Moving Charges & Magnetism — StudyBeacon
1️⃣ Magnetic Force on a Moving Charge
Lorentz Force:
$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$
In pure magnetic field:
$$F = qvB\sin\theta$$
Special Cases:- θ = 0° → F = 0
- θ = 90° → F = qvB (maximum)
$$qvB = \frac{mv^2}{r}$$
$$r = \frac{mv}{qB}$$
$$T = \frac{2\pi m}{qB}$$
$$f = \frac{qB}{2\pi m}$$
⚠ Trap: Frequency is independent of velocity. ---2️⃣ Helical Motion
If velocity has components:$$v_\parallel = v\cos\theta$$
$$v_\perp = v\sin\theta$$
Radius:$$r = \frac{mv_\perp}{qB}$$
Pitch:$$p = v_\parallel T$$
---3️⃣ Magnetic Force on Current Carrying Conductor
$$\vec{F} = I \vec{L} \times \vec{B}$$
Magnitude:$$F = ILB\sin\theta$$
For parallel wires:$$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$
⚠ Same direction currents → Attraction ⚠ Opposite → Repulsion ---4️⃣ Biot–Savart Law
$$dB = \frac{\mu_0}{4\pi}\frac{Idl\sin\theta}{r^2}$$
Circular loop (center):$$B = \frac{\mu_0 I}{2R}$$
N turns:$$B = \frac{\mu_0 N I}{2R}$$
Arc:$$B = \frac{\mu_0 I \theta}{4\pi R}$$
---5️⃣ Ampere’s Law
$$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$$
Long straight wire:$$B = \frac{\mu_0 I}{2\pi r}$$
Solenoid:$$B = \mu_0 n I$$
Toroid:$$B = \frac{\mu_0 N I}{2\pi r}$$
⚠ Inside ideal solenoid → uniform ⚠ Outside → approx 0 ---6️⃣ Force on Charge in Electric + Magnetic Field
Velocity Selector:$$qE = qvB$$
$$v = \frac{E}{B}$$
---7️⃣ Cyclotron
Cyclotron Frequency:$$f = \frac{qB}{2\pi m}$$
Maximum Energy:$$E = \frac{q^2 B^2 R^2}{2m}$$
⚠ Does NOT work for relativistic speeds. ---8️⃣ Torque on Current Loop
Magnetic moment:$$\vec{\mu} = NIA$$
Torque:$$\tau = \mu B \sin\theta$$
Potential Energy:$$U = -\mu B \cos\theta$$
---9️⃣ Motion in Crossed Fields
If both fields present:$$F = q(E + vB)$$
For no deflection:$$v = \frac{E}{B}$$
---📊 JEE Mains Question Profiling
- Radius comparison problems
- Helical pitch calculation
- Force between wires
- Solenoid/toroid magnetic field
- Velocity selector numerical
- Cyclotron frequency
- Torque & magnetic moment orientation
- Ratio type questions
⚠ Common Traps
- Confusing sin and cos in torque
- Using wrong direction in cross product
- Forgetting μ₀ value: $$4\pi \times 10^{-7}$$
- Using radius formula without resolving velocity components
- Ignoring relativistic limitation in cyclotron
🧠StudyBeacon Strategy
- Memorize core formulas (≈20)
- Practice direction-based questions daily
- Master circular motion derivation
- Revise solenoid/toroid distinctions
Master motion of charge → Master half of Magnetism.
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