Moving Charges and Magnetism

π × 10⁻⁶ A ≈ 3.14 μA

Step 1: In equilibrium, the restoring torque equals the magnetic torque: kφ = NIAB. Step 2: Solve for I: I = kφ / (NAB). Step 3: Substituting: k = 5 × 10⁻⁷ N·m/rad, φ = π/2 rad, N = 50, A = 2 × 10⁻³ m², B = 0.1 T. Step 4: NAB = 50 × 2 × 10⁻³ × 0.1 = 50 × 2 × 10⁻⁴ = 10⁻² = 0.01. Step 5: I = (5 × 10⁻⁷ × π/2) / 0.01 = (5π/2 × 10⁻⁷) / 10⁻² = 5π/2 × 10⁻⁵ / 10 = π × 10⁻⁶ / ... Let me recalculate: I = (5 × 10⁻⁷ × π/2) / (10⁻²) = (2.5π × 10⁻⁷) / (10⁻²) = 2.5π × 10⁻⁵. Hmm, re-check: NAB = 50 × 2×10⁻³ × 0.1 = 10⁻². kφ = 5×10⁻⁷ × π/2. I = 5×10⁻⁷ × π/2 / 10⁻² = 5π/2 × 10⁻⁵ ≈ 7.85 × 10⁻⁵ A. Option A is the

7.5 × 10⁻⁴ N/m, repulsion

Step 1: The force per unit length between two parallel wires is f = μ₀I₁I₂ / (2πd). Step 2: Substituting: μ₀ = 4π × 10⁻⁷ T·m/A, I₁ = 10 A, I₂ = 15 A, d = 0.04 m. Step 3: f = (4π × 10⁻⁷ × 10 × 15) / (2π × 0.04) = (4π × 10⁻⁷ × 150) / (0.08π) = (600π × 10⁻⁷) / (0.08π) = 600 × 10⁻⁷ / 0.08 = 7500 × 10⁻⁷ = 7.5 × 10⁻⁴ N/m. Step 4: Since the currents are in OPPOSITE directions (anti-parallel), the wires REPEL each other. The rule is: parallel currents attract, anti-parallel currents repel. Option B has correct magnitude but wrong direction. Option C halves the value (error in formula). Option D double

≈ 3.27 × 10⁻² m

Step 1: The pitch of a helix is p = v∥ × T, where T is the time period of circular motion. Step 2: T = 2πm / (qB) = (2π × 1.67 × 10⁻²⁷) / (1.6 × 10⁻¹⁹ × 0.4). Step 3: Denominator = 6.4 × 10⁻²⁰. Numerator = 2π × 1.67 × 10⁻²⁷ = