Moving Charges and Magnetism Study Notes

Historical Introduction to Magnetism and Electromagnetism

For over 2000 years, electricity and magnetism were considered independent phenomena. In 1820, Danish physicist Hans Christian Oersted discovered their intimate connection during a lecture demonstration. He observed that a current passing through a straight wire caused a magnetic compass needle to deflect.

Oersted’s investigations revealed that the compass needle aligns itself tangential to imaginary concentric circles centered on the wire, within a plane perpendicular to the wire. This effect is most prominent when the current is large or the needle is close to the conductor, allowing the earth’s magnetic field to be neglected. Reversing the current direction reverses the needle’s orientation. Furthermore, increasing the current or decreasing the distance to the wire increases the deflection. Iron filings sprinkled around such a wire also arrange themselves in concentric circles. Oersted concluded that moving charges or currents generate a magnetic field in the surrounding space.

Following Oersted, intense experimentation led to major milestones:

• In 1864, James Maxwell unified the laws of electricity and magnetism, realizing that light consists of electromagnetic waves.

• By the late 19th century, Hertz discovered radio waves, which were subsequently produced by J.C. Bose and G. Marconi.

• The 20th century saw rapid technological progress due to advanced understanding of electromagnetism and the invention of devices for the production, amplification, transmission, and detection of electromagnetic waves.

Field Conventions and Forces

To represent vectors in three dimensions on a two-dimensional surface, the following conventions are adopted:

• A dot ($\odot$) represents a current or field emerging out of the plane of the paper (like the tip of an arrow).

• A cross ($\otimes$) represents a current or field going into the plane of the paper (like the feathered tail of an arrow).

Sources and Fields

An electric field $\mathbf{E}$ is produced by a source charge $Q$. The field is a vector field defined as:

$E = \frac{Q \hat{r}}{4\pi \epsilon_0 r^2}$

where $\hat{r}$ is the unit vector along $\mathbf{r}$. A test charge $q$ interacting with this field experiences a force:

$F = q \mathbf{E} = \frac{q Q \hat{r}}{4\pi \epsilon_0 r^2}$

The electric field conveys energy and momentum and propagates at a finite speed. Similarly, moving charges or currents produce a vector magnetic field $\mathbf{B}(\mathbf{r})$. Like the electric field, magnetic fields obey the principle of superposition, where the total field from multiple sources is the vector sum of their individual fields.

The Lorentz Force

A point charge $q$ moving with velocity $\mathbf{v}$ in the presence of both an electric field $\mathbf{E}(\mathbf{r})$ and a magnetic field $\mathbf{B}(\mathbf{r})$ experiences a total force known as the Lorentz force:

$F = q [\mathbf{E}(\mathbf{r}) + \mathbf{v} \times \mathbf{B}(\mathbf{r})] = \mathbf{F}_{electric} + \mathbf{F}_{magnetic}$

Key features of the magnetic force $\mathbf{F}_{mag} = q(\mathbf{v} \times \mathbf{B})$ include:

1. Dependence: It depends on the charge magnitude, velocity, and the magnetic field strength. The force on a negative charge is opposite to that on a positive charge.

2. Vector Product: The force vanishes if the velocity is parallel or anti-parallel to the magnetic field. The force always acts in a direction perpendicular to both the velocity and the magnetic field, following the right-hand screw rule.

3. Motion: The magnetic force is zero if the charge is stationary ($|v| = 0$). Only moving charges experience this force.

Magnetic Field Units

The SI unit of magnetic field is the tesla ($T$), named after Nikola Tesla. One tesla is defined as the field in which a charge of $1\,C$ moving perpendicular to the field at $1\,m/s$ experiences a force of $1\,N$.

$[B] = [F / qv]$

Units for $B$ are also expressed as Newton second / (coulomb metre). A smaller, non-SI unit is the gauss ($G$), where:

$1\,G = 10^{-4}\,T$

The earth’s magnetic field is approximately $3.6 \times 10^{-5}\,T$.

