Summary

• Lorentz Force: Force on a charge $q$ moving with velocity $\vec{v}$ in electric $\vec{E}$ and magnetic $\vec{B}$ fields: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$. For a current-carrying conductor, $\vec{F} = I (\vec{l} \times \vec{B})$.

• Biot–Savart Law: Magnetic field due to a current element $I d\vec{l}$ at distance $r$: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$. For a long straight wire: $B = \frac{\mu_0 I}{2\pi r}$.

• Ampere’s Circuital Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$. Use symmetry: solenoid $B = \mu_0 n I$ (inside), toroid $B = \frac{\mu_0 N I}{2\pi r}$.

• Force Between Parallel Wires: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ — attractive if currents same direction, repulsive if opposite.

• Torque on a Current Loop: $\vec{\tau} = \vec{m} \times \vec{B}$, where magnetic moment $\vec{m} = N I A \hat{n}$. Potential energy: $U = -\vec{m} \cdot \vec{B}$.

• Moving Coil Galvanometer: Current sensitivity $\propto N B A / k$; converted to ammeter by shunt $S = \frac{I_g G}{I - I_g}$, to voltmeter by series resistor $R = \frac{V}{I_g} - G$.

• Cyclotron: Frequency $f = \frac{qB}{2\pi m}$ independent of speed; maximum kinetic energy $K_{\text{max}} = \frac{q^2 B^2 R^2}{2m}$.

• Key Constants: $\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}$; permeability of free space.

Physical quantities, symbols, dimensions and units used in this chapter.

| Physical Quantity | Symbol | Nature | Dimensions | Units | Remarks |

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