}

Direction: use a right-hand rule:

  • Curl fingers in the direction of current around the loop
  • Thumb points in the direction of \vec{B} through the center

This model helps you understand how coils produce strong, directed magnetic fields.

Magnetic field inside a long solenoid

A solenoid is a long coil of wire. For an ideal long solenoid with N total turns over length L, the turn density is

n = \frac{N}{L}

The magnetic field inside is approximately uniform and has magnitude

B = \mu_0 n I

This is a key engineering idea: by wrapping wire into a solenoid, you can get a fairly uniform field in a region, which is useful for electromagnets and inductors.

Direction: same loop rule; the solenoid has a “north end” and “south end,” behaving like a bar magnet.

Superposition and multi-wire situations

Magnetic fields add as vectors. If multiple currents contribute, you find each field (magnitude and direction) at the point, then add them.

A common conceptual trap is to add magnitudes only. Direction is often what the question is really testing.

Magnetic materials (qualitative but important)

Ferromagnetic materials (like iron) can greatly increase magnetic field strength because microscopic magnetic dipoles (from electron behavior) can align into domains. For AP Physics 2:

  • You should know that inserting an iron core into a solenoid increases the field.
  • You should not treat the iron core as “creating” field from nothing; it amplifies the field produced by current.
Exam Focus
  • Typical question patterns:
    • Compute B at a point due to a long straight wire, loop center, or solenoid interior.
    • Determine the direction of \vec{B} using the right-hand grip rule (often using into/out of page symbols).
    • Combine fields from multiple wires (superposition with direction).
  • Common mistakes:
    • Using B = \mu_0 I/(2\pi r) for situations where the wire is not effectively long/straight in the model.
    • Mixing up which right-hand rule to use (wire circles vs loop/solenoid axial field).
    • Adding magnitudes when fields oppose (sign/direction errors).

Magnetic Flux

What magnetic flux means physically

Magnetic flux measures how much magnetic field “passes through” a surface. The symbol is \Phi_B. Flux is not just about field strength; it also depends on the area and the orientation of the surface relative to the field.

Flux matters because changing flux is what creates induced emf in Faraday’s law. In other words, flux is the bridge between magnetism and electricity.

Definition of magnetic flux

For a uniform magnetic field passing through a flat surface of area A:

\Phi_B = BA\cos\theta

where \theta is the angle between \vec{B} and the surface’s area vector (a vector perpendicular to the surface).

  • If \vec{B} is perpendicular to the surface, then \theta = 0 and \Phi_B = BA.
  • If \vec{B} is parallel to the surface, then \theta = 90^\circ and \Phi_B = 0.

Units: \Phi_B is measured in webers (Wb), where

1\ \text{Wb} = 1\ \text{T·m}^2

The area vector and why the angle can feel “backwards”

Many students incorrectly use the angle between the field and the surface itself. The formula uses the angle between \vec{B} and the normal vector to the surface.

A quick way to avoid this mistake:

  • If the field goes straight “through” the surface, you want maximum flux, so \cos\theta = 1.
  • If the field skims along the surface, you want zero flux, so \cos\theta = 0.

Flux through multiple turns

If a coil has N identical turns, each with flux \Phi_B, the total flux linkage is N\Phi_B. This is what appears in Faraday’s law for induced emf.

Example: flux changing by rotation

A loop of area A = 0.020\ \text{m}^2 is in a uniform field B = 0.30\ \text{T}. The loop rotates so that \theta changes from 0^\circ to 60^\circ. Find the change in flux.

Initial flux:

\Phi_i = BA\cos 0^\circ = (0.30)(0.020)(1) = 0.0060\ \text{Wb}

Final flux:

\Phi_f = BA\cos 60^\circ = (0.30)(0.020)(0.5) = 0.0030\ \text{Wb}

Change:

\Delta\Phi = \Phi_f - \Phi_i = -0.0030\ \text{Wb}

The negative sign tells you flux decreased.

Exam Focus
  • Typical question patterns:
    • Compute magnetic flux for a surface at an angle (often with a “which orientation gives maximum flux” conceptual part).
    • Determine whether flux is increasing/decreasing given a changing B, changing area, or rotation.
    • Use flux change as an intermediate step before Faraday’s law.
  • Common mistakes:
    • Using \sin\theta instead of \cos\theta because the angle was measured from the plane rather than the normal.
    • Forgetting that zero flux can occur even with a strong field if the surface is parallel to the field.
    • Ignoring the sign of \Delta\Phi when using Lenz’s law later.

