Magnetic Effects of Current and Magnetism — Practice Questions with Answers

The Biot-Savart law gives the small magnetic field $d\vec{B}$ produced at a point by a small current element $Id\vec{l}$, stating that it is proportional to the current, the length of the element, the sine of the angle between the element and the position vector to the point, and inversely proportional to the square of the distance.

$d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec{l}\times\hat{r}}{r^2}$

Ampere's circuital law states that the line integral of the magnetic field around any closed loop equals $\mu_0$ times the total current enclosed by that loop: $\oint\vec{B}\cdot d\vec{l} = \mu_0I_{enc}$.

It greatly simplifies field calculations for highly symmetric current distributions — for example, finding the magnetic field inside a long, straight current-carrying solenoid, where symmetry allows the loop integral to be evaluated algebraically rather than through direct, complex Biot-Savart integration.

The magnetic force provides the centripetal force for circular motion: $qvB = \dfrac{mv^2}{r}$

Solving for $r$: $r = \dfrac{mv}{qB}$

This shows the radius is directly proportional to momentum ($mv$) and inversely proportional to both charge and field strength.

Each current-carrying wire produces its own circular magnetic field around itself (by the right-hand rule). At the location of the second wire, the first wire's field points in a specific direction (perpendicular to the line joining the wires).

The second wire, carrying current within this external field, experiences a force $F = I_2lB_1$ (by $\vec{F}=I\vec{l}\times\vec{B}$) — careful application of the right-hand rule shows that when both currents flow in the same direction, this resulting force pulls the wires toward each other (attraction); if currents are in opposite directions, the force pushes them apart (repulsion).

A moving coil galvanometer consists of a rectangular coil of many turns, suspended in a strong radial magnetic field (usually from permanent magnets), with the coil free to rotate against a restoring spring (torsion) force.

When current flows through the coil, it experiences a torque due to the magnetic field ($\tau = NIAB$), causing it to rotate until this magnetic torque is balanced by the restoring torque of the spring — the resulting deflection angle is directly proportional to the current, allowing the device to measure current (or, with appropriate modification, voltage or resistance).

Diamagnetic materials are weakly repelled by an external magnetic field, and lose their (very weak) induced magnetisation once the field is removed. Example: bismuth, copper.

Paramagnetic materials are weakly attracted by an external field, with induced magnetisation aligned with the field, but also lose it once the field is removed. Example: aluminium, sodium.

Ferromagnetic materials are strongly attracted by an external field, and can retain significant magnetisation even after the field is removed (permanent magnetism). Example: iron, cobalt, nickel.

The magnetic dipole moment of a current loop, $m = NIA$ (for $N$ turns, current $I$, area $A$), is a vector quantity representing the loop's overall magnetic strength and orientation, directed perpendicular to the loop's plane (by the right-hand rule).

When placed in a uniform external field $B$, the loop experiences a torque: $\vec{\tau} = \vec{m}\times\vec{B}$, or in magnitude, $\tau = mB\sin\theta$, where $\theta$ is the angle between the dipole moment and the field — this torque tends to align the loop's magnetic moment with the external field.

A cyclotron is a device used to accelerate charged particles to high speeds/energies, using a combination of a uniform magnetic field (to keep particles moving in a circular path) and an alternating electric field (to repeatedly accelerate them).

Charged particles move in a spiral path within two D-shaped hollow electrodes ("dees"), gaining kinetic energy each time they cross the gap between the dees (accelerated by the alternating electric field), while the magnetic field curves their path into an increasing spiral of radius, until they reach the desired high energy and exit the device.

Inside an ideal (very long) solenoid, the magnetic field contributions from the many tightly wound circular current loops combine constructively along the axis, resulting in a strong, essentially uniform field parallel to the axis throughout the interior — this uniformity arises from the solenoid's high symmetry and can be shown rigorously using Ampere's law.

Outside the solenoid (but still relatively close to it), the field contributions from opposite sides of the winding largely cancel each other out (a consequence of the field pattern of many closely spaced loops), leaving a near-zero field just outside an ideal, tightly wound, very long solenoid.

