Comprehensive Study Notes on Moving Charges and Magnetism

Origins of Electromagnetism and Oersted's Discovery

• Historical Context: Electricity and magnetism were studied as separate phenomena for over 2000 years. Their intimate connection was established in 1820.

Hans Christian Oersted (1820): During a lecture demonstration, the Danish physicist noticed that an electric current in a straight wire caused a deflection in a nearby magnetic compass needle.

• Experimental Observations:

  • The alignment of the needle is tangential to an imaginary circle centered on the wire, with the plane of the circle perpendicular to the wire.

  • The effect is most noticeable when the current is large and the needle is close enough to the wire to neglect the earth’s magnetic field.

  • Reversing the current direction reverses the orientation of the needle.

  • Increasing the current or decreasing the distance to the wire increases the deflection.

  • Iron filings sprinkled around the wire arrange themselves in concentric circles centered on the wire.

• Conclusions: Oersted concluded that moving charges or currents produce a magnetic field in the surrounding space.

• Unification and Progress:

  • James Maxwell (1864): Unified the laws of electricity and magnetism, realizing that light consists of electromagnetic waves.

  • Heinrich Hertz: Discovered radio waves.

  • J.C. Bose and G. Marconi: Produced radio waves by the end of the 19th century.

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

Notation and Conventions for Fields and Currents

• Out of Plane: A current or field (electric or magnetic) emerging out of the plane of the paper is depicted by a dot symbol ($\cdot$), representing the tip of an arrow pointed toward the viewer.

• Into Plane: A current or field going into the plane of the paper is depicted by a cross symbol ($\otimes$\text{ or } $\times$), representing the feathered tail of an arrow moving away from the viewer.

The Concept of Magnetic Field and Lorentz Force

• Electric Field Recapitulation: A source charge $Q$ produces an electric field $E$ defined as:     $E = \frac{Q}{4\pi\epsilon_0 r^2} \hat{r}$     where $\hat{r}$ is the unit vector along $r$. A charge $q$ interacting with this field experiences a force:     $F = q E$

• Role of Fields: The field is a physical entity that conveys energy and momentum. It propagates at a finite speed and can vary with space and time.

• Magnetic Field ($B$): Just as static charges produce an electric field, moving charges or currents produce a magnetic field ($B(r)$).

• Principle of Superposition: Like electric fields, magnetic fields of several sources add vectorially.

• Lorentz Force: The total force on a point charge $q$ moving with velocity $v$ in the presence of both an electric field $E$ and a magnetic field $B$ is:     $F = q [ E(r) + v \times B(r) ]$

  • The force due to the magnetic field is $F_{magnetic} = q [ v \times B ]$.

  • Features of Magnetic Force:

    1. It depends on the charge $q$, velocity $v$, and magnetic field $B$. The force on a negative charge is opposite to that on a positive charge.

    2. The force vanishes if the velocity is parallel or anti-parallel to the magnetic field because the vector product is zero.

    3. The force acts in a direction perpendicular to both the velocity and the magnetic field. The direction is determined by the right-hand screw rule.

    4. The magnetic force is zero if the charge is stationary ($|v| = 0$).

Units and Dimensions of Magnetic Field

• Definition of Tesla: The magnitude of the magnetic field $B$ is 1 SI unit ($T$) when the force acting on a unit charge ($1\,C$) moving perpendicular to $B$ at a speed of $1\,m/s$ is one newton ($1\,N$).

• Dimensions: $[B] = [F / qv]$.

• Units:

  • $1\,tesla (T) = 1\,Newton\,second / (coulomb\,metre)$.

  • Named after Nikola Tesla.

  • Gauss: A smaller non-SI unit. $1\,gauss = 10^{-4}\,tesla$.

  • Earth's magnetic field is approximately $3.6 \times 10^{-5}\,T$.

Magnetic Force on a Current-Carrying Conductor

• Derivation: For a straight rod of cross-sectional area $A$, length $l$, and carrier number density $n$, the total number of carriers is $nlA$. If each carrier has a drift velocity $v_d$ and charge $q$:     $F = (nlA) q v_d \times B$

  • Since current density $j = n q v_d$ and current $I = |j| A$, the force is:     $F = [ (n q v_d) l A ] \times B = [ j A l ] \times B = I l \times B$

  • Here, $l$ is a vector with magnitude equal to the length of the rod and direction identical to the current $I$.

• Arbitrary Shapes: For a wire of arbitrary shape, the total Lorentz force is calculated by summing linear segments $dl_j$:     $F = \sum_j I dl_j \times B$     This is typically converted into an integral.

Motion of a Charged Particle in a Magnetic Field

• Work and Energy: Because the magnetic force is always perpendicular to the velocity ($F \cdot v = 0$), the magnetic force does no work. It changes the direction of the velocity but not its magnitude (speed remains constant).

• Circular Motion ($v \perp B$): The magnetic force acts as a centripetal force:     $\frac{m v^2}{r} = q v B$

  • Radius of Path: $r = \frac{m v}{q B}$

  • Angular Frequency: $\omega = 2\pi \nu = \frac{q B}{m}$

  • Cyclotron Frequency: The frequency $\nu$ is independent of the particle's speed or energy, which is a principle used in cyclotrons.

• Helical Motion: If the velocity has a component parallel to the magnetic field ($v_{\parallel}$), the particle moves in a helix.

  • The circular part of the motion has radius $r$.

