Magnetic Fields

Magnetic Force

On a moving charge: $\vec{F} = q\vec{v} \times \vec{B}$ Magnitude: $F = qvB\sin\theta$.

On a current-carrying wire: $\vec{F} = I\vec{L} \times \vec{B}$ Magnitude: $F = BIL\sin\theta$.

Charged Particle in a Magnetic Field

• Circular motion: $r = mv/(qB)$.
• Period: $T = 2\pi m/(qB)$ (independent of speed).
• Helical motion when $\vec{v}$ has a component parallel to $\vec{B}$.

Biot-Savart Law

$d\vec{B} = \frac{\mu_0}{4\pi}\frac{Id\vec{l} \times \hat{r}}{r^2}$

Field at center of circular loop: $B = \frac{\mu_0 I}{2R}$

Field on axis of circular loop (distance $x$): $B = \frac{\mu_0 IR^2}{2(x^2 + R^2)^{3/2}}$

Long straight wire: $B = \frac{\mu_0 I}{2\pi r}$

Ampère's Law

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

Useful for high symmetry:

• Long straight wire: $B = \mu_0 I/(2\pi r)$
• Inside a solenoid: $B = \mu_0 n I$ ($n$ = turns per unit length)
• Toroid: $B = \mu_0 NI/(2\pi r)$

Force Between Parallel Wires

$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$

Parallel currents attract; antiparallel currents repel.

Magnetic Torque on a Current Loop

$\vec{\tau} = \vec{\mu} \times \vec{B}, \quad \mu = NIA$ where $\vec{\mu}$ is the magnetic dipole moment.

Problem: Use the Biot-Savart law to derive the magnetic field at the center of a circular loop of radius $R$ carrying current $I$.

Solution:

For each element $dl$ of the loop, $|d\vec{l} \times \hat{r}| = dl$ (since they are perpendicular) and $r = R$ for all elements.

$B = \frac{\mu_0 I}{4\pi R^2}\oint dl = \frac{\mu_0 I}{4\pi R^2}(2\pi R) = \frac{\mu_0 I}{2R}$

• Biot-Savart law: $d\vec{B} = (\mu_0/4\pi)(Id\vec{l}\times\hat{r})/r^2$.
• Ampère's law: $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$ — for symmetric current distributions.
• Solenoid: $B = \mu_0 nI$; center of loop: $B = \mu_0 I/(2R)$.
• Magnetic force is perpendicular to velocity — it does no work.