Comprehensive Study Notes: Moving Charges and Magnetism

INTRODUCTION

• Historical Context: Historically, it was believed that electricity and magnetism were independent phenomena with no connection between them.

• Christian Oersted’s Observation (1820): Hans Christian Oersted discovered that when a compass needle is placed very close to a current-carrying conductor, the needle deflects. This led to the conclusion that just as static charges produce an electric field, a current (or moving charges) produces a magnetic field in addition to any existing electric field.

MAGNETIC FIELD AND INDUCTION

• Definition of Magnetic Field: It is the region surrounding a magnetic dipole or a current-carrying conductor within which other magnetic dipoles or current-carrying conductors experience a force of attraction or repulsion.

• Magnetic Induction Vector ($B$): The strength and direction of a magnetic field are defined by a vector quantity called magnetic induction. It is also referred to as:

  • Magnetic Flux Density

  • Magnetic Field ($B$)

• Units of Magnetic Field:

  • $\text{weber/metre}^2$

  • $\text{tesla (T)}$

• Magnetic Field Lines: These are imaginary lines within a magnetic field representing the path along which a hypothetical, isolated north pole would move.

• Magnetic Flux ($\Phi_B$): This represents the total number of magnetic field lines passing through a specific unit area in a direction perpendicular to that area. It is defined mathematically as:     $\Phi_B = \int B \cdot da$

MAGNETIC LORENTZ FORCE

• Definition: When a charge $q$ moves with a velocity $V$ inside a magnetic field $B$, it experiences a magnetic force ($F_B$) known as the Magnetic Lorentz Force:     $F_B = q(V \times B)$

• Key Characteristics of Magnetic Lorentz Force:

  1. The force depends on the magnitude of the charge $q$, the velocity $V$, and the magnetic field $B$. Its direction is perpendicular to both $V$ and $B$, determined by the right-hand thumb rule.

  2. The force acting on a negative charge is in the exact opposite direction of the force acting on a positive charge.

  3. $F = 0$ if the velocity $V = 0$ (stationary charge) or if the magnetic field $B = 0$.

  4. $F = 0$ if the velocity $V$ and magnetic field $B$ are parallel ($\theta = 0^{\circ}$) or anti-parallel ($\theta = 180^{\circ}$).

• Total Lorentz Force: This is the total force acting on a charged particle moving through a region where both electric ($E$) and magnetic ($B$) fields coexist:     $F = F_E + F_B = qE + q(V \times B)$     $F = q(E + V \times B)$

WORK DONE BY MAGNETIC FORCE

• Principles of Work: Given the magnetic force $F_B = q(V \times B)$, the work done ($dW$) by this force on a charged particle is calculated as:     $dW = F_B \cdot dr$

• Conclusion: Since the displacement $dr$ and velocity $V$ are in the same direction and both are always perpendicular to the magnetic force $F_B$, the dot product is zero:     $dW = F_B \cdot dr = 0$     Therefore, the magnetic force does no work on the charged particle.

MOTION OF A CHARGED PARTICLE IN A MAGNETIC FIELD

• Case 1: Movement Parallel to the Field ($V \parallel B$)

  • $\theta = 0^{\circ}$ or $180^{\circ}$

  • $F_B = qVB \sin(0) = 0$

  • The particle experiences no force and continues to move in a straight line, unaffected by the field.

• Case 2: Movement Perpendicular to the Field ($V \perp B$)

  • $\theta = 90^{\circ}$

  • $F = qVB \sin(90^{\circ}) = qVB$

  • This force acts perpendicular to the velocity at every point, serving as a centripetal force. Consequently, the particle executes circular motion perpendicular to the magnetic field.

  • Derivation of Radius ($r$):         $\frac{mv^2}{r} = qVB \implies r = \frac{mV}{qB}$

  • Time Period ($T$):         $T = \frac{2\pi r}{V} = \frac{2\pi m}{qB}$

  • Frequency of Rotation ($\nu$):         $\nu = \frac{1}{T} = \frac{qB}{2\pi m}$

  • Observation: Frequency is independent of velocity. This principle is utilized in the design of the cyclotron. As momentum ($mV$) increases, the radius ($r$) also increases.

• Case 3: Entry at an Arbitrary Angle ($\theta$)

  • The velocity has two components:

    • Perpendicular Component ($V_{\perp} = V \sin(\theta)$): Rotates the particle.

