Moving Charges

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Moving Charges and Magnetism — Key Facts Magnetic force on charge: $F = qvB \sin\theta$; direction given by Fleming’s left-hand rule Lorentz force: $F = q(E + v \times B)$; perpendicular to both velocity and magnetic field Cyclotron: velocity perpendicular to B → circular motion; radius $r = \frac{mv}{qB}$; frequency $f = \frac{qB}{2\pi m}$ Force on current-carrying conductor: $F = BIL \sin\theta$ Parallel conductors: $\frac{F}{L} = \frac{\mu_0}{4\pi} \cdot \frac{2I_1 I_2}{d}$ — attraction if currents in same direction ⚡ Exam tip: Magnetic force does NO work (always perpendicular to velocity); it only changes direction — kinetic energy remains constant

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of moving charges in magnetic fields.

Moving Charges and Magnetism — CUET Physics Study Guide

When an electric charge moves through a magnetic field, it experiences a force that is maximum when the charge moves perpendicular to the field lines and zero when moving parallel. This is the fundamental principle behind the cyclotron, a particle accelerator invented by Lawrence and Livingston in 1934. In a cyclotron, charged particles (like protons or alpha particles) are accelerated by an alternating voltage while spiralling outward in a magnetic field. The radius of each orbit increases with momentum, and the frequency remains constant since $f = \frac{qB}{2\pi m}$ is independent of speed (non-relativistic case).

Biot-Savart Law describes the magnetic field produced by a small current element: $dB = \frac{\mu_0}{4\pi} \cdot \frac{I , dl \times \hat{r}}{r^2}$. For a long straight wire, this gives $B = \frac{\mu_0 I}{2\pi d}$ at distance $d$ from the wire. The direction is given by the right-hand grip rule — thumb points in current direction, fingers curl in field direction.

Force on a current-carrying conductor in a uniform magnetic field: $F = BIL \sin\theta$. When $\theta = 90°$, the force is maximum at $F_{\max} = BIL$. Two parallel wires carrying currents $I_1$ and $I_2$ separated by distance $d$ attract each other with force per unit length $\frac{F}{L} = \frac{\mu_0}{4\pi} \cdot \frac{2I_1 I_2}{d}$.

Moving coil galvanometer works on the torque principle: $\tau = NIAB \sin\theta$. The coil experiences torque proportional to current, deflecting against a spring. To convert to ammeter, a shunt resistor $R_s = \frac{I_g R_g}{I - I_g}$ is connected in parallel. To convert to voltmeter, a multiplier resistor $R_m = \frac{V}{I_g} - R_g$ is added in series.

Example Problem: A proton moves with speed $v = 3 \times 10^6$ m/s perpendicular to a magnetic field $B = 0.2$ T. Find the radius of the circular path. Radius $r = \frac{mv}{qB} = \frac{(1.67 \times 10^{-27})(3 \times 10^6)}{(1.6 \times 10^{-19})(0.2)} = 0.156$ m ≈ 15.6 cm.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious preparation with derivations and applications.

Moving Charges and Magnetism — Complete CUET Physics Notes

Helical Motion: When a charged particle enters a uniform magnetic field at an angle $\theta$ (not perpendicular), it traces a helix. The velocity component parallel to B ($v_\parallel = v \cos\theta$) remains constant (no force), while the perpendicular component ($v_\perp = v \sin\theta$) causes circular motion with radius $r = \frac{mv_\perp}{qB}$. The pitch (distance between consecutive turns) is $p = v_\parallel T = v_\parallel \times \frac{2\pi m}{qB}$.

Torque on Rectangular Coil: Consider a rectangular coil of dimensions $l \times b$ carrying current $I$ in a uniform magnetic field $B$, with its plane making angle $\theta$ with the field. Torque $\tau = NIAB \sin\theta$, where $N$ is number of turns. The magnetic dipole moment $\mu = NIA$, and $\tau = \mu B \sin\theta$. This is the working principle of electric motors. Work done in rotating the coil from angle $\theta_1$ to $\theta_2$ is $W = NIAB(\cos\theta_1 - \cos\theta_2)$.

Magnetic Field Due to Straight Wire: At a point P at distance $r$ from a finite wire carrying current $I$, making angles $\theta_1$ and $\theta_2$ with lines to the ends, $B = \frac{\mu_0 I}{4\pi r}(\sin\theta_2 - \sin\theta_1)$. For an infinitely long wire, $\theta_1 \to -\pi/2$ and $\theta_2 \to +\pi/2$, giving $B = \frac{\mu_0 I}{2\pi r}$.

Solenoid: A solenoid of length $L$ with $N$ turns has $n = N/L$ turns per unit length. Inside (near centre), $B = \mu_0 nI$. The field is uniform inside and negligible outside. For a long solenoid, $B \approx \mu_0 nI$ throughout the interior.

Ampere’s Circuital Law: $\oint B \cdot dl = \mu_0 I_{\text{enc}}$. For an infinite straight wire, this gives $B(2\pi r) = \mu_0 I$, so $B = \frac{\mu_0 I}{2\pi r}$. For a solenoid, applying Ampere’s law gives $B = \mu_0 nI$ inside.

Ballistic Galvanometer: Used to measure charge in a current pulse. The charge $Q = C \theta$, where $C$ is the galvanometer constant and $\theta$ is the maximum deflection. The deflection is proportional to the total charge passed, not the current.

CUET Exam Patterns (2022–2024):

• Questions on Lorentz force and cyclotron appear every year (1–2 marks)
• Force on parallel conductors tested in 2023 (1 mark)
• Torque on current loop: numerical problems in Section B
• Common mistake: forgetting that magnetic force does zero work; students often try to calculate work done by magnetic force

⚡ Key insight: In electromagnetic induction problems, always check whether the magnetic force does work or not. If a charge moves at constant speed in a magnetic field, no work is done because $F \perp v$. But if the magnetic field itself changes (as in transformers), then induced EMF appears due to change in flux.

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