Faraday’s Law and Lenz’s Law (Electromagnetic Induction)

The core phenomenon: changing flux induces an emf

Electromagnetic induction is the process where a changing magnetic environment produces an electric effect—specifically an induced emf (electromotive force), which can drive current.

This matters because it is the foundation of:

  • Electric generators (mechanical energy to electrical energy)
  • Transformers (changing voltage levels efficiently)
  • Inductors and many circuits (opposing changes in current)

The key idea is not “motion causes current” but more general:

  • A changing magnetic flux through a loop induces an emf around the loop.

Faraday’s law (mathematical statement)

For a coil with N turns, Faraday’s law is

\varepsilon = -N\frac{\Delta\Phi_B}{\Delta t}

The negative sign is not just a convention; it encodes Lenz’s law (the direction of the induced emf).

You may also see the instantaneous form

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

In algebra-based problems, you’re commonly given a steady change so \Delta form is enough.

What the negative sign means: Lenz’s law

Lenz’s law says:

  • The induced current (and the magnetic field it produces) acts to oppose the change in magnetic flux that caused it.

This is really an energy conservation statement. If induction made changes “run away” (amplify themselves), you could get energy from nowhere.

A reliable way to apply Lenz’s law is a two-step logic chain:

  1. Decide whether the external flux through the loop is increasing or decreasing (and in which direction through the loop).
  2. The induced field must oppose that change (it tries to keep flux from changing).

Then use a right-hand rule for a current loop to get the current direction.

How to reason about “direction of induced current” problems

These are some of the most common conceptual questions in the unit, and the most common place to get lost. Here is a structured method.

Step 1: Define the positive flux direction.
Often the diagram shows a loop and a magnetic field “into the page” or “out of the page.” Choose that as the flux direction.

Step 2: Determine the change in flux.
Flux can change because:

  • B changes (field gets stronger/weaker)
  • A changes (loop area changes)
  • \theta changes (rotation)
  • The loop enters/leaves a field region (effective area in field changes)

Step 3: Induced field opposes the change (not the field itself).
For example:

  • If external flux into the page is increasing, the induced field is out of the page.
  • If external flux into the page is decreasing, the induced field is into the page.

Step 4: Use the loop right-hand rule to get current direction.

  • Fingers curl with conventional current
  • Thumb gives induced \vec{B} direction through the loop

A classic misconception is to say “induced field opposes the external field.” That is only true if the external flux is increasing. If the external field is decreasing, the induced field points the same direction as the external field (to oppose the decrease).

Induced emf vs induced current

Faraday’s law gives induced emf. Whether current actually flows depends on whether there is a closed conducting path.

If the loop has resistance R, then the induced current magnitude is

I = \frac{|\varepsilon|}{R}

So “bigger induced emf” and “bigger induced current” are not the same claim unless resistance is fixed.

Energy perspective: why induction often produces a force opposing motion

If you move a magnet toward a conducting loop and an induced current appears, that current’s magnetic field interacts with the magnet and produces a force that resists your push or pull. You must do work, and that work becomes electrical energy (and often thermal energy) in the circuit.

This is why induction is a mechanism for converting mechanical energy into electrical energy.

Example: induced emf from changing field

A coil has N = 200 turns, each of area A = 4.0\times 10^{-4}\ \text{m}^2. A uniform field perpendicular to the coil changes from 0.10\ \text{T} to 0.40\ \text{T} in 0.050\ \text{s}. Find induced emf magnitude.

Flux per turn is \Phi = BA (since \theta = 0).

Change in flux per turn:

\Delta\Phi = A\Delta B = (4.0\times 10^{-4})(0.30) = 1.2\times 10^{-4}\ \text{Wb}

Induced emf magnitude:

|\varepsilon| = N\frac{|\Delta\Phi|}{\Delta t} = 200\frac{1.2\times 10^{-4}}{0.050}

|\varepsilon| = 200(2.4\times 10^{-3}) = 0.48\ \text{V}

So the induced emf is about 0.48\ \text{V}.

Example: Lenz’s law direction (conceptual)

A circular loop is in a region where the magnetic field points into the page. The field strength increases.

  • Flux into the page is increasing.
  • Induced field must oppose the increase, so it points out of the page.
  • A loop current that produces out-of-page field is counterclockwise (right-hand rule).

