Force on Moving Charge and Cyclotron

• Apply the Lorentz force law F = q(v × B) to find magnitude and direction of force
• Explain why a magnetic force does no work on a moving charge
• Derive the radius and time period of circular motion in a uniform magnetic field
• Explain the working principle of the cyclotron and the resonance condition
• Analyse the velocity selector (crossed E and B fields)
• Calculate the force per unit length between two parallel current-carrying conductors

Level 1 — Foundations

Lorentz Force:

$\vec{F} = q(\vec{v} \times \vec{B})$

Magnitude: F = qvB sinθ, where θ is the angle between v and B.

Special cases:

• v ∥ B (θ = 0°): F = 0 — no force; particle moves in a straight line.
• v ⊥ B (θ = 90°): F = qvB (maximum) — particle moves in a circle.

Direction: Use the right-hand rule for v × B, then apply the sign of q. For an electron (q negative), the force is opposite to v × B.

No work done by magnetic force: F is always perpendicular to v, so the work done W = F · v dt = 0. Therefore:

• Magnetic force cannot change the kinetic energy (speed) of a charged particle.
• It can only change the direction of motion.

Circular motion in a uniform B field: The magnetic force provides centripetal force:

$qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB}$

Time period (independent of v!):

$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$

Cyclotron frequency:

$f = \frac{1}{T} = \frac{qB}{2\pi m}$

Level 2 — JEE depth

Helical motion: If v has a component v∥ along B and v⊥ perpendicular to B:

• The ⊥ component produces circular motion (radius r = mv⊥/qB).
• The ∥ component is unaffected (no force along B).
• Combined motion is a helix with pitch p = v∥ × T = v∥(2πm/qB).

Cyclotron working principle:

• Two D-shaped hollow cavities (dees) placed in a uniform B field.
• An alternating electric field is applied across the gap between the dees.
• A charged particle (e.g., proton) starting at the centre accelerates across the gap, curves in a semicircle inside the dee, and returns to the gap for another acceleration.
• Resonance condition: the frequency of the alternating field = cyclotron frequency f = qB/(2πm).
• Since T is independent of speed, the resonance condition remains satisfied as the particle spirals outward.
• Maximum kinetic energy: KE_max = q²B²R²/(2m), where R = radius of dee.

Cyclotron limitation: Fails at relativistic speeds (v → c) because mass increases (relativistic mass m → γm), making the cyclotron frequency dependent on speed. Synchrotrons solve this by varying B or frequency.

Velocity Selector (crossed E and B fields): Electric force (qE, downward) balanced by magnetic force (qvB, upward):

$qE = qvB \implies v = \frac{E}{B}$

Only particles with exactly this speed pass straight through, regardless of charge or mass — hence useful for mass spectrometers.

Force between two parallel current-carrying conductors: Wire 1 (current I₁) creates B₁ = μ₀I₁/(2πd) at wire 2. Force on length L of wire 2:

$F = I_2 L B_1 = \frac{\mu_0 I_1 I_2 L}{2\pi d}$

Force per unit length:

$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$

• Currents in the same direction → wires attract.
• Currents in opposite directions → wires repel.
• Definition of the SI ampere: 1 A is the current in each of two parallel wires 1 m apart that produces a force of 2 × 10⁻⁷ N per metre.

JEE traps:

• The cyclotron cannot accelerate neutrons or neutral particles (q = 0 → no force).
• Magnetic force does no work — so it cannot increase KE; the electric field in the gap does the work in a cyclotron.
• When computing r = mv/(qB), q must be in coulombs (use 1.6 × 10⁻¹⁹ C for a proton/electron).
• For a proton and an electron with the same KE entering B: r_proton/r_electron = √(m_p/m_e) ≈ 43 (since r = √(2mKE)/(qB)).

