Unit 4: Magnetic Fields

Magnetic field (\vec{B})

A vector field that describes magnetic forces on moving electric charges and on currents; a stationary charge (in the source’s rest frame) feels no magnetic force.

Tesla (T)

SI unit of magnetic field: 1 T = 1 N/(A·m) = 1 N·s/(C·m).

Magnetic field lines

Visualization tool where the tangent gives the direction of \vec{B} and the line density indicates relative field strength; magnetic field lines form closed loops.

Into/out-of-page notation (dot/cross)

A dot (•) indicates a vector pointing out of the page; a cross (×) indicates a vector pointing into the page.

Conventional current

Current direction defined as the direction positive charge would move (opposite electron flow); used in right-hand rules and in \vec{L} for wire forces.

Right-hand rule (straight current-carrying wire)

Point your right-hand thumb along the conventional current; curled fingers show the direction of circular magnetic field lines around the wire.

Right-hand rule (circular current loop)

Curl right-hand fingers with the current around the loop; thumb points along the loop’s axis in the direction of \vec{B} through the center (and of \hat{n}).

Solenoid

A coil of wire that produces a magnetic field when current flows; a long ideal solenoid has a strong, nearly uniform internal field parallel to its axis and a small external field.

Ideal solenoid field (B = \mu_0 n I)

Magnetic field inside a long (ideal) solenoid: B = \mu_0 n I, where n is turns per unit length (n = N/\ell).

Bar magnet field direction (outside magnet)

Outside a bar magnet, field lines point from the north pole to the south pole; the field is strongest near the poles.

Gauss’s law for magnetism

(\oint \vec{B}\cdot d\vec{A} = 0): net magnetic flux through any closed surface is zero, reflecting no isolated magnetic monopoles in standard treatment.

Divergence-free magnetic field (\nabla\cdot\vec{B}=0)

Vector-calculus form of Gauss’s law for magnetism; conceptually indicates magnetic field lines do not start or end (no sources/sinks).

Superposition (magnetic fields)

The net magnetic field from multiple currents/sources is the vector sum of each individual field contribution.

Electric force on a charge (\vec{F}=q\vec{E})

Force on a charge q in an electric field \vec{E}; points along \vec{E} for positive charges and opposite \vec{E} for negative charges.

Lorentz magnetic force (\vec{F}=q\vec{v}\times\vec{B})

Magnetic force on a charge q moving with velocity \vec{v} in a magnetic field \vec{B}; direction is perpendicular to both \vec{v} and \vec{B}.

Total Lorentz force (electric + magnetic)

Combined force when both fields are present: (\vec{F}=q\vec{E}+q\vec{v}\times\vec{B}).

Magnetic force magnitude (F = |q|vB\sin\theta)

Magnitude of the magnetic part of the Lorentz force, where \theta is the angle between \vec{v} and \vec{B}.

Zero magnetic force condition

Magnetic force is zero when \vec{v} is parallel or antiparallel to \vec{B} (\sin\theta = 0).

Negative-charge sign flip (direction)

The right-hand rule gives force direction for a positive charge; for a negative charge, the magnetic force points in the opposite direction.

Magnetic force does no work

Because \vec{F}_B \perp \vec{v}, the power from magnetic force is (P=\vec{F}\cdot\vec{v}=0); magnetic fields change direction of motion but not speed (in magnetostatics).

Force on a current-carrying wire segment (\vec{F}=I\vec{L}\times\vec{B})

Magnetic force on a straight wire segment of vector length \vec{L} (in direction of conventional current) in a uniform magnetic field \vec{B}.

Wire force magnitude (F = ILB\sin\theta)

Magnitude of the force on a wire segment, where \theta is the angle between the current direction (\vec{L}) and \vec{B}.

Net force vs torque on a current loop

In a uniform \vec{B}, forces on opposite sides of a closed loop can cancel (net force ≈ 0) while still producing a nonzero torque (a couple) that causes rotation (motor principle).

Magnetic dipole moment (\vec{\mu}=NIA\hat{n})

Vector that characterizes a current loop’s magnetic strength and orientation; \hat{n} is perpendicular to the loop’s plane set by the current right-hand rule.