Force on a Current-Carrying Conductor

For a straight rod of cross-sectional area $A$ and length $l$ carrying a steady current $I$ with mobile charge carriers of charge $q$ and number density $n$, the total number of carriers is $nlA$. Each carrier moves with an average drift velocity $\mathbf{v}_d$. In an external magnetic field $\mathbf{B}$, the force is:

$F = (nlA) q \mathbf{v}_d \times \mathbf{B}$

Substituting the current density $\mathbf{j} = n q \mathbf{v}_d$ and the relation $I = |j| A$, the force on a straight rod is:

$F = I \mathbf{l} \times \mathbf{B}$

here $\mathbf{l}$ is a vector with magnitude equal to the length of the rod and direction matching the current flow. For wires of arbitrary shape, the total force is the integral of the forces on infinitesimal segments $d\mathbf{l}$.

Example 4.1: Suspension in Mid-Air

A wire of mass $200\,g$ ($0.2\,kg$) and length $1.5\,m$ carries a current of $2\,A$. To suspend it in mid-air, the upward magnetic force must balance gravity:

$mg = IlB \implies B = \frac{mg}{Il}$

$B = \frac{0.2 \times 9.8}{2 \times 1.5} \approx 0.65\,T$

Motion of Charged Particles in a Magnetic Field

Since the magnetic force is always perpendicular to the velocity, the work done on the particle is zero ($W = \mathbf{F} \cdot d\mathbf{r} = 0$). Thus, the particle's kinetic energy and the magnitude of its velocity remains constant, though its direction changes.

Circular and Helical Motion

1. Perpendicular Entry: If $\mathbf{v}$ is perpendicular to $\mathbf{B}$, the magnetic force acts as a centripetal force ($\frac{mv^2}{r}$). The radius $r$ of the circular path is:

$r = \frac{mv}{qB}$

2. Angular Frequency: The angular frequency $\omega$ (cyclotron frequency) is:

$\omega = 2\pi \nu = \frac{qB}{m}$

This frequency is independent of the particle's speed or energy, a principle exploited in the cyclotron.

3. General Entry: If velocity has a component parallel to the field ($v_{\parallel}$), that component remains unchanged. The resulting path is a helix. The distance moved along the field per rotation is the pitch $p$:

$p = v_{\parallel} T = \frac{2\pi m v_{\parallel}}{qB}$

The Biot-Savart Law

The Biot-Savart law relates a current element to the magnetic field it produces. For an infinitesimal element $d\mathbf{l}$ carrying current $I$, the magnetic field $d\mathbf{B}$ at distance $\mathbf{r}$ is:

$dB = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{r}}{r^3}$

The magnitude is given by:

$dB = \frac{\mu_0}{4\pi} \frac{I dl \sin(\theta)}{r^2}$

where $\mu_0 = 4\pi \times 10^{-7}\,Tm/A$ is the permeability of free space.

Comparison with Coulomb’s Law

• Commonality: Both are long-range and follow the inverse-square law ($1/r^2$). Both obey the principle of superposition and are linear in their sources.

• Differences: The electrostatic source is scalar (charge), whereas the magnetic source is a vector ($I d\mathbf{l}$). The electric field is along the displacement vector, but the magnetic field is perpendicular to both the current element and the displacement vector. The Biot-Savart law involves an angular dependence ($\sin(\theta)$).

A critical relationship exists between the speed of light $c$, permittivity $\epsilon_0$, and permeability $\mu_0$:

$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

Magnetic Field on the Axis of a Circular Current Loop

For a circular loop of radius $R$ carrying current $I$, the magnetic field at a point $P$ on the x-axis at distance $x$ from the center is:

$B = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}} \mathbf{\hat{i}}$

At the center of the loop ($x=0$), the field simplifies to:

$B = \frac{\mu_0 I}{2R} \mathbf{\hat{i}}$

The direction of the field is determined by curled fingers in the direction of the current; the thumb then indicates the field's direction.

Ampere’s Circuital Law

Ampere's law relates the magnetic field along a closed loop (Amperian loop) to the current passing through the surface bounded by that loop:

$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$

For systems with high symmetry, the law simplifies to:

$BL = \mu_0 I_e$

where $L$ is the length of the path where $B$ is tangential and constant, and $I_e$ is the enclosed current.