    • Parallel Component ($V_{\parallel} = V \cos(\theta)$): Translates the particle along the field lines.

  • Result: The particle executes helical motion.

  • Pitch ($P$): The distance moved along the magnetic field during one full rotation.         $P = T \times V_{\parallel}$         $P = \frac{2\pi m V \cos(\theta)}{qB}$

NUMERICAL EXAMPLE: ELECTRON PATH

• Problem: Calculate the radius of the path, frequency, and energy (in keV) for an electron ($m = 9 \times 10^{-31}\,kg$, $q = 1.6 \times 10^{-19}\,C$) moving at a speed of $3 \times 10^7\,m/s$ in a magnetic field of $6 \times 10^{-4}\,T$ perpendicular to it.

• Radius Calculation:     $r = \frac{mV}{qB} = \frac{9 \times 10^{-31} \times 3 \times 10^7}{1.6 \times 10^{-19} \times 6 \times 10^{-4}} = 26 \times 10^{-2}\,m$

• Frequency Calculation:     $\nu = \frac{qB}{2\pi m} = 2 \times 10^7\,Hz$

• Energy Calculation:     $E = \frac{1}{2}mV^2 = 40.5 \times 10^{-17}\,J$     Conversion: $1\,eV = 1.6 \times 10^{-19}\,J$     $E = 2.5\,keV$

FORCE ON A CURRENT-CARRYING CONDUCTOR

• Scenario: A wire with cross-section $A$, length $l$, free electron density $n$, carrying current $I$ in an external field $B$.

• Derivation:

  • Total charge in the wire: $q = nAle$

  • Average electron velocity: Drift velocity ($V_d$)

  • Total force ($F_B$): $nAle(V_d \times B)$

  • Since $I = neAV_d$, the equation simplifies for a straight conductor to:         $F_B = I(l \times B)$

• Arbitrary Shapes: For a wire of non-linear shape, the force on a small element is:     $F_B = I(dl \times B)$

BIOT-SAVART LAW

• Definition: The magnetic field ($dB$) at a point due to a small current element ($Idl$) is:

  1. Directly proportional to current ($I$).

  2. Directly proportional to the length of the element ($dl$).

  3. Directly proportional to the sine of the angle ($\theta$) between the element and the displacement vector $r$.

  4. Inversely proportional to the square of the distance ($r^2$).

• Mathematical Formula:     $dB = \frac{\mu_0}{4\pi} \frac{I dl \sin(\theta)}{r^2}$     Where $\frac{\mu_0}{4\pi} = 10^{-7}\,Tm/A$ is the proportionality constant.

• Vector Form:     $dB = \frac{\mu_0}{4\pi} \frac{I(dl \times r)}{r^3}$

MAGNETIC FIELD ON THE AXIS OF A CIRCULAR LOOP

• Setup: Loop of radius $R$ in the YZ plane, centered at the origin, carrying current $I$. Axis is the X-axis.

• Calculation at distance $x$ from center:

  1. The magnetic field $dB$ from an element has X-components ($dB\cos(\theta)$) and perpendicular components ($dB\sin(\theta)$).

  2. Perpendicular components cancel out when summed over the loop.

  3. Net field $B = \int dB\cos(\theta)$.

  4. Resulting formula:         $B = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}}$

• Special Cases:

  1. For $N$ turns: $B = \frac{\mu_0 N I R^2}{2(x^2 + R^2)^{3/2}}$

  2. At the center of the loop ($x=0$): $B = \frac{\mu_0 I}{2R}$

AMPERE’S CIRCUITAL LAW

• Definition: The line integral of the magnetic field around any closed loop is equal to $\mu_0$ times the net current ($I_{enc}$) enclosed by that loop:     $\oint B \cdot dl = \mu_0 I_{enc}$

• Application: Long Straight Conductor:

  • For a point at distance $r$, $\oint B \cdot dl = B(2\pi r)$.

  • By Ampere's Law: $B(2\pi r) = \mu_0 I$

  • Formula: $B = \frac{\mu_0 I}{2\pi r}$

MAGNETIC FIELD OF A SOLENOID

• Definition: A solenoid is a long insulated wire wound in a helix where turns are closely spaced. It behaves like a bar magnet when current flows.

• Ideal Solenoid: Field outside is nearly zero; field inside is parallel to the axis and uniform.

• Magnetic Field Formula:     $B = \mu_0 n I$     Where $n = \frac{N}{L}$ (turns per unit length).

