Moving Charges

Moving Charges

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Moving Charges and Magnetism links electric currents to the magnetic fields they produce and the forces they experience. The Biot–Savart law gives the field of a current element: dB = (μ₀/4π) · (I dl × r̂)/r², where μ₀ = 4π × 10⁻⁷ T·m/A. Ampère’s circuital law simplifies symmetric problems: ∮ B·dl = μ₀ I_enc. The Lorentz force on a charge q moving with velocity v in fields E and B is F = q(E + v × B), whose magnitude F = qvB sinθ vanishes when v ∥ B. In a uniform perpendicular field, the charge executes uniform circular motion with radius r = mv/(qB) and period T = 2πm/(qB) — independent of speed, the operating principle of the cyclotron. A current loop has magnetic moment μ = IA, experiencing torque τ = μ × B. JEE Main typically asks one direct formula question (numerical on r, T, or force between parallel wires) and one conceptual item on field direction.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Field of a Current Element

A small current-carrying segment I dl at position r from the field point produces dB = (μ₀/4π)(I dl × r̂)/r². The cross product fixes direction via the right-hand rule: fingers point along dl, curl toward r, and the thumb gives dB. The constant μ₀/4π = 10⁻⁷ T·m/A. This is the magnetic analogue of Coulomb’s law, but B is produced only by moving charge and the field lines always close on themselves.

Ampère’s Circuital Law

For any closed Amperian loop, ∮ B·dl = μ₀ I_enclosed (only currents piercing the loop count). It yields the field inside a long solenoid directly: B = μ₀ n I, where n = N/L is turns per unit length. Outside an ideal solenoid B ≈ 0. A toroid gives B = μ₀ N I/(2π r) inside its core.

Force on a Moving Charge and Current

F = q(v × B); with an electric field included, F = q(E + v × B). The magnitude F = qvB sinθ is maximum when v ⊥ B and zero when parallel. On a current element in an external field, dF = I(dl × B), which yields the force per unit length between two parallel wires F/L = μ₀ I₁ I₂/(2π d) — attractive for like currents, repulsive for unlike.

Circular Motion in a Uniform Field

When v ⊥ B, qvB = mv²/r, giving r = mv/(qB) and T = 2πm/(qB). Because m and q are fixed for a given particle, T is independent of both v and r — raising v only inflates r. The cyclotron accelerates particles by an alternating voltage matched to T/2.

Magnetic Dipole

A planar loop of area A carrying current I behaves as a dipole with μ = IA (direction by right-hand rule). In a uniform B: torque τ = μ × B and potential energy U = −μ·B. Moving coils in galvanometers exploit this torque.

Typical JEE Main Question Patterns

• Numerical: radius or frequency of a charged particle in a given B.
• Assertion–Reason on direction of B or force.
• Compare fields inside vs. outside a solenoid or toroid.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Velocity Selector and Edge Cases

Crossed E and B fields form a velocity selector: only charges with v = E/B travel undeflected, because qE = qvB. Heavier ions take longer to traverse, so the device filters by speed, not mass. This same balance underlies mass spectrometry when the exit region has only B, bending ions into circles of radius r ∝ m.

Mechanism Behind the Period Independence

The period T = 2πm/(qB) emerges because the centripetal force qvB ∝ v, while the inertial term mv²/r ∝ v² — increasing v enlarges r linearly so angular speed ω = qB/m stays constant. This is why cyclotrons fall out of resonance at relativistic speeds: mass grows, ω drops, and the AC frequency must drop too (synchrocyclotron).

Connections to Adjacent Topics

• Electromagnetic induction: A changing magnetic flux produces an EMF, the time-derivative counterpart to the Biot–Savart field of steady currents.
• Alternating current: rms values of current in solenoids connect directly to the inductance derived from the flux linkage NΦ = LI.
• Optics (electron beam): A moving-charge setup with crossed E and B fields can deflect, focus, or steer electron beams in CRT and electron microscopes.

Common Mistakes

• Writing F = qvB without sinθ; when v ∥ B, force is zero even with non-zero charge and field.
• Confusing B inside (μ₀nI) and outside (≈0) a solenoid — questions often ask the contrast.
• Misreading Ampère’s law: only enclosed currents contribute; the field on the loop need not be uniform.
• Treating magnetic field lines like electric field lines — they have no sources or sinks and always close.

Worked Micro-Example

A proton (m = 1.67 × 10⁻²⁷ kg, q = 1.6 × 10⁻¹⁹ C) enters B = 0.5 T perpendicular to the field with v = 2 × 10⁶ m/s. Radius: r = mv/(qB) = (1.67 × 10⁻²⁷ × 2 × 10⁶)/(1.6 × 10⁻¹⁹ × 0.5) = 4.18 × 10⁻² m ≈ 4.2 cm. Period: T = 2πm/(qB) = 2π(1.67 × 10⁻²⁷)/(1.6 × 10⁻¹⁹ × 0.5) ≈ 1.31 × 10⁻⁷ s.

Practice Prompts

1. Two long parallel wires 5 cm apart carry 3 A each in the same direction. Compute the force per unit length and state whether it is attractive or repulsive.
2. A deuteron and a proton both enter the same uniform B perpendicularly with equal speeds. Find the ratio of their radii and periods.

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