Proton in Magnetic Field — Radius and Time Period

Given: B = 0.5 T, v = 2 × 10⁶ m/s (perpendicular to B)
       Mass of proton m = 1.67 × 10⁻²⁷ kg
       Charge of proton q = 1.6 × 10⁻¹⁹ C

Step 1: Radius of circular path
  r = mv / (qB)
  r = (1.67 × 10⁻²⁷ × 2 × 10⁶) / (1.6 × 10⁻¹⁹ × 0.5)
  r = (3.34 × 10⁻²¹) / (8.0 × 10⁻²⁰)
  r = 0.04175 m ≈ 4.18 cm

Step 2: Time period (independent of v)
  T = 2πm / (qB)
  T = (2π × 1.67 × 10⁻²⁷) / (1.6 × 10⁻¹⁹ × 0.5)
  T = (1.05 × 10⁻²⁶) / (8.0 × 10⁻²⁰)
  T ≈ 1.31 × 10⁻⁷ s ≈ 131 ns

Answer: r ≈ 4.18 cm, T ≈ 131 ns.
Note: Doubling v would double r but T stays the same.

Force Between Parallel Wires

Given: I₁ = I₂ = 10 A, d = 0.1 m, currents in same direction
Formula: F/L = μ₀I₁I₂ / (2πd)

Step 1: Substitute
  F/L = (4π × 10⁻⁷ × 10 × 10) / (2π × 0.1)

Step 2: Simplify
  = (4π × 10⁻⁷ × 100) / (0.2π)
  = (400π × 10⁻⁷) / (0.2π)
  = 400 × 10⁻⁷ / 0.2
  = 2000 × 10⁻⁷
  = 2.0 × 10⁻⁴ N/m

Answer: Force per unit length = 2.0 × 10⁻⁴ N/m (attractive,
        since currents flow in the same direction).

• Q1: An electron moves at 3 × 10⁶ m/s perpendicular to B = 0.2 T. Find r and compare to a proton at the same speed in the same field.
• Q2: A cyclotron operates with B = 1.0 T for protons. Find the cyclotron frequency and the radius when KE = 1 MeV.
• Q3: In a velocity selector, E = 4 × 10⁴ V/m and B = 0.2 T. What speed of particles passes through?
• Q4: Two wires 0.5 m apart carry currents 6 A and 4 A in opposite directions. Find F/L and state whether attractive or repulsive.
• Stretch: Q5: A proton enters a cyclotron dee of radius 30 cm in B = 1.5 T. Find (a) maximum KE in MeV and (b) the number of revolutions needed to reach this energy starting from rest, if each gap gives 200 V.

NCERT Chapter 4 link: Sections 4.3–4.5 cover the Lorentz force, circular motion in B, and the force between parallel conductors (which defines the ampere). Section 4.4 discusses the cyclotron in detail. All diagrams in the NCERT text are essential for direction analysis.

Exam connections: JEE consistently asks: (1) MCQs on work done by magnetic force, (2) problems where a charged particle enters with v at an angle (helix problems), (3) cyclotron maximum energy formula, (4) comparing r for proton vs alpha particle vs electron with same speed or same KE.

Study strategy: Master the r = mv/(qB) formula by substituting numbers for common particles (proton, electron, alpha) mentally. Practice direction problems with a pen: hold it as v, fingers curl to B, palm faces the force direction — do this physically 5 times per problem set until it is automatic.

• Use the Cyclotron / Charged Particle live world: launch protons and electrons with adjustable v and B, observe circular and helical paths, measure r directly on screen.
• Home activity: Roll a ball in a curved path on a rotating turntable to feel how a centripetal force keeps it in circular motion — analogous to the magnetic force in a cyclotron.
• AI Mentor voice reflection: "Explain to a younger sibling why a magnetic field can bend a particle's path but never speed it up."

• "Explain the Lorentz force using the example of a cricket ball spinning through the air — the Magnus force is surprisingly similar in concept to the magnetic force on a charge."
• "What is one common mistake students make about the cyclotron and its resonance condition, and how would you catch yourself making it?"
• Stretch: "How does the principle behind the cyclotron connect to cancer treatment (proton therapy), particle physics (CERN), and the design of fusion reactors like ITER?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• Build a simple speaker from a paper cone, a coil, and a magnet — voice current in the coil experiences a force from the permanent magnet, driving the cone; connect to phone audio and test.
• Direct link to AI Mastery (particle tracking in detectors) and Green Tech (electric motor and generator design both rely on F = qvB).
• Coding extension: Simulate the circular path of a charged particle in Python (Euler method), updating position and velocity each time step, and plot the trajectory.

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.