Moving Charges

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Key Formulas:

• Magnetic force on moving charge: F = qvB sinθ
• Lorentz force: F = q(E + v × B)
• Cyclotron radius: r = mv/(qB)
• Cyclotron frequency: f = qB/(2πm) (independent of v and r!)
• Biot-Savart law: dB = (μ₀/4π)(Idl × r̂)/r²
• Ampere’s law: ∮ B·dl = μ₀I_enc
• Magnetic field of straight wire: B = μ₀I/(2πd)
• Magnetic field of circular loop at centre: B = μ₀I/(2R)
• Solenoid: B = μ₀nI
• Force on current element: F = BIL sinθ
• Force between parallel wires: F/L = (μ₀/4π)(2I₁I₂/d)
• Torque on current loop: τ = NIAB sinθ
• Magnetic dipole moment: m = NIA

Exam Tips:

• Use Fleming’s left-hand rule for motor effect (force on current in magnetic field)
• Use Fleming’s right-hand rule for generators (induced current direction)
• Cyclotron frequency is independent of speed and radius — this is a key fact
• Magnetic force does no work (always perpendicular to velocity)
• v × B uses vector cross product — direction via right-hand rule
• Cyclotron is a particle accelerator, not a energy generating device

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of magnetic forces and fields.

Lorentz Force and Vector Cross Product:

The magnetic force on a charged particle is F = q(v × B), giving magnitude F = qvB sinθ, where θ is the angle between v and B. The direction is perpendicular to both v and B, given by the right-hand rule for positive charges (left-hand rule for negative charges). The Lorentz force combines electric and magnetic effects: F = q(E + v × B). The magnetic component always does zero work because F ⊥ v, making magnetic fields useful for steering charged particles without changing their kinetic energy.

Motion of Charged Particle in Uniform Magnetic Field:

When a charged particle enters a uniform B field with velocity v at angle θ to B, decompose v into parallel (v_parallel = v cosθ) and perpendicular (v_perp = v sinθ) components. The parallel component is unchanged (no magnetic force along B), while the perpendicular component causes circular motion in the plane perpendicular to B. The radius of circular motion is r = mv_perp/(qB) = mv sinθ/(qB). The angular frequency (cyclotron frequency) is ω = qB/m, giving frequency f = qB/(2πm). Crucially, f is independent of v and r — all particles of the same charge-to-mass ratio spiral with the same frequency regardless of their speed.

The combination of uniform motion along B and circular motion perpendicular to B produces a helical trajectory. The pitch (distance between successive turns) is p = v_parallel × T = v_parallel × (2πm/qB). A uniform magnetic field can therefore be used to confine charged particles in helical paths — the basis for magnetic mirrors and plasma confinement.

Worked Example: A proton (q = 1.6×10⁻¹⁹ C, m = 1.67×10⁻²⁷ kg) enters a magnetic field B = 0.5 T with velocity v = 4×10⁶ m/s perpendicular to B. Find the radius and frequency.

• r = mv/(qB) = (1.67×10⁻²⁷ × 4×10⁶)/(1.6×10⁻¹⁹ × 0.5) = 0.0835 m = 8.35 cm
• f = qB/(2πm) = (1.6×10⁻¹⁹ × 0.5)/(2π × 1.67×10⁻²⁷) ≈ 7.6 MHz

Force on Current Element:

For a small current element Idl in a magnetic field B, the force is dF = I(dl × B), giving magnitude dF = BIL sinθ where θ is the angle between the element direction and B. For a straight wire of length L in uniform B, the total force is F = BIL sinθ. This is the motor effect: converting electrical energy to mechanical force. Applications include electric motors, loudspeakers, and torque in moving coil instruments.

Force Between Parallel Current-Carrying Wires:

Two parallel wires carrying currents I₁ and I₂ separated by distance d exert forces on each other. Wire 1 produces magnetic field B₁ = μ₀I₁/(2πd) at wire 2 (perpendicular to both wires). The force on wire 2 length L is F = I₂L B₁ = (μ₀/4π)(2I₁I₂/d)L. This is attractive for currents in the same direction and repulsive for opposite directions. The ampere is defined using this force: one ampere is the current that, when flowing in each of two parallel wires one metre apart, produces a force of 2×10⁻⁷ N per metre of length.

Torque on Current Loop:

Consider a rectangular coil of length L, breadth b, carrying current I in a uniform magnetic field. When the plane of the coil makes angle θ with the field direction, the torque magnitude is τ = NIAB sinθ, where A = Lb is the area of one turn and N is the number of turns. The torque is maximum (τ = NIAB) when the coil plane is parallel to B (θ = 90°) and zero when the plane is perpendicular to B (θ = 0°). The torque tends to rotate the coil to align its magnetic moment with the field.

