Moving Charges and Magnetism - Comprehensive Notes

4.1 Introduction

• Electricity and magnetism have been known for >2000 years; relationship realized around 1820 via Oersted’s experiment.
• Observation: A current in a straight wire deflects a nearby magnetic compass needle.
• Needle alignment: tangential to an imaginary circle with the wire at its centre; plane perpendicular to the wire.
• Reversing current reverses needle deflection; larger current or closer needle increases deflection.
• Iron filings around a wire form concentric circles about the wire.
• Oersted concluded moving charges (currents) produce a magnetic field in surrounding space.
• In 1864, Maxwell unified electricity and magnetism; light is electromagnetic waves; later developments include Hertz, Bose, Marconi leading to radio waves and tech progress.
• This chapter explores how magnetic fields exert forces on moving charges, how currents produce magnetic fields, particle acceleration in cyclotron, and detection with galvanometer.
• Conventions used: emerging out of the plane is represented by a dot ( "+"), going into the plane by a cross (⊗); figures 4.1(a) and (b) illustrate two opposite current directions.

4.2 Magnetic Force

• 4.2.1 Sources and fields
  • Electric field E(r) from charges is given by static field relations; superposition applies: if multiple charges, fields add vectorially.
  • For a point charge Q, the electric field is
    $\mathbf{E} = \frac{Q}{4\pi \varepsilon_0}\frac{\hat{\mathbf{r}}}{r^2}.$
  • A test charge q experiences a force $\mathbf{F} = q\mathbf{E} = q\frac{Q}{4\pi \varepsilon_0}\frac{\hat{\mathbf{r}}}{r^2}.$
  • The electric field conveys energy and momentum; it propagates with finite speed (not instantaneous).
  • Magnetic field B(r) is produced by moving charges (currents) and has identical properties to E in being a vector field, time-dependent in general, and obeys superposition.
• 4.2.2 Magnetic Field, Lorentz Force
  • Lorentz force on a charge q moving with velocity v in fields E and B is
    $\mathbf{F} = q\big( \mathbf{E}(\mathbf{r}) + \mathbf{v} \times \mathbf{B}(\mathbf{r}) \big).$
  • Key features:
  • Force depends on q, v, and B; opposite charges experience opposite forces (sign of q).
  • The magnetic term is a vector product; the force is perpendicular to both v and B (right-hand rule).
  • The magnetic force vanishes if the charge is at rest (|v| = 0) or if v is parallel or antiparallel to B (v × B = 0).
  • Magnitude relation: $|\mathbf{F}_\text{magnetic}| = q v B \sin\theta,$ where θ is the angle between v and B.
  • Unit of magnetic field: defined so that a unit charge moving perpendicularly to B at speed 1 m/s experiences a force of 1 N; unit is the tesla (T): $1\,\text{T} = \frac{\text{N}}{\text{A} \cdot \text{m}} = \frac{\text{N}}{\text{C} \cdot \text{m} \cdot \text{s}^{-1}}.$ Small unit gauss $ = 10^{-4}$ T.
  • Earth's field ~ 3.6 × 10^{-5} T (often neglected in basic illustrations).
• 4.2.3 Magnetic force on a current-carrying conductor
  • For a straight rod of length l carrying current I (drift velocity v_d of carriers):
    $\mathbf{F} = I \mathbf{l} \times \mathbf{B}.$
  • Derivation: total charge carriers nlA with drift velocity vd; current I = (n q vd) A l; thus F = (n q v_d) l A × B = I (l × B).
  • For a wire of arbitrary shape, the force is obtained by summing/integrating over small elements: $\mathbf{F} = \sum I \, d\mathbf{l} \times \mathbf{B} \quad (\text{or } \mathbf{F} = \int I \, d\mathbf{l} \times \mathbf{B}).$

4.3 Motion in a Magnetic Field

• Magnetic force on a charge is always perpendicular to velocity; does no work on the charge, so the speed |v| stays constant (magnetic field does not do work).
• Consider uniform B and velocity components parallel and perpendicular to B.
• If v ⟂ B, the magnetic force acts as a centripetal force, producing circular motion of radius
  $r = \frac{m v}{q B}.$
• Angular frequency for circular motion: $\omega = \frac{q B}{m},$ and the rotation frequency is $\nu = \frac{\omega}{2\pi} = \frac{q B}{2\pi m}.$
• If there is a component along B, that component is unaffected by B, leading to helical motion: circular motion in a plane perpendicular to B plus linear motion along B.
• Pitch of the helix (distance moved along B in one revolution):
  $p = v<em>{\parallel} T = \frac{2 \pi v</em>{\parallel} m}{q B} = \frac{v<em>{\parallel}}{\nu}.$ (Where $v{\parallel}$ is the component of velocity along B.)
• Significance: cyclotron frequency is independent of particle energy, which is exploited in cyclotrons.

