Magnetic Effects of Current
Why This Chapter Matters
Magnetics is a major JEE chapter — 10-14 marks. Biot-Savart law, Ampere's law, force on current, and moving charges in magnetic fields are all tested with numerical problems.
Core Concepts
1. Biot-Savart Law
dB = (μ₀/4π)(Idl × r̂)/r²
μ₀ = 4π×10⁻⁷ T·m/A
Key results:
At center of circular loop (radius R, current I): B = μ₀I/2R
On axis of loop at distance x: B = μ₀IR²/[2(R²+x²)^(3/2)]
Infinite straight wire at distance d: B = μ₀I/2πd
Finite wire (angles α₁, α₂): B = μ₀I(sinα₁+sinα₂)/4πd
Direction: Right-hand rule. Curl fingers in direction of current → thumb points along B.
2. Ampere's Circuital Law
∮B⃗·dl⃗ = μ₀I_enclosed
Applications:
Infinite solenoid (n turns/m, current I): B = μ₀nI (inside), B = 0 (outside)
Toroid (N turns, mean radius R): B = μ₀NI/2πR (inside), B = 0 (outside)
3. Force on Current-Carrying Conductor
F = IL × B = BIL sinθ
Maximum when I ⊥ B (sinθ=1). Zero when I ∥ B.
Direction: Fleming's Left Hand Rule (or F = IL⃗ × B⃗)
Force per unit length between parallel wires (separation d):
F/L = μ₀I₁I₂/2πd
Same direction currents → attract. Opposite → repel.
4. Force on Moving Charge (Lorentz Force)
F = q(v⃗ × B⃗) = qvB sinθ
Magnetic force does NO work (always ⊥ to velocity)
→ Speed unchanged, direction changes → circular motion
For charge q in magnetic field B (velocity v):
Radius: r = mv/qB
Time period: T = 2πm/qB (independent of speed! — used in cyclotron)
5. Cyclotron
Charged particles accelerated in D-shaped magnets.
Resonance condition: frequency of AC = cyclotron frequency = qB/2πm
Max KE: KE_max = q²B²R²/2m (R = max radius)
Limitation: at high speeds, relativistic mass increase changes cyclotron frequency → resonance lost.
6. Magnetic Dipole
Bar magnet / current loop = magnetic dipole
Dipole moment: m = NIA (N=turns, I=current, A=area)
Torque in uniform B: τ = m × B = mB sinθ
Potential energy: U = -m·B = -mB cosθ
PYQs
2024: Proton (mass m, charge q) enters magnetic field B perpendicular to it at speed v. Radius?
r = mv/qB
2023: Two long parallel wires carry currents I and 2I in same direction, separated by 3 cm. Force per unit length?
F/L = μ₀I(2I)/2πd = μ₀(2I²)/2π(0.03) = force per unit length (attractive)
2022: Magnetic field at center of square loop (side a, current I)?
Square has 4 sides. Each side at distance a/2 from center.
B from each side: μ₀I sin45°×2/(4π×a/2) = μ₀I√2/πa... [each side contributes]
Total: B = 4 × μ₀I sinα/(πa) = 2√2μ₀I/πa