So the induced current is counterclockwise.

Exam Focus
  • Typical question patterns:
    • Given a graph of B vs time (or flux vs time), determine induced emf magnitude and/or when it is zero.
    • Determine induced current direction using Lenz’s law for magnets moving toward/away from loops.
    • Compare scenarios: which produces greater emf (faster change, more turns, larger area, more perpendicular orientation).
  • Common mistakes:
    • Saying induced field always opposes external field rather than opposing the change in flux.
    • Using Faraday’s law but forgetting to multiply by N for multiple turns.
    • Confusing induced emf (a loop integral effect) with electric field from static charges (a conservative field).

Motional EMF and Induction from Moving Conductors

What “motional emf” is

Motional emf occurs when a conductor physically moves through a magnetic field and charges in the conductor experience magnetic forces. Those forces separate charges, creating a potential difference across the conductor.

This is not a different law from Faraday’s law; it is a common special case of induction where the flux changes because geometry changes as something moves.

Deriving the basic motional emf formula

Consider a straight conducting rod of length L moving with speed v perpendicular to a uniform magnetic field B. Charges in the rod move with the rod, so they have velocity \vec{v}. Each charge feels magnetic force magnitude

F_B = |q|vB

This pushes positive and negative charges to opposite ends, creating an internal electric field E until electric and magnetic forces balance:

|q|E = |q|vB

So

E = vB

The potential difference across the rod is

\Delta V = EL = vBL

That potential difference is an emf:

\varepsilon = B L v

This form assumes motion is perpendicular to \vec{B} and the rod is oriented so that charges are pushed along its length.

Sliding rod on rails: induced current and magnetic force

A classic setup is a rod sliding on conducting rails, forming a rectangular loop with a resistor. As the rod moves, the loop’s area changes, so the flux changes:

  • Area A = Lx if the rail separation is L and the rod position is x.
  • Flux \Phi = BA = BLx (if field is perpendicular to loop).
  • Then

\varepsilon = \left|\frac{\Delta\Phi}{\Delta t}\right| = BL\frac{\Delta x}{\Delta t} = BLv

So Faraday’s law and motional emf give the same result.

If the circuit has resistance R, induced current magnitude is

I = \frac{\varepsilon}{R} = \frac{BLv}{R}

Lenz’s law and magnetic drag:
The induced current in the rod experiences a magnetic force F = ILB that typically opposes the motion. So you must pull with an external force to keep constant speed. The mechanical power you supply becomes electrical power dissipated in the resistor.

Mechanical power input:

P_{mech} = Fv

Electrical power in resistor:

P_{elec} = I^2R

In the ideal model (no other losses), these match.

Eddy currents and magnetic braking

When a solid conductor (like an aluminum plate) moves through a magnetic field, circulating currents called eddy currents can form. By Lenz’s law, the magnetic effects of these currents oppose the motion that created them, producing a resistive “magnetic drag.”

Real-world applications:

  • Magnetic braking in some trains and exercise equipment
  • Damping in analog meters
  • Induction cooktops (eddy currents produce heating)

A common misunderstanding is thinking eddy currents require a wire loop. They can form in any bulk conductor because charges can circulate in closed paths within the material.

Example: motional emf and current

A rod of length L = 0.50\ \text{m} moves at v = 3.0\ \text{m/s} perpendicular to a magnetic field B = 0.80\ \text{T}. The circuit resistance is R = 2.0\ \Omega. Find \varepsilon and I.

\varepsilon = BLv = (0.80)(0.50)(3.0) = 1.2\ \text{V}

I = \frac{\varepsilon}{R} = \frac{1.2}{2.0} = 0.60\ \text{A}

Example: magnetic force opposing motion

Using the same numbers, the magnetic force on the rod is

F = ILB = (0.60)(0.50)(0.80) = 0.24\ \text{N}

That force direction is opposite the motion (Lenz’s law), so an external pull of 0.24\ \text{N} is required to maintain constant speed.

Exam Focus
  • Typical question patterns:
    • Compute motional emf \varepsilon = BLv and induced current given a resistance.
    • Use energy reasoning to relate mechanical work/power to electrical dissipation in a resistor.
    • Determine induced current direction for a rod moving through a field (often into/out of page field symbols).
  • Common mistakes:
    • Forgetting that motional emf requires the correct geometry (velocity must have a component that causes charges to be pushed along the conductor).
    • Getting induced current direction wrong by opposing the field rather than opposing the change in flux.
    • Treating the magnetic force on the rod as aiding motion rather than opposing it when current is induced.