The magnetic field at the centre of a circular current loop, $B = \dfrac{\mu_0NI}{2R}$, depends on:

• The current $I$ flowing through the loop (directly proportional).
• The number of turns $N$, if the loop consists of multiple turns (directly proportional).
• The radius $R$ of the loop (inversely proportional — a smaller loop concentrates the field more strongly at the centre).

$B = \dfrac{\mu_0I}{2\pi r} = \dfrac{4\pi\times10^{-7}\times10}{2\pi\times0.05}$

The $\pi$ cancels: $B = \dfrac{4\times10^{-7}\times10}{2\times0.05} = \dfrac{4\times10^{-6}}{0.1}$

$B = \mathbf{4\times10^{-5}\text{ T}}$

$B = \dfrac{\mu_0NI}{2R} = \dfrac{4\pi\times10^{-7}\times50\times2}{2\times0.1}$

$B = \dfrac{4\pi\times10^{-7}\times100}{0.2} = \dfrac{400\pi\times10^{-7}}{0.2}$

$B = 2000\pi\times10^{-7} \approx \mathbf{6.28\times10^{-4}\text{ T}}$

Since velocity is perpendicular to the field, $\sin\theta=1$: $F = qvB$

$F = 2\times10^{-6}\times3\times10^5\times0.5$

$F = 2\times3\times0.5\times10^{-1} = 3\times10^{-1}$

$F = \mathbf{0.3\text{ N}}$

Force per unit length: $\dfrac{F}{l} = \dfrac{\mu_0I_1I_2}{2\pi d}$

$\dfrac{F}{l} = \dfrac{4\pi\times10^{-7}\times5\times8}{2\pi\times0.1}$

The $\pi$ cancels: $\dfrac{F}{l} = \dfrac{4\times10^{-7}\times40}{2\times0.1} = \dfrac{1.6\times10^{-5}}{0.2}$

$\dfrac{F}{l} = \mathbf{8\times10^{-5}\text{ N/m}}$ (attractive, since currents are in the same direction)

$r = \dfrac{mv}{qB} \Rightarrow v = \dfrac{qBr}{m}$

$v = \dfrac{1.6\times10^{-19}\times0.2\times0.5}{1.67\times10^{-27}}$

Numerator: $1.6\times0.2\times0.5 = 0.16$, so $0.16\times10^{-19} = 1.6\times10^{-20}$

$v = \dfrac{1.6\times10^{-20}}{1.67\times10^{-27}} \approx \mathbf{9.58\times10^6\text{ m/s}}$

At equilibrium, magnetic torque = restoring torque: $NIAB = k\theta$

$I = \dfrac{k\theta}{NAB} = \dfrac{10^{-6}\times0.5}{100\times2\times10^{-3}\times0.1}$

$I = \dfrac{5\times10^{-7}}{2\times10^{-2}} = \mathbf{2.5\times10^{-5}\text{ A} = 25\,\mu\text{A}}$

$F = BIl\sin\theta$, with $\theta=90°$, $\sin90°=1$:

$F = BIl = 0.3\times4\times0.5$

$F = \mathbf{0.6\text{ N}}$

Total resistance needed: $R_{total} = \dfrac{V}{I_g} = \dfrac{10}{5\times10^{-3}} = 2000\,\Omega$

Series resistance required: $R_{series} = R_{total} - G = 2000 - 50 = \mathbf{1950\,\Omega}$

$B = \mu_0nI$, where $n = N/l$ (turns per unit length)

$n = \dfrac{1000}{0.5} = 2000\text{ turns/m}$

$B = 4\pi\times10^{-7}\times2000\times3$

$B = 4\pi\times6\times10^{-4} = 24\pi\times10^{-4}$

$B \approx \mathbf{7.54\times10^{-3}\text{ T}}$

Maximum torque occurs when the dipole moment is perpendicular to the field ($\theta=90°$, so $\sin\theta=1$): $\tau_{max} = NIAB$

Area: $A = \pi r^2 = \pi(0.05)^2 = \pi\times0.0025 \approx 0.00785\text{ m}^2$

$\tau_{max} = 20\times4\times0.00785\times0.6$

$\tau_{max} = 80\times0.00785\times0.6 \approx \mathbf{0.377\text{ N·m}}$