Unit 4 Magnetism: Understanding Magnetic Fields and the Forces They Produce

Magnetic field (\vec{B})

A vector field that exists in a region of space and exerts forces on moving electric charges and on currents; measured in teslas (T).

Tesla (T)

The SI unit of magnetic field strength (magnetic flux density) used for $\vec{B}$.

Conventional current

The direction positive charge would move (from + to −); AP right-hand rules use conventional current, not electron drift direction.

Magnetic field lines

A visual representation of \vec{B}: the field direction at a point is tangent to the line, and closer spacing indicates stronger field.

Permeability of free space (\mu_0)

A constant in magnetic field formulas: \mu_0 = 4\pi\times 10^{-7}\ \text{T·m/A}.

Right-hand rule (straight wire)

For a long straight current-carrying wire: point your right thumb with conventional current; curled fingers show the direction of $\vec{B}$ circling the wire.

Magnetic field of a long straight wire

Magnitude at distance r from a long straight wire: $B = \frac{\mu_0 I}{2\pi r}$; field lines are concentric circles around the wire.

Magnetic field at center of a circular loop

For a single circular loop of radius R: $B = \frac{\mu_0 I}{2R}$ at the center (direction from a right-hand rule around the loop).

Ideal solenoid

A long coil of wire that produces an approximately uniform magnetic field inside; outside field is much weaker for an ideal long solenoid.

Turns per unit length (n)

The number of turns per meter in a solenoid (turns/m), used in $B = \mu_0 n I$.

Lorentz force (magnetic part)

Vector magnetic force on a moving charge: $\vec{F} = q\vec{v}\times\vec{B}$.

Cross product ( imes)

A vector operation giving a result perpendicular to both input vectors; magnitude depends on $\sin\theta$ and direction follows a right-hand rule.

Magnetic force magnitude on a charge

Magnitude of the magnetic force: $F = |q| v B \sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.

Right-hand rule for \vec{v} imes\vec{B} (positive charge)

Point fingers along $\vec{v}$, curl toward $\vec{B}$ (smaller angle), thumb gives $\vec{F}$ for a positive charge.

Negative charge force direction

For a negative charge (e.g., electron), the magnetic force direction is opposite the right-hand-rule thumb direction for \vec{v} imes\vec{B}.

Magnetic force does no work

In the ideal magnetic-only case, $\vec{F}$ is perpendicular to velocity, so it changes direction but not speed; kinetic energy stays constant.

Circular motion radius in uniform \vec{B}

If $\vec{v} \perp \vec{B}$, the path is circular with radius $r = \frac{mv}{|q|B}$. Larger $B$ gives a smaller $r$.

Cyclotron period in uniform \vec{B}

Time for one revolution when $\vec{v} \perp \vec{B}$: $T = \frac{2\pi m}{|q|B}$, which does not depend on speed.

Helical motion

Motion in a uniform magnetic field when velocity has components both perpendicular and parallel to \vec{B}: circular motion plus constant forward motion (a helix).

Velocity selector (crossed \vec{E} and \vec{B})

With perpendicular electric and magnetic fields, a particle goes straight when forces balance: $|q|E = |q|vB$, so $v = \frac{E}{B}$.

Force on a current-carrying wire segment

Magnetic force on a straight wire segment in uniform $\vec{B}$: $\vec{F} = I\vec{L}\times\vec{B}$; magnitude $F = I L B \sin\theta$.

Length vector (\vec{L}) in \vec{F}=I\vec{L} imes\vec{B}

A vector pointing along conventional current with magnitude equal to the length of wire actually in the magnetic field region.

Magnetic torque on a current loop

A loop in a uniform magnetic field experiences torque: $\tau = N I A B \sin\theta$, where $\theta$ is between the loop’s area vector and $\vec{B}$.

Magnetic dipole moment (\mu = NIA)

A property of a current loop equal to number of turns × current × area; determines how strongly the loop tends to align with a magnetic field.

Force per unit length between parallel wires

Two long parallel wires separated by r exert force per length: $\frac{F}{L} = \mu_0 I_1 I_2/(2\pi r)$; same current directions attract, opposite directions repel.