Applications of Ampere’s Law

1. Infinite Straight Wire: At distance $r$ from a long wire carrying current $I$:

$B(2\pi r) = \mu_0 I \implies B = \frac{\mu_0 I}{2\pi r}$

2. Thick Wire of radius $a$:

• Outside ($r > a$): $B = \frac{\mu_0 I}{2\pi r}$

• Inside ($r < a$): Assuming uniform current distribution:

$B = \left( \frac{\mu_0 I}{2\pi a^2} \right) r$

Solenoids and Toroids

A solenoid is a long wire wound into a helix. For a long solenoid (length much greater than radius) with $n$ turns per unit length carrying current $I$:

• The field inside is uniform and along the axis.

• The field outside is ideally zero. Using an Amperian loop of length $h$, the enclosed current is $I(nh)$. Ampere's law gives:

$Bh = \mu_0 n h I \implies B = \mu_0 n I$

Force Between Two Parallel Currents

Two long parallel conductors separated by distance $d$ carrying currents $I_a$ and $I_b$ exert forces on one another. The field produced by wire $a$ at the position of wire $b$ is $B_a = \frac{\mu_0 I_a}{2\pi d}$. The force per unit length $f_{ba}$ on wire $b$ is:

$f_{ba} = \frac{\mu_0 I_a I_b}{2\pi d}$

Key Principles:

• Parallel currents in the same direction attract.

• Antiparallel (opposite) currents repel.

Definition of the Ampere

The ampere ($A$) is defined as the steady current which, when maintained in two infinitely long parallel conductors $1\,m$ apart in vacuum, produces a force of $2 \times 10^{-7}\,N/m$ on each conductor. This definition from 1946 provides the basis for defining the coulomb ($C$): $1\,C$ is the charge flowing in $1\,s$ when the current is $1\,A$.

Torque on Current Loops and Magnetic Dipoles

A rectangular loop of area $A = ab$ carrying current $I$ in a uniform magnetic field $\mathbf{B}$ experiences a torque. If the normal to the loop makes an angle $\theta$ with the field:

$\tau = IAB \sin(\theta)$

Defining the magnetic moment $\mathbf{m} = I\mathbf{A}$, the torque is:

$\tau = \mathbf{m} \times \mathbf{B}$

For a coil with $N$ turns, $m = NIA$. The dimensions of magnetic moment are $[L^2 A]$, with units $Am^2$ or $J/T$.

• Stable equilibrium occurs when $\mathbf{m}$ is parallel to $\mathbf{B}$ ($\theta = 0$).

• Unstable equilibrium occurs when $\mathbf{m}$ is antiparallel to $\mathbf{B}$ ($\theta = \pi$).

The Moving Coil Galvanometer (MCG)

The MCG is a device used to detect and measure small currents. It consists of a coil with $N$ turns suspended in a uniform radial magnetic field produced by a cylindrical soft iron core. This core ensures the field is always parallel to the plane of the coil ($\sin(\theta) = 1$).

When current $I$ flows, the magnetic torque is $NIAB$. This is balanced by a restoring torque $k\phi$ from a spring with torsional constant $k$. At equilibrium:

$k\phi = NIAB \implies \phi = \left( \frac{NAB}{k} \right) I$

Sensitivity and Conversion

1. Current Sensitivity: Deflection per unit current:

$\frac{\phi}{I} = \frac{NAB}{k}$

2. Voltage Sensitivity: Deflection per unit voltage:

$\frac{\phi}{V} = \frac{NAB}{kR}$

3. Ammeter Conversion: A low resistance (shunt) $r_s$ is connected in parallel with the galvanometer ($R_G$). The effective resistance becomes very low.

4. Voltmeter Conversion: A high resistance $R$ is connected in series with the galvanometer. The effective resistance becomes very high.

Increasing current sensitivity (by increasing $N$) does not necessarily increase voltage sensitivity because resistance $R$ also increases proportionally with the number of turns.