4.4 Magnetic Field Due to a Current Element, Biot–Savart Law

• Magnetic fields arise from currents and magnetic moments; the Biot–Savart law gives the field from a current element dl.
• For a current I, an infinitesimal element dl at position gives a field dB at point P (vector r from element to P):
  $d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}$
• Alternatively (using r̂):
  $d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}.$
• μ0/4π is the constant with vacuum permeability; μ0 = 4π × 10^{-7} T m/A.
• Direction of dB is perpendicular to the plane containing dl and r (Right-Hand Rule for cross product).
• Similarities and differences with Coulomb’s law:
  • Both long-range and obey superposition.
  • Electrostatic field is from a scalar source; magnetic field from a vector source I dl.
  • Magnetic field is perpendicular to the plane defined by dl and r, not along r.
  • Both include angular (sin θ) dependence via the cross product.
• Relation between constants: μ0 ε0 c^2 = 1, connecting μ0, ε0, and the speed of light c (c = 1/√(μ0 ε0)).

4.5 Magnetic Field on the Axis of a Circular Current Loop

• Circular loop of radius R carrying current I lies in the x–y plane with axis along the x-axis. Point P on axis is a distance x from the centre.
• Magnetic field from a differential element dl is integrated; the resulting on-axis field magnitude is
  $B<em>x = \frac{\mu</em>0 I R^2}{2 (R^2 + x^2)^{3/2}}.$
• At the centre (x = 0):
  $B = \frac{\mu_0 I}{2R}.$
• Direction: along the axis; right-hand rule: curl fingers along current direction; thumb gives B direction.
• Field lines of a current loop form closed loops around the wire; the loop’s upper side can be associated with a north pole and the lower with a south pole.

4.6 Ampère’s Circuital Law

• Ampère’s law relates the line integral of the magnetic field around a closed loop to the current passing through the loop:
  $\oint<em>C \mathbf{B}\cdot d\mathbf{l} = \mu</em>0 I_{\text{enc}}.$
• For symmetric cases, a simplified form is used: if the loop is chosen so that B is tangent and constant in magnitude along the loop, then
  $B L = \mu<em>0 I</em>{\text{enc}},$
  where L is the length of the portion of the loop where B is tangent.
• Example: straight, infinite current-carrying wire yields the familiar field
  $B = \frac{\mu_0 I}{2 \pi r},$
  with field lines forming concentric circles about the wire.
• Ampère’s law holds for steady currents; time-varying currents require full Maxwell–Ampère correction (displacement current) for consistency in electrodynamics (not detailed here).
• Right-hand rule to determine the direction of B around a current-carrying wire.

4.7 The Solenoid

• A long solenoid is a helical winding of closely spaced turns; treat each turn as a circular loop; the net field is the sum of fields from all turns.
• Inside a long solenoid, the magnetic field is approximately uniform and parallel to the axis; outside, field is weak and tends to zero (idealized).
• Using Ampère’s law with a rectangular Amperian loop, the field inside is
  $B = \mu_0 n I,$
  where n is the number of turns per unit length and I is the current per turn.
• Example: A solenoid of length 0.5 m, radius 1 cm, with 500 turns carrying 5 A has n = 1000 turns/m, giving
  $B = \mu_0 n I = \left(4\pi \times 10^{-7}\right) \cdot 1000 \cdot 5 \approx 6.28 \times 10^{-3} \ \text{T}.$
• The field inside is parallel to the axis; a soft iron core can further increase the field in practical devices.

4.8 Force Between Two Parallel Currents (Ampère)

• Two long parallel conductors A and B, separated by distance d, carrying currents IA and IB in the same direction.
• The conductor A creates a magnetic field at B: $B<em>A = \frac{\mu</em>0 I_A}{2\pi d}.$
• The current in B experiences a sideways force per unit length due to BA: $f</em>{BA} = \frac{\mu<em>0 I</em>A I_B}{2\pi d}.$
• Force on each conductor is equal in magnitude and opposite in direction: attractive for parallel (same direction) currents, repulsive for antiparallel currents.
• Ampère’s law and Biot–Savart law give equivalent descriptions in this symmetric case.