Torque on a dipole/loop (\vec{\tau}=\vec{\mu}\times\vec{B})

Torque on a magnetic dipole in a uniform magnetic field; tends to rotate \vec{\mu} to align with \vec{B}.

Magnetic dipole potential energy (U = -\vec{\mu}\cdot\vec{B})

Potential energy of a magnetic dipole in a magnetic field; minimum when \vec{\mu} is parallel to \vec{B} and maximum when antiparallel.

Mechanical torque (\tau = Fr\sin\theta)

General mechanics definition of torque magnitude from a force F applied at lever arm r, with \theta between \vec{r} and \vec{F}.

Torsional stiffness (k = \tau/\theta)

Measure of how much torque \tau is needed to twist an object by angle \theta (radians); depends on material and geometry.

Circular motion in a uniform magnetic field

When \vec{v} \perp \vec{B}, the magnetic force acts as a centripetal force, producing circular motion at constant speed.

Radius of curvature (r = mv/(|q|B))

Radius of a charged particle’s circular path in a uniform magnetic field when motion is perpendicular to \vec{B}.

Cyclotron angular frequency (\omega = |q|B/m)

Angular frequency of circular motion of a nonrelativistic charged particle in a uniform magnetic field.

Cyclotron period (T = 2\pi m/(|q|B))

Time for one full revolution in a uniform magnetic field; in the nonrelativistic model it depends on m, |q|, and B (not on speed or radius).

Helical motion

Trajectory when a particle has both perpendicular and parallel velocity components relative to \vec{B}: circular motion from v\perp plus constant motion along \vec{B} from v\parallel.

Velocity components relative to \vec{B} (v\perp, v\parallel)

Decomposition of velocity: v\perp causes magnetic deflection (circular part), while v\parallel is unaffected by \vec{B} and remains constant.

Helix pitch (p = v_\parallel T)

Distance advanced along the magnetic field direction in one revolution of helical motion.

Velocity selector

Crossed electric and magnetic fields configured so only particles with a specific speed pass through undeflected (electric and magnetic forces cancel).

Selected speed in a velocity selector (v = E/B)

Speed for which qE and qvB magnitudes are equal (with perpendicular fields), causing zero net deflection.

Biot–Savart law

Rule for the magnetic field contribution from a steady current element: (d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{\ell}\times \hat{r}}{r^2}).

Vacuum permeability (\mu_0)

Magnetic constant in SI units used in many AP calculations (often treated as (4\pi\times10^{-7}\ \text{T·m/A})).

Long straight wire field (B = \mu_0 I/(2\pi r))

Magnitude of the magnetic field a distance r from a very long straight wire carrying current I; direction is tangential to circles centered on the wire.

Circular loop center field (B = \mu_0 I/(2R))

Magnetic field magnitude at the center of a single circular loop of radius R carrying current I.

N-turn loop center field (B = \mu_0 N I/(2R))

Magnetic field magnitude at the center of a circular coil with N turns (each of radius R) carrying current I.

Magnetic field on the axis of a loop

On-axis field of a single circular loop (radius R) a distance x from center: (B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}).

Ampère’s law (integral form)

For magnetostatics: (\oint \vec{B}\cdot d\vec{\ell} = \mu0 I{\text{enc}}); most useful when symmetry makes \vec{B} constant along the chosen loop.

Toroid field (B = \mu_0 N I/(2\pi r))

Magnetic field inside an ideal toroid (in the core region) at radius r from the center; outside is much smaller (idealized as ~0).

Force per unit length between parallel wires

For two long parallel wires separated by distance d with currents I1 and I2: (\frac{F}{L} = \frac{\mu0 I1 I_2}{2\pi d}). Same-direction currents attract; opposite-direction currents repel.

Magnetic flux (\Phi_B)

Scalar measure of magnetic field through an area: (\PhiB=\int \vec{B}\cdot d\vec{A}); for uniform \vec{B} through flat area A, (\PhiB=BA\cos\theta) where \theta is between \vec{B} and the area normal.

Diamagnetism

Material response where an induced magnetic moment opposes an applied magnetic field (typically weak repulsion).

Paramagnetism

Material response where magnetic moments tend to align with an applied field, producing weak attraction.

Ferromagnetism

Strong magnetic behavior due to domain alignment; can produce persistent magnetization and strong attraction (basis of many permanent magnets).