• Ways to Increase Magnetic Field ($B$):

  1. Increase the number of turns per unit length.

  2. Increase the current flow.

  3. Insert a soft iron core into the solenoid.

FORCE BETWEEN PARALLEL CONDUCTORS (AMPERE FORCE)

• Scenario: Two straight conductors $P$ and $Q$ separated by distance $r$ carrying currents $I_1$ and $I_2$.

• Force per unit length ($f$):     $f = \frac{\mu_0 I_1 I_2}{2\pi r}$

• Interaction Rules:

  • Currents in the same direction: Forces are attractive.

  • Currents in opposite directions: Forces are repulsive.

• Definition of One Ampere: An ampere is the constant current which, if maintained in two straight parallel conductors of infinite length placed one metre apart in vacuum, produces a force per unit length of $2 \times 10^{-7}\,N/m$ between them.

TORQUE ON A CURRENT LOOP AND MAGNETIC DIPOLE

• Torque ($\tau$) in Uniform Magnetic Field:     $\tau = m \times B = m B \sin(\theta)$     For a rectangular loop with turns $N$, current $I$, and area $A$:     $\tau = NIAB \sin(\theta)$

• Magnetic Dipole Moment ($m$):     $m = IA$     For $N$ turns: $m = NIA$

• Comparison to Electric Dipole:

  • Magnetic Field at axial distance $x$: $B = \frac{\mu_0}{4\pi} \frac{2m}{x^3}$

  • Electric Field at axial distance $x$: $E = \frac{1}{4\pi\epsilon_0} \frac{2p}{x^3}$

  • This confirms a circular loop acts as a magnetic dipole.

MOVING COIL GALVANOMETER

• Purpose: To detect the presence and direction of small electric currents.

• Construction: Light rectangular coil with $N$ turns and area $A$ wound on a soft iron cylinder. It rotates in a uniform radial magnetic field created by concave magnetic poles. A spring provides a restoring torque.

• Working Principle: Torque produced by current ($NIAB$) is balanced by restoring torque from the spring ($k\phi$):     $NIAB = k\phi \implies I = (\frac{k}{NAB}) \phi$     Thus, current is directly proportional to deflection ($I \propto \phi$).

• Sensitivities:

  • Current Sensitivity: Deflection per unit current = $\frac{\phi}{I} = \frac{NAB}{k}$

  • Voltage Sensitivity: Deflection per unit voltage = $\frac{\phi}{V} = \frac{NAB}{kR}$

• Note: Doubling the number of turns ($N$) doubles current sensitivity but might not change voltage sensitivity because doubling $N$ often doubles resistance ($R$).

AMMETER AND VOLTMETER CONVERSION

• Ammeter (Current Measurement):

  • Purpose: High sensitivity current detection must be modified to measure large currents without damage.

  • Conversion: Connect a small resistance (shunt $S$) in parallel with the galvanometer ($G$).

  • Formula: $S = \frac{I_g G}{I - I_g}$

  • Note: Ideal ammeter resistance is zero. It is always connected in series.

• Voltmeter (Potential Difference Measurement):

  • Purpose: Measure potential difference between two points without affecting current flow.

  • Conversion: Connect a high resistance ($R$) in series with the galvanometer ($G$).

  • Formula: $R = \frac{V}{I_g} - G$

  • Note: Ideal voltmeter resistance is infinite. It is always connected in parallel.

QUESTIONS & DISCUSSION

• Question (March 2020): Why might increasing the current sensitivity not increase the voltage sensitivity?

  • Response: If resistance increases proportionally with the number of turns (e.g., doubling $N$), the ratio $\frac{N}{R}$ remains constant, hence voltage sensitivity remains unchanged.

• Question (March 2024): Why is an ammeter always connected in series?

  • Response: To ensure that the entire circuit current passes through the instrument for measurement while maintaining a low resistance so as not to change the circuit's total resistance.

• Question (March 2025): What will be the direction of magnetic field with respect to the velocity of a charged particle moving in a circular path?

  • Response: The magnetic field must be perpendicular ($90^{\circ}$) to the velocity of the charged particle to sustain circular motion.

• Question (March 2025): Is there any change in kinetic energy of the charged particle in a magnetic field?

  • Response: No, because the magnetic force is always perpendicular to the velocity, meaning it does no work. Therefore, the speed (and kinetic energy) remains constant, even though the direction of motion changes.