4.9 Torque on a Current Loop, Magnetic Dipole

• A rectangular loop of width b and height a carrying current I in a uniform magnetic field B experiences a torque but no net force when the field is uniform.
• Forces on the two arms AB and CD are F1 and F2, equal in magnitude (I b B) and opposite in direction, creating a couple that tends to rotate the loop.
• Torque magnitude for the loop when the plane of the loop makes angle θ with the field (or equivalently, when the angle between the magnetic moment and B is θ):
  $\tau = I A B \sin\theta,$ where A = ab is the area.
• For a loop with N turns, the magnetic moment is
  $\mathbf{m} = N I \mathbf{A}, <br /> \quad |\mathbf{m}| = N I A.$
• The general torque expression is the vector product
  $\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B},$
  with magnitude $|\boldsymbol{\tau}| = m B \sin\theta.$
• If m and B are parallel, torque is zero; this is an equilibrium condition. Parallel (same direction) gives stable equilibrium; antiparallel can be unstable for small rotations.
• If the loop has N turns, the formula remains valid with m = N I A.
• The concept of a magnetic dipole moment extends to magnetic dipoles from loops and intrinsic moments (e.g., electrons); magnetic monopoles are not known to exist.

4.9.2 Circular current loop as a magnetic dipole

• A planar loop behaves as a magnetic dipole at large distances, with moment m = I A.
• On-axis field of a circular loop at large distance x (x ≫ R) approximates to a dipole form:
  $B<em>x \approx \frac{\mu</em>0}{4\pi} \frac{2 m}{x^3}, <br /> \quad \text{with } m = I A.$
• The magnetic dipole field in the plane far from the loop has the same functional form as the electric dipole field with p replaced by m and ε0 replaced by 1/µ0 (qualitative analogy).
• This analogy underpins Ampère’s suggestion that magnetism arises from circulating currents; nevertheless, intrinsic magnetic moments exist (e.g., electron) which are not just loop currents.

4.10 The Moving Coil Galvanometer (MCG)

• Structure: a coil with many turns, free to rotate about a fixed axis in a radial magnetic field; an iron core concentrates the field.
• When current I flows, a torque acts on the coil:$\boldsymbol{\tau} = N I A B \hat{\text{axis}}.$ Along the plane of the coil, sin θ = 1 for a radial field, giving
  $\tau = N A B I.$
• Balance of torques in equilibrium with a restoring spring (torsion constant k):
  $k \phi = N I A B, \Rightarrow \phi = \frac{N A B}{k} I.$
• Sensitivities:
  • Current sensitivity (deflection per unit current): $\frac{\phi}{I} = \frac{N A B}{k}.$
  • To measure current, galvanometer is placed in series with a shunt rs to divert most current away from the coil when measuring large currents; total circuit resistance is approximated by rs when RG ≫ rs.
  • For voltmeter measurements, a large resistor R is placed in series with the galvanometer to keep the coil current small; the voltage sensitivity is
    $\frac{\phi}{V} = \frac{N A B}{k} \frac{1}{R + R<em>G},$ which, for R ≫ RG, reduces to $\frac{\phi}{V} \approx \frac{N A B}{k} \frac{1}{R}.$
• Converting between galvanometer, ammeter, and voltmeter involves resistance changes and trade-offs between current sensitivity and voltage sensitivity.

4.11–4.12 Exercises and Examples (highlights)

