Magnetic Fields due to Electric Current

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The Lorentz force law states that the force on a charge q moving with velocity v⃗ in electric field E⃗ and magnetic field B⃗ is: F⃗ = q[E⃗ + (v⃗ × B⃗)]. Key implications: (1) If v⃗ ∥ B⃗, magnetic forc…

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For a charged particle in uniform magnetic field B⃗ ⊥ to velocity: Magnetic force provides centripetal force: qvB = mv²/R. Therefore, R = mv/qB (cyclotron radius). The cyclotron frequency is fc = qB/2…

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Given: B = 1.5 T, v = 2 × 10⁶ m/s, q = e = 1.6 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg Radius: R = mv/qB = (1.67 × 10⁻²⁷ × 2 × 10⁶)/(1.6 × 10⁻¹⁹ × 1.5) = 0.0139 m = 1.39 cm Frequency: fc = qB/2πm = (1.6 × 10⁻¹…

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Cyclotron uses crossed electric and magnetic fields to accelerate charged particles. Two semicircular 'dees' have alternating voltage. Magnetic field (⊥ to dees) causes circular motion while electric …

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When a charged particle has velocity components both parallel (v∥) and perpendicular (v⊥) to magnetic field B⃗: v∥ is unaffected (no magnetic force), v⊥ causes circular motion. Result: helical motion …

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For straight wire of length L carrying current I in magnetic field B⃗: F⃗ = IL⃗ × B⃗. Derivation: Charge flowing in time t = It = IL/vd. Force on this charge: F⃗ = q(v⃗d × B⃗) = (IL/vd)(vd × B⃗) = IL⃗…

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For a closed loop in uniform magnetic field B⃗: F⃗ = I∮dl⃗ × B⃗ = I(∮dl⃗) × B⃗. Since ∮dl⃗ = 0 for any closed path (vector sum of displacement around closed loop is zero), the net force F⃗ = 0. Howeve…

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For rectangular loop (length l₁, width l₂) at angle θ to field B⃗: Forces on sides perpendicular to B⃗ are F = Il₁B (equal and opposite), creating torque. Moment arm = ½l₂sinθ. Total torque: τ = 2 × I…