Magnetic Dipole Moment:

The magnetic dipole moment vector is m = NIA (direction perpendicular to coil plane, given by right-hand rule). A current loop acts like a magnetic dipole with north and south poles. The torque on the loop in a magnetic field is τ = m × B. The potential energy of a magnetic dipole in a field is U = -m·B = -mB cosθ, minimum when m is aligned with B.

Moving Coil Galvanometer:

A moving coil galvanometer uses the torque on a current-carrying coil in a radial magnetic field to measure current. The coil is suspended by a spring; the restoring torque is proportional to angular displacement. At equilibrium, NIAB sin90° = kθ, giving θ ∝ I. The galvanometer can be converted to an ammeter by connecting a shunt resistance S = G/(n-1) in parallel, where G is coil resistance and n = full-scale current/I_g. For a voltmeter, a multiplier resistance R = (V_fs - I_gG)/I_g is connected in series.

Magnetic Field Patterns:

Straight wire: Concentric circles centred on the wire, direction given by right-hand grip rule (thumb points current direction, fingers curl in field direction).

Circular loop: Field lines enter one side and leave the other. At the centre of a loop of radius R carrying current I: B = μ₀I/(2R). Along the axis at distance x from centre: B = (μ₀IR²)/(2(R² + x²)^(3/2)).

Solenoid: Inside a long solenoid (length >> radius), the field is uniform and parallel to the axis: B = μ₀nI, where n = N/L is the turns per unit length. Outside, the field resembles that of a bar magnet.

Velocity Selector:

A velocity selector uses perpendicular electric and magnetic fields to transmit only particles with a specific velocity. With E field between two plates and B field perpendicular to the page, a particle moving perpendicular to both experiences electric force F_E = qE (downward if q > 0) and magnetic force F_B = qvB (upward if using right-hand rule). Only particles with v = E/B pass through undeflected; faster particles deflect one way, slower particles deflect the other. This is used as a pre-filter in mass spectrometers and particle accelerators.

Mass Spectrometer Principle:

A mass spectrometer separates ions by their mass-to-charge ratio. Ions are first accelerated through a potential difference, then pass through a velocity selector (E and B perpendicular) to obtain mono-velocity ions. These enter a region with magnetic field perpendicular to their velocity; they follow circular paths with radius r = p/(qB), where p = mv is the momentum. Since v = E/B from the selector, r = mE/(qB²). Ions of different m/q strike different positions on the detector, allowing identification and quantification of isotopes.

Worked NEET Example: In a velocity selector with E = 10⁵ V/m and B = 0.5 T, what velocity particles pass undeflected?

• v = E/B = 10⁵/0.5 = 2 × 10⁵ m/s

If these particles then enter a region with B’ = 0.5 T and are detected at radius r = 0.1 m, find m/q.

• From r = mv/(qB’): m/q = rB’/v = 0.1 × 0.5/(2×10⁵) = 2.5 × 10⁻⁷ kg/C

🔴 Extended — Deep Study (3mo+)

Comprehensive derivation-based theory for complete mastery of moving charges and magnetic fields.

Ampere’s Circuital Law Derivation:

Ampere’s circuital law states that the line integral of magnetic field around any closed loop equals μ₀ times the total current enclosed: ∮ B·dl = μ₀I_enc. This is the magnetic analogue of Gauss’s law for electricity. It can be derived from the Biot-Savart law or treated as a fundamental law of electromagnetism (one of Maxwell’s equations for static fields).

Applications of Ampere’s Law:

Infinite straight wire: Choose a circular Amperian loop of radius r centred on the wire. By symmetry, B is constant and tangential along the loop. ∮ B·dl = B(2πr) = μ₀I. Therefore B = μ₀I/(2πr). This agrees with the Biot-Savart calculation.

Infinite current sheet: For an infinite sheet of surface current density K (current per unit width perpendicular to flow), choosing a rectangular Amperian loop straddling the sheet gives B = μ₀K/2 on each side, directed parallel to the sheet. The field is uniform and independent of distance from the sheet.

Solenoid: For an ideal solenoid (infinite length, tight winding), choose a rectangular Amperian loop with one side inside the solenoid (along the axis) and one outside. The field outside is zero for an ideal solenoid; inside, B is uniform and parallel to the axis. The loop’s contribution from the inside is B × length; from the sides (perpendicular to B) is zero; outside is zero. Ampere’s law gives B × L = μ₀NI, so B = μ₀nI where n = N/L.