• Example 4.1: A straight wire of mass 0.2 kg, length 1.5 m, current 2 A suspended in a horizontal magnetic field B: balancing magnetic and gravitational forces yields
  $B = \frac{m g}{I l} = \frac{0.2 \times 9.8}{2 \times 1.5} \approx 0.65 \text{ T}.$
• Example 4.2: For a velocity along x and B along y, Lorentz force is perpendicular to both, giving along ±z depending on sign of charge: electron (−e) gives −z, proton (+e) gives +z.
• Example 4.3: Electron with mass m = 9×10^{-31} kg, charge e = 1.6×10^{-19} C, speed v = 3×10^7 m/s in B = 6×10^{-4} T (perp to B). Radius
  $r = \frac{m v}{q B} \approx 0.28 \text{ m},$
  frequency $\nu = \frac{q B}{2 \pi m} \approx 1.7 \times 10^7\,\text{Hz}$. Energy E = (1/2) m v^2 ≈ 4.0×10^{-16} J ≈ 2.5 \text{ keV}. $</li> <li>Example 4.4: Field due to a small current element at distance 0.5 m on the y-axis yields a small magnetic field magnitude ~4×10^{-8} T; direction along +z.</li> <li>Example 4.5: A straight current-bent semicircular arc of radius 2.0 cm has B at the center: contributions from straight segments cancel; semicircular arc gives B ≈ 1.9×10^{-4} T into the page.</li> <li>Example 4.6: A 100-turn coil of radius 10 cm, current 1 A gives center field B ≈ 6.28×10^{-3} T.</li> <li>Example 4.8: Long solenoid with N turns, current I, B ≈ μ0 n I; for a 0.5 m long, 1 cm radius solenoid with 500 turns and I = 5 A: B ≈ 6.28×10^{-3} T.</li> <li>The Ampere’s law and Biot–Savart law yield consistent results for steady currents; time-varying currents require more careful treatment due to field momentum.</li> </ul> <h3 id="413summaryofkeyformulas">4.13 Summary of Key Formulas</h3> <ul> <li>Lorentz force on a moving charge:<br />$ \mathbf{F} = q (\mathbf{v} \times \mathbf{!B} + \mathbf{E}). $</li> <li>Magnetic force on a straight conductor of length l carrying current I in a uniform field B:<br />$ \mathbf{F} = I \mathbf{l} \times \mathbf{B}. $</li> <li>Cyclotron relations (uniform B):<ul> <li>Circular motion:$ r = \frac{m v}{q B}, \quad \omega = \frac{q B}{m}, \quad \nu = \frac{q B}{2\pi m}. $</li></ul></li> <li>Biot–Savart law for a current element:<br />$ d\mathbf{B} = \frac{\mu0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} = \frac{\mu0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}. $</li> <li>Magnetic field on axis of a circular loop of radius R at distance x:<br />$ Bx = \frac{\mu0 I R^2}{2 (R^2 + x^2)^{3/2}}. $<br /> Center value:$ B = \frac{\mu_0 I}{2 R}. $</li> <li>Ampère’s circuital law (integral form):<br />$ \ointC \mathbf{B} \cdot d\mathbf{l} = \mu0 I_{\text{enc}}. $</li> <li>Magnetic field of a long straight wire:$ B = \frac{\mu_0 I}{2\pi r}. $</li> <li>Magnetic field inside a long solenoid:$ B = \mu_0 n I. $</li> <li>Force between two parallel currents (per unit length):<br />$ f = \frac{\mu0 IA I_B}{2\pi d} \, , $<br /> with total force F = f L for length L.</li> <li>Magnetic dipole moment of a loop:$ \mathbf{m} = I A \hat{\mathbf{n}}; \quad m = I A \; (\text{or } m = N I A \text{ for } N \text{ turns}). $</li> <li>Torque on a current loop:$ \boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}; \quad |\boldsymbol{\tau}| = m B \sin\theta. $</li> <li>Torque in a moving-coil galvanometer:$ \phi = \frac{N A B}{k} I; \quad \frac{\phi}{I} = \frac{N A B}{k}. $</li> <li>Voltmeter sensitivity (with series resistor R and galvanometer resistance RG):<br />$ \frac{\phi}{V} = \frac{N A B}{k} \frac{1}{R + RG} \quad(\text{for } R \gg RG \Rightarrow \frac{\phi}{V} \approx \frac{N A B}{k R}). $</li> </ul> <h3 id="pointstoponder">Points to Ponder</h3> <ul> <li>Electrostatic field lines originate and terminate on charges; magnetic field lines form closed loops (no magnetic monopoles).</li> <li>The chapter assumes steady currents for Ampère’s law; time-varying currents require field momentum considerations for Newton’s third law to hold.</li> <li>The Lorentz force involves both E and B; a frame moving with velocity v can render the magnetic part of the force frame-dependent, illustrating deeper connections between electricity and magnetism (electromagnetism).</li> <li>Ampère’s law and Biot–Savart law are not independent; Ampère’s law can be derived from Biot–Savart in steady-state cases; their relation mirrors Gauss’s law vs Coulomb’s law.</li> </ul> <h3 id="summaryquickreference">Summary (quick reference)</h3> <ul> <li>Lorentz force:$ \mathbf{F} = q (\mathbf{v} \times \mathbf{B} + \mathbf{E}). $</li> <li>Magnetic force on a current-carrying wire:$ \mathbf{F} = I \mathbf{l} \times \mathbf{B}. $</li> <li>Magnetic field around a long straight wire:$ B = \frac{\mu_0 I}{2\pi r}. $</li> <li>Magnetic field inside a long solenoid:$ B = \mu_0 n I. $</li> <li>On-axis field of a circular loop:$ B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}}. $</li> <li>Dipole form at large distance:$ Bz \approx \frac{\mu0}{4\pi} \frac{2 m}{x^3}, \quad m = I A. $</li> <li>Torque on loop:$ \boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}, \quad |\boldsymbol{\tau}| = m B \sin\theta. $</li> <li>Moving-coil galvanometer:$ \phi = \frac{N A B}{k} I; \quad \frac{\phi}{I} = \frac{N A B}{k}. $$
• The magnetic field does no work on a moving charge; it only changes direction of motion, not its speed (magnitude of v).