  • Pitch ($p$): The distance traveled along the magnetic field in one rotation:     $p = v_{\parallel} T = \frac{2\pi m v_{\parallel}}{q B}$

The Biot-Savart Law

• Definition: Relates a current element to the magnetic field it produces. For a current element $I dl$ at a distance $r$ from point $P$:     $dB = \frac{\mu_0}{4\pi} \frac{I dl \times r}{r^3}$

• Magnitude:     $dB = \frac{\mu_0}{4\pi} \frac{I dl \sin(\theta)}{r^2}$     where $\theta$ is the angle between $dl$ and $r$.

• Permeability of Free Space ($\mu_0$):     $\mu_0 / 4\pi = 10^{-7}\,T\,m/A$

• Comparison with Coulomb’s Law:

  1. Both are long-range (inverse square law).

  2. Magnetic field is produced by a vector source ($I dl$); electrostatic field by a scalar source ($q$).

  3. Magnetic field is perpendicular to the displacement vector; electrostatic field is along it.

  4. Biot-Savart law has an angle dependence ($\sin(\theta)$).

• Relation to Speed of Light ($c$):     $\mu_0 \epsilon_0 = \frac{1}{c^2}$

Magnetic Field on the Axis of a Circular Current Loop

• Setup: Loop of radius $R$ in the $y-z$ plane with current $I$. Point $P$ is at distance $x$ on the axis.

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

• At the Center ($x = 0$):     $B = \frac{\mu_0 I}{2R} \hat{i}$

• Right-Hand Thumb Rule: Curl the fingers of the right hand in the direction of the current; the thumb points in the direction of the magnetic field.

Ampere’s Circuital Law

• Statement: The line integral of the magnetic field around a closed loop is equal to $\mu_0$ times the total current passing through the surface bounded by the loop:     $\oint B \cdot dl = \mu_0 I$

• Amperian Loop Simplified Version: If $B$ is tangential and constant over a length $L$, then:     $B L = \mu_0 I_e$     where $I_e$ is the enclosed current.

• Field of an Infinite Straight Wire: At distance $r$:     $B = \frac{\mu_0 I}{2\pi r}$

  • The field exhibits cylindrical symmetry.

  • Field lines form closed concentric circles.

The Solenoid

• Structure: A long wire wound in a helix where turns are closely spaced. Neighboring turns are insulated using enamel.

• Interior Field: For a very long solenoid, the field inside is uniform, strong, and parallel to the axis. The field outside is nearly zero.

• Formula: Using an Amperian loop of length $h$ and turns per unit length $n$:     $B = \mu_0 n I$

Force Between Two Parallel Currents

• Interaction: Two long parallel conductors $a$ and $b$ separated by distance $d$ with currents $I_a$ and $I_b$.

• Force Magnitude: The force per unit length ($f_{ba}$) is:     $f_{ba} = \frac{\mu_0 I_a I_b}{2\pi d}$

• Directions:

  • Parallel currents attract.

  • Anti-parallel currents repel.

• Definition of the Ampere: The ampere is the steady current which, if maintained in two very long, straight, parallel conductors of negligible cross-section placed one meter apart in vacuum, produces a force of $2 \times 10^{-7}\,newtons$ per metre of length.

Torque on Current Loops and the Magnetic Dipole

• Torque Calculation: A rectangular loop ($ab$) with area $A = ab$ and current $I$ in uniform field $B$ experiences torque:     $\tau = m \times B$     where the magnetic moment is $m = I A$.

• Scalar Magnitude: $\tau = m B \sin(\theta)$, where $\theta$ is the angle between the magnetic moment (normal to the loop) and the magnetic field.

• Equilibrium:

  • Stable: $m$ and $B$ are parallel ($\theta = 0$).

  • Unstable: $m$ and $B$ are anti-parallel ($\theta = \pi$).

• Dipole Analogy: A circular current loop at large distances ($x \gg R$) produces a field:     $B \simeq \frac{\mu_0}{4\pi} \frac{2m}{x^3}$     This is identical in form to the electric field of an electric dipole.

The Moving Coil Galvanometer (MCG)

• Principle: A coil in a radial magnetic field experiences torque $NIAB$. This is balanced by a restoring torque from a spring with torsional constant $k$.

• Deflection ($\phi$):     $k \phi = N I A B \implies \phi = \frac{N A B}{k} I$

• Sensitivities:

  • Current Sensitivity: $\phi / I = N A B / k$

  • Voltage Sensitivity: $\phi / V = N A B / (k R)$

• Conversion to Ammeter: Connect a small shunt resistance ($r_s$) in parallel with the MCG.

• Conversion to Voltmeter: Connect a large resistance ($R$) in series with the MCG.

Key Physical Quantities and Units

• Permeability of Free Space ($\mu_0$): $4\pi \times 10^{-7}\,T\,m\,A^{-1}$

• Magnetic Field ($B$): Tesla ($T$). Dimensions: $[M T^{-2} A^{-1}]$

• Magnetic Moment ($m$): $A\,m^2$ or $J/T$. Dimensions: $[L^2 A]$

• Torsion Constant ($k$): $N\,m\,rad^{-1}$. Dimensions: $[M L^2 T^{-2}]$

Questions & Discussion

• Could a loop turn about a vertical axis in a horizontal field?: No, torque $\tau = I A \times B$ is in the plane of the loop if the area vector $A$ is vertical.

• Orientation for stable equilibrium?: The area vector $A$ points in the direction of the external magnetic field $B$, maximizing total flux.

• Why does a flexible loop become circular in a field?: The circle encloses the maximum area for a given perimeter, thus maximizing the magnetic flux through the loop.