Self-Inductance, Inductors, and RL Circuits

What an inductor is and why it exists

An inductor is a circuit element (often a coil) that resists changes in current. When current changes in a coil, the magnetic field produced by that current changes, which changes the magnetic flux through the coil itself. By Faraday’s law, that changing self-flux induces an emf in the coil that opposes the change.

This is called self-induction, and it is the circuit-level version of Lenz’s law.

Inductors matter because they:

  • Smooth current changes in power supplies
  • Store energy in magnetic fields (unlike capacitors, which store energy in electric fields)
  • Appear in transformers and AC circuits

Inductance and induced emf

The induced emf across an inductor is modeled as

\varepsilon_L = -L\frac{dI}{dt}

where L is the inductance (unit: henry, H).

If the current changes approximately linearly over time, you can use an average-rate form:

\varepsilon_L \approx -L\frac{\Delta I}{\Delta t}

Meaning of the negative sign: If current is increasing, the inductor’s induced emf acts like a “back emf” opposing that increase. If current is decreasing, the inductor acts to keep current going.

Energy stored in an inductor

Inductors store energy in their magnetic fields. The energy stored when the current is I is

U = \frac{1}{2}LI^2

This formula is important for energy conservation questions and for understanding what happens when you open a circuit containing an inductor.

A real-world warning: inductors can generate large voltages if you try to change current very quickly (large dI/dt). That’s why car ignition systems can create sparks, and why circuits often include protection diodes.

RL circuits: current growth and decay

An RL circuit contains a resistor R and an inductor L.

  • When you connect a battery, current does not jump instantly to V/R. The inductor initially opposes current increase.
  • When you disconnect the battery, current does not drop instantly to zero. The inductor induces an emf that keeps current going (often causing a spark if there’s a gap).

The characteristic time scale is the time constant

\tau = \frac{L}{R}

For a series RL circuit connected to a battery of voltage V, the steady-state current is

I_{max} = \frac{V}{R}

The current as a function of time during growth is

I(t) = I_{max}\left(1 - e^{-t/\tau}\right)

During decay (battery removed, current decreasing), if the initial current is I_0, then

I(t) = I_0 e^{-t/\tau}

AP Physics 2 typically emphasizes interpreting these behaviors (initial vs final states, qualitative graphs, time constant meaning) more than deriving the exponential forms.

Key checkpoints you should always know:

  • At t = 0 right after switching on: inductor behaves like an open circuit (current near zero in the ideal model).
  • At long time after switching on: inductor behaves like a wire (nearly zero voltage across it), so current approaches V/R.

Induced emf size in switching situations

Because \varepsilon_L depends on dI/dt, fast changes produce large induced voltages. This is why opening a switch in an inductive circuit can cause arcing. In the AP model, this often becomes an explanation question: “Why does a spark occur when opening the switch?”

Example: time constant and interpretation

An inductor L = 0.40\ \text{H} is in series with a resistor R = 8.0\ \Omega.

Time constant:

\tau = \frac{L}{R} = \frac{0.40}{8.0} = 0.050\ \text{s}

Interpretation: after one time constant, the current has risen to about 63\% of its final value during growth, or fallen to about 37\% of its initial value during decay.

Example: energy stored

If the current through that inductor reaches I = 2.0\ \text{A}, energy stored is

U = \frac{1}{2}LI^2 = \frac{1}{2}(0.40)(2.0)^2 = 0.80\ \text{J}

Exam Focus
  • Typical question patterns:
    • Interpret RL switching graphs: identify initial and final currents and voltages across R and L.
    • Use \tau = L/R to compare how fast circuits respond.
    • Compute energy stored U = (1/2)LI^2 and connect to sparks/back emf explanations.
  • Common mistakes:
    • Treating an inductor like a resistor (assuming instantaneous current changes).
    • Forgetting that the inductor’s induced emf changes sign depending on whether current is increasing or decreasing.
    • Confusing “inductor stores energy” with “inductor dissipates energy” (ideal inductors store and return energy; resistors dissipate).

