AP Physics 2- 12.3 Magnetism and Current- Carrying Wires- Study Notes- New Syllabus

Magnetic Field Produced by a Current-Carrying Wire

A moving charge or current produces a magnetic field in the surrounding space. The direction of the magnetic field around a straight current-carrying wire is determined by the right-hand rule: if the thumb points in the direction of current, the curled fingers show the circular magnetic field lines.

Biot–Savart Law (General Expression):

The magnetic field at a point due to a small current element is given by:

\( \delta \vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I \, \delta \vec{l} \times \hat{r}}{r^2} \)

• \(\mu_0\): permeability of free space \((4\pi \times 10^{-7} \, \mathrm{T \cdot m/A})\).
• \(I\): current in the wire.
• \(\delta \vec{l}\): small element of the wire in the direction of current.
• \(\hat{r}\): unit vector pointing from the current element to the field point.
• \(r\): distance between the current element and the field point.

Magnetic Field Due to a Long Straight Wire:

Using the Biot–Savart law and symmetry, the magnetic field at a perpendicular distance \(\mathrm{r}\) from a long straight wire is: 

\( B = \dfrac{\mu_0 I}{2\pi r} \)

• Direction is given by the right-hand rule.
• Field lines form concentric circles around the wire.

Magnetic Field of a Circular Current Loop:

At the center of a circular loop of radius \(\mathrm{R}\) carrying current \(\mathrm{I}\):

\( B = \dfrac{\mu_0 I}{2R} \)

• Field direction: given by the right-hand rule (curl fingers along current → thumb gives field direction at center).
• For \(N\) turns: \( B = \dfrac{\mu_0 N I}{2R} \).

Magnetic Field Inside a Long Solenoid:

A solenoid is a coil of many turns of wire carrying current. The magnetic field inside an ideal solenoid is nearly uniform:

\( B = \mu_0 n I \)

• \(n = \dfrac{N}{L}\): number of turns per unit length.
• Outside the solenoid, the field is very weak (ideal solenoid approximation).