Lorentz Magnetic Force & Biot-Savart Law
- Lorentz Force on a Moving Charge:
$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$
When moving purely in perpendicular magnetic field: $F = qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB}$.
- Biot-Savart Law:
$$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I (d\vec{l} \times \hat{r})}{r^2}$$
- Center of circular current loop of radius $R$: $B = \frac{\mu_0 I}{2R}$
- Straight infinite wire at distance $r$: $B = \frac{\mu_0 I}{2\pi r}$
Ampere's Circuital Law & Galvanometer
$$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$$
- Inside a long solenoid ($n$ turns per unit length): $B = \mu_0 n I$.
Moving Coil Galvanometer:
Deflection $\theta$ is directly proportional to current: $I = \left(\frac{k}{NAB}\right)\theta$.
- Converting to Ammeter: Connect a small resistance (shunt, $S$) in parallel: $S = \frac{I_g R_g}{I - I_g}$.
- Converting to Voltmeter: Connect a large resistance ($R$) in series: $R = \frac{V}{I_g} - R_g$.