Generators, Transformers, and AC Applications of Induction

From induction to electric power: the unifying idea

Generators and transformers are essentially engineered ways to create a changing magnetic flux so that Faraday’s law produces useful emfs. The “trick” is controlling the change:

  • Generators: change flux by rotating a coil in a magnetic field (or rotating a magnet relative to a coil)
  • Transformers: change flux by using an alternating current to create a changing magnetic field in a core

AC generator: rotating a coil in a magnetic field

Consider a coil with N turns and area A rotating in a uniform magnetic field of magnitude B at angular speed \omega. If the angle between \vec{B} and the coil’s area vector is \theta(t), then

\Phi_B(t) = BA\cos\theta(t)

If the coil rotates so that

\theta(t) = \omega t

then

\Phi_B(t) = BA\cos(\omega t)

Faraday’s law gives a sinusoidal emf. The commonly used result for an ideal generator is

\varepsilon(t) = NBA\omega\sin(\omega t)

The peak emf is

\varepsilon_{max} = NBA\omega

Conceptual payoff: the generator converts mechanical work (turning the coil against magnetic effects) into electrical energy.

A frequent misconception is that the emf is maximum when the coil is “most aligned” with the field. Actually, emf depends on how fast flux changes. Flux is maximum when aligned, but its rate of change is zero there; emf is maximum when flux is changing fastest.

Transformers: changing voltage using changing magnetic flux

A transformer uses two coils:

  • Primary coil with N_p turns connected to an AC source
  • Secondary coil with N_s turns connected to a load

A changing current in the primary produces a changing magnetic field (often guided by an iron core), which changes flux through the secondary, inducing an emf.

For an ideal transformer (no energy losses), the voltage ratio equals the turns ratio:

\frac{V_s}{V_p} = \frac{N_s}{N_p}

This is one of the most important relationships in power distribution.

  • If N_s > N_p, it’s a step-up transformer (increases voltage).
  • If N_s < N_p, it’s a step-down transformer (decreases voltage).

Power and current in an ideal transformer

In an ideal transformer, power is conserved:

P_p = P_s

Using P = IV, that means

I_p V_p = I_s V_s

So the current ratio is the inverse of the voltage ratio:

\frac{I_s}{I_p} = \frac{V_p}{V_s} = \frac{N_p}{N_s}

Why this matters in real life: Electrical transmission lines use high voltage to reduce current for a given power, which reduces resistive losses

P_{loss} = I^2R

So stepping up voltage for transmission reduces I, drastically reducing I^2R heating losses.

Why transformers require changing current (AC, not steady DC)

A transformer relies on changing magnetic flux. With steady DC, after a brief transient when you first connect it, the current becomes constant, flux becomes constant, and the induced emf in the secondary goes to zero. That’s why transformers are designed for AC (or for rapidly switching DC in modern electronics, which is effectively AC-like).

Real-world notes (kept conceptual)

Real transformers have losses due to:

  • Resistance of the coils (heating)
  • Eddy currents in the core (mitigated by laminated cores)
  • Hysteresis losses in magnetic materials

AP questions may mention these qualitatively, especially eddy currents and why laminated cores help.

Example: transformer turns ratio

An ideal transformer has N_p = 500 turns and N_s = 50 turns. The primary is connected to V_p = 120\ \text{V} AC. Find V_s.

\frac{V_s}{V_p} = \frac{N_s}{N_p}

V_s = V_p\frac{N_s}{N_p} = 120\frac{50}{500} = 12\ \text{V}

So it steps down to 12\ \text{V}.

Example: current relationship

If the secondary delivers I_s = 2.0\ \text{A} at 12\ \text{V}, power is

P_s = I_sV_s = (2.0)(12) = 24\ \text{W}

Ideal primary current is

I_p = \frac{P_p}{V_p} = \frac{24}{120} = 0.20\ \text{A}

Notice how stepping down voltage steps up current.

Exam Focus
  • Typical question patterns:
    • Use V_s/V_p = N_s/N_p to find missing voltage or turns in an ideal transformer.
    • Use power conservation to relate primary and secondary currents.
    • Conceptual questions about why AC is required for transformer action and why high-voltage transmission reduces losses.
  • Common mistakes:
    • Treating transformer ratios backwards (mixing up primary vs secondary in the turns ratio).
    • Forgetting that an ideal transformer conserves power (so higher voltage implies lower current, and vice versa).
    • Thinking a transformer “creates” power; it changes voltage/current tradeoffs but does not add energy.