Toroid: A toroid is a solenoid bent into a doughnut shape. For a toroid with N turns and mean radius r, choosing a circular Amperian loop along the field gives B × 2πr = μ₀NI, so B = μ₀NI/(2πr). The field is confined to the toroid’s interior.

Cyclotron Details and Limitations:

A cyclotron consists of two hollow D-shaped electrodes (dees) in a uniform magnetic field, with an AC voltage applied between them. Ions are released at the centre and spiral outward as they gain energy each time they cross the gap between dees. The cyclotron frequency f = qB/(2πm) must match the AC frequency for resonance. The radius of the nth orbit is r_n = mv_n/(qB), and the speed after n turns is v_n = nqV/m, where V is the gap voltage.

The key limitation is relativistic mass increase: as particles approach the speed of light, their mass m increases, causing f to decrease. The ion gets out of phase with the accelerating voltage, limiting the maximum energy achievable. The synchrocyclotron addresses this by modulating the frequency downward as particles become more relativistic. Modern cyclotrons also use azimuthal varying field (AVF) to provide internal focusing and handle relativistic effects.

Magnetic Mirror and Plasma Confinement:

When a charged particle moves into a region of increasing magnetic field, its perpendicular velocity component increases (because the magnetic moment m_perp = ½mv_perp²/B is adiabatically invariant). This causes the pitch angle to increase until θ approaches 90° and the particle reflects. This is the magnetic mirror effect, used to confine plasma in devices like mirror machines and Earth’s Van Allen radiation belts.

The magnetic bottle concept uses two magnetic mirrors (regions of stronger field at each end) to confine charged particles between them. Particles bounce back and forth, trapped in the central region of lower field. This principle is used in experimental fusion reactors where plasma must be confined without material walls.

Grad-B and Curvature Drifts:

In non-uniform magnetic fields, particles undergo additional drifts. The gradient-B drift occurs when the magnetic field strength varies perpendicular to B: the curvature and gradient of B create a drift velocity v_grad = (mv_perp²/(2qB)) × (B × ∇B)/B². The curvature drift occurs because guiding centre motion along curved field lines produces a centrifugal force that causes drift: v_c = (mv_parallel²/(qB)) × (B × R_c)/R_c²B². The net drift across field lines is the sum of these effects. These drifts are important for understanding particle transport in space plasmas and laboratory confinement devices.

Torque and Work Done on Current Loop:

When a current loop rotates in a magnetic field, work is done by the external agent against the magnetic torque. The differential work done in rotating the loop by dθ is dW = τ dθ = NIAB sinθ dθ. Integrating from initial angle θ₁ to final angle θ₂: W = NIAB(cosθ₁ - cosθ₂). This work equals the change in potential energy: U = -m·B = -NIAB cosθ. The work done by the agent to rotate the loop increases its potential energy (aligns the magnetic moment with the field).

Consistency with Orbital Angular Momentum Quantisation:

The magnetic moment m = IA = (q/T) × (πr²) = (qω/2π) × πr² = (½qωr²). For an electron in a circular orbit, the orbital angular momentum is L = mvr = mωr². Therefore m = (q/2m)L. This shows the magnetic moment is directly proportional to mechanical angular momentum. In quantum mechanics, the orbital angular momentum is quantised (L = nℏ), giving m = (qℏ/2m)n = μ_B n, where μ_B = eℏ/2m is the Bohr magneton. This consistency between classical and quantum pictures is important in atomic physics.

Common NEET/JEE Mistakes to Avoid:

1. Confusing cyclotron frequency with the frequency of revolution. The revolution frequency f_rev = v/(2πr) decreases as radius increases (for constant B), while cyclotron frequency f_c = qB/(2πm) is constant.

2. Sign conventions: The magnetic force on a negative charge is opposite to that on a positive charge — always check the sign.

3. Using Fleming’s left-hand rule for generators: In electromagnetic induction (generators), the motion-induced emf produces a current that experiences a magnetic force opposing motion (Lenz’s law). Use Fleming’s right-hand rule for the induced current direction in a moving conductor.

4. Forgetting that magnetic force does no work — energy changes in magnetic fields come from electric fields or non-conservative effects.

5. In the Biot-Savart law, the direction of dl × r̂ gives the direction of dB — using the wrong direction leads to the wrong answer.

Previous Year NEET/JEE Question Patterns:

• Velocity selector questions: finding the specific velocity that passes undeflected, then using that in subsequent calculations
• Cyclotron radius and frequency calculations with different particles (proton, deuteron, alpha particle)
• Force between parallel wires, definition of ampere
• Torque on current loop in a magnetic field, finding maximum torque or work done
• Magnetic field at centre of circular loop or along axis of solenoid
• Applications: mass spectrometer identifying isotopes, cyclotron limitations due to relativistic effects