AP Physics C: E&M Unit 4 Notes — How Currents Create Magnetic Fields

Biot–Savart law

A “from-the-source” method for magnetostatics that gives the magnetic field contribution from a small current element and then integrates (vector-sums) over the entire current distribution.

Permeability of free space ($\mu_0$)

A physical constant that sets the scale of magnetic fields produced by currents in vacuum; it appears as the proportionality constant in Biot–Savart and Ampere’s law.

Current element ($I\,d\vec{\ell}$)

The source term in Biot–Savart: a tiny wire segment vector $d\vec{\ell}$ pointing along the current direction, multiplied by the current $I$.

Position vector in Biot–Savart ($\vec{r}$)

The vector drawn from the current element (source point) to the observation (field) point; its magnitude r is the distance between them.

Unit vector ($\hat{r}$)

The unit (length-1) vector in the direction of $\vec{r}$, used to specify direction from the current element to the field point.

Cross product ($d\vec{\ell} \times \hat{r}$)

The vector operation in Biot–Savart that determines the direction of $d\vec{B}$ (perpendicular to both $d\vec{\ell}$ and $\hat{r}$) and includes a $\sin(\theta)$ factor for the angle between them.

Right-hand rule (for $d\vec{\ell} \times \hat{r}$)

A direction rule for the Biot–Savart cross product: point fingers along $d\vec{\ell}$, curl toward $\hat{r}$, and the thumb gives the direction of $d\vec{B}$.

Vector superposition (magnetic fields)

Magnetic fields add as vectors: the net field is the vector sum/integral of all individual contributions, so direction and sign matter (not just magnitudes).

Ampere’s law (integral form)

For steady currents, the circulation of the magnetic field around any closed loop equals $\mu_0$ times the net current enclosed: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc}$.

Amperian loop

A chosen closed path used in Ampere’s law; it is selected to exploit symmetry so $\vec{B}$ is tangent and/or constant along parts of the loop.

Enclosed current ($I_{enc}$)

The net current passing through the surface bounded by an Amperian loop, including sign based on orientation (currents in opposite directions can cancel).

Circulation ($\oint \vec{B} \cdot d\vec{\ell}$)

A closed-loop line integral measuring the tangential component of $\vec{B}$ along a loop; it is the left-hand side of Ampere’s law.

Volume current density ($\vec{J}$)

Current per unit cross-sectional area flowing through a conductor volume (units $A/m^2$); used to compute enclosed current via area integration.

Surface current density ($\vec{K}$)

Current per unit width flowing along a surface (units A/m); commonly used for idealized current sheets.

Turns per unit length (n)

Coil winding density for a solenoid (units 1/m); used in the ideal solenoid field formula $B = \mu_0 n I$.

Number of turns (N)

The total number of wire loops in a coil (dimensionless); magnetic fields from identical turns scale linearly with N (e.g., loop or toroid results).

Magnetic field of an infinite straight wire

A standard Ampere’s law result: at distance r from a long straight wire, $B = \mu_0 I/(2\pi r)$, tangent to circles centered on the wire (right-hand rule).

Magnetic field inside a uniform solid cylindrical conductor

For r < R in a solid wire of radius R with uniform current density, the field increases linearly: $B = (\mu_0 I/(2\pi R^2)) r$ (direction is circular around the axis).

Magnetic field at the center of a circular loop

A Biot–Savart result for a loop of radius R: $B = \mu_0 I/(2R)$, directed along the loop’s axis (perpendicular to the loop’s plane) by the right-hand rule.

Magnetic field on the axis of a circular loop

At distance x along the axis of a loop (radius R): $B = \mu_0 I R^2 / (2(R^2 + x^2)^{3/2})$, directed along the axis by the right-hand rule.

Magnetic field of a finite straight wire segment

A Biot–Savart result for a point a perpendicular distance r from the wire: $B = (\mu_0 I/(4\pi r))(\sin(\theta_1) + \sin(\theta_2))$, where $\theta_1$ and $\theta_2$ are angles to the endpoints measured from the perpendicular.

Long (ideal) solenoid magnetic field

Using Ampere’s law with symmetry for a long, tightly wound solenoid: inside, $B \approx \mu_0 n I$ (nearly uniform and along the axis); outside is approximately small for the ideal case.

Toroid magnetic field (inside the windings)

For a toroid with N turns, at radius r within the core region: $B = \mu_0 N I/(2\pi r)$, tangent to circular paths; an ideal toroid has approximately zero field outside.

Infinite current sheet magnetic field

An ideal infinite sheet with uniform surface current density K produces uniform fields of magnitude $B = \mu_0 K/2$ on each side, with opposite directions determined by a right-hand rule.

Maxwell–Ampere law (displacement current correction)

The generalized Ampere’s law valid with time-varying electric fields: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0(I_{enc} + \epsilon_0 d\Phi_E/dt)$, where $\epsilon_0 d\Phi_E/dt$ is the displacement current term.