Key Points

Hans Christian Oersted discovered in 1820 that a moving charge or an electric current produces a magnetic field in the surrounding space. This established the intimate relationship between electricity and magnetism.

The total force on a charge $q$ moving with velocity $\mathbf{v}$ in combined electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields is the Lorentz force, given by $\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$.

The magnetic force $\mathbf{F}_m = q(\mathbf{v} \times \mathbf{B})$ is always perpendicular to both velocity $\mathbf{v}$ and magnetic field $\mathbf{B}$. Consequently, it does no work on the charged particle and only changes its direction of motion, not its speed.

A charged particle $q$ moving perpendicular to a uniform magnetic field $\mathbf{B}$ follows a circular path. The radius of this path is given by $r = \frac{mv}{qB}$, where $m$ and $v$ are the mass and speed of the particle.

The frequency of revolution for a charged particle in a uniform magnetic field is called the cyclotron frequency, given by $\nu_c = \frac{qB}{2\pi m}$. This frequency is independent of the particle's velocity and the radius of its orbit.

The Biot-Savart law gives the magnetic field $d\mathbf{B}$ produced by an infinitesimal current element $I d\mathbf{l}$ as $d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{r}}{r^3}$. Here, $\mu_0$ is the permeability of free space.

Ampere's law states that the line integral of the magnetic field $\mathbf{B}$ around any closed loop is proportional to the total current $I_{\text{enc}}$ passing through the loop: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}$.

The magnitude of the magnetic field at a perpendicular distance $r$ from a long, straight wire carrying a current $I$ is given by $B = \frac{\mu_0 I}{2\pi r}$. The magnetic field lines are concentric circles around the wire.

The magnetic field inside a long solenoid is uniform and strong, given by the formula $B = \mu_0 n I$. Here, $n$ is the number of turns per unit length and $I$ is the current. The field outside is negligible.

A straight conductor of length $l$ carrying a current $I$ placed in a uniform external magnetic field $\mathbf{B}$ experiences a force given by $\mathbf{F} = I (\mathbf{l} \times \mathbf{B})$.

Two parallel wires carrying currents in the same direction attract each other, while wires with currents in opposite directions repel. The force per unit length between them is $f = \frac{\mu_0 I_a I_b}{2\pi d}$.

A planar loop with $N$ turns carrying current $I$ and having area $\mathbf{A}$ has a magnetic dipole moment $\mathbf{m} = NI\mathbf{A}$. Its unit is ampere-meter squared ($\text{A m}^2$).

When a current loop with magnetic moment $\mathbf{m}$ is placed in a uniform magnetic field $\mathbf{B}$, it experiences a torque given by $\tau = \mathbf{m} \times \mathbf{B}$. The net force on the loop is zero.

A moving coil galvanometer (MCG) works on the principle that a current-carrying coil experiences a torque in a magnetic field. In equilibrium, the magnetic torque $NIAB$ is balanced by the restoring torque $k\phi$ of a spring, so $\phi = (\frac{NAB}{k})I$.

To convert a galvanometer into an ammeter, a low resistance called a shunt ($r_s$) is connected in parallel with the galvanometer coil. This allows most of the circuit current to bypass the galvanometer.

To convert a galvanometer into a voltmeter, a high resistance ($R$) is connected in series with the galvanometer coil. This limits the current drawn from the circuit, ensuring an accurate voltage measurement.