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

Magnetic field (\u2192B)

A vector field that describes magnetic forces in space; measured in teslas (T).

Tesla (T)

The SI unit of magnetic field strength; 1 T = 1 N/(A\u00b7m).

Key difference: electric field vs magnetic field

Electric fields act on charges whether moving or not; magnetic fields exert force only on moving charges (or currents).

Magnetic field lines

Visual representation of $\rightarrow B$: direction is tangent to the line at each point; closer lines indicate stronger field.

Closed-loop nature of magnetic field lines

Magnetic field lines form continuous closed loops and do not begin or end (no monopoles in the standard model).

Magnetic monopole (in AP model)

A hypothetical isolated north or south pole; not included in the standard AP Physics 2 model.

Magnetic force on a moving charge

$\rightarrow F_B = q \rightarrow v \times \rightarrow B$; force is perpendicular to both velocity and magnetic field.

Magnitude of magnetic force on a charge

$F_B = |q|vB \sin(\theta)$, where $\theta$ is the angle between $\rightarrow v$ and $\rightarrow B$.

Angle factor (sin$\theta$) in magnetic force

Accounts for orientation: force is zero if motion is parallel to $\rightarrow B$ and maximum if motion is perpendicular to $\rightarrow B$.

Right-hand rule for \u2192F_B (positive charge)

Point fingers along \u2192v, curl toward \u2192B, thumb gives direction of \u2192F_B.

Negative charge direction in a magnetic field

Force direction is opposite the right-hand-rule result for a positive charge.

Magnetic force does no work (on a point charge)

Because $\rightarrow F_B \perp \rightarrow v$, it changes direction but not speed; kinetic energy stays constant if only magnetic forces act.

Uniform circular motion in a magnetic field

If \u2192v \u22a5 \u2192B in a uniform field, magnetic force provides centripetal force, producing circular motion.

Cyclotron (radius) formula

r = mv/(|q|B) for a charged particle moving perpendicular to a uniform magnetic field.

Cyclotron (period) formula

T = $\frac{2\pi m}{|q|B}$; the period is independent of speed v in the ideal model.

Magnetic force on a current-carrying wire

$F = ILB \sin(\theta)$ for a straight wire segment of length L in a uniform magnetic field.

Vector form for force on a wire

$\rightarrow F = I \rightarrow L \times \rightarrow B$, where $\rightarrow L$ points in the direction of conventional current.

Torque on a current loop

A current loop in a magnetic field experiences a torque that tends to rotate it to align with the field.

Magnetic dipole moment (\u2192$\mu$)

For a loop: magnitude $\mu = NIA$; direction is perpendicular to the loop (right-hand rule).

Torque magnitude on a loop

\u03c4 = \u03bcBsin\u03b8, where \u03b8 is the angle between \u2192\u03bc and \u2192B.

Net force vs net torque on a loop (uniform field)

In a uniform magnetic field, a current loop has zero net force but can have a nonzero net torque (a couple).

Moving charges as sources of magnetic fields

Currents (moving charges) produce magnetic fields; key quantitative sources: long wires, loops, and solenoids.

Permeability of free space (\u03bc0)

A constant in magnetic field equations: \u03bc0 = 4\u03c0\u00d710\u22127 T\u00b7m/A.

Magnetic field of a long straight wire

B = $\frac{\mu_0 I}{2\pi r}$, where r is distance from the wire.

Right-hand grip rule (straight wire)

Thumb points along conventional current; curled fingers show circular direction of \u2192B around the wire.

Magnetic field at the center of a circular loop

For one loop: B = (\u03bc0 I)/(2R); for N turns: B = (\u03bc0 N I)/(2R).

Right-hand rule (current loop field direction)

Curl fingers with current around loop; thumb points in direction of \u2192B through the center.

Solenoid

A long coil of wire that produces an approximately uniform magnetic field inside when current flows.

Turn density (n)

For a solenoid: n = N/L, where N is total turns and L is solenoid length.

Magnetic field inside an ideal long solenoid

B = $\mu_0 n I$; field is approximately uniform inside the solenoid.

Magnetic superposition

Magnetic fields add as vectors; you must account for direction (not just magnitudes).

Ferromagnetic material

Material (e.g., iron) that can greatly increase magnetic field strength by aligning magnetic domains.

Iron core effect in a solenoid

An iron core amplifies the magnetic field produced by current (it doesn\u2019t create field from nothing).

Magnetic flux (\u03a6_B)

A measure of how much magnetic field passes through a surface; depends on B, area, and orientation.

Magnetic flux formula (uniform field)

$\Phi_B = BA \cos(\theta)$, where $\theta$ is between $\rightarrow B$ and the area vector (surface normal).

Area vector (surface normal)

A vector perpendicular to a surface used in flux calculations; its direction sets the flux angle $\theta$.

Weber (Wb)

The SI unit of magnetic flux; $1\,\text{Wb} = 1\,\text{T}\cdot\text{m}^2$.

Flux linkage (N\u03a6_B)

Total linked flux for a coil with N identical turns; appears in Faraday\u2019s law.

Electromagnetic induction

Production of an emf due to a changing magnetic flux through a loop or coil.

Faraday\u2019s law

$\varepsilon = -N \left(\frac{\Delta \Phi_B}{\Delta t}\right)$ (or $-N \frac{d\Phi_B}{dt}$); changing flux induces emf.

Lenz\u2019s law

The induced current creates a magnetic field that opposes the change in magnetic flux that produced it.

Induced emf vs induced current

Faraday\u2019s law gives emf; current flows only if there is a closed conducting path (I = |\u03b5|/R).

Motional emf

An induced emf caused by a conductor moving through a magnetic field, leading to charge separation.

Motional emf formula

\u03b5 = BLv for a rod of length L moving at speed v perpendicular to a uniform field B (with proper geometry).

Sliding rod on rails (induced current)

A moving rod changes loop area, changing flux; induced current magnitude is $I = \frac{BLv}{R}$.

Magnetic drag (induction braking)

A resistive force opposing motion caused by induced currents; mechanical work converts to electrical/thermal energy.

Eddy currents

Circulating currents induced in bulk conductors by changing flux; they oppose motion/flux change and can cause heating.

Inductor

A circuit element (often a coil) that resists changes in current by inducing an emf opposing \u0394I/\u0394t.

Inductance (L)

A property of an inductor that sets the induced emf for a given rate of change of current; unit is henry (H).

RL circuit time constant (\u03c4)

$\tau = \frac{L}{R}$; sets the timescale for current growth/decay in a series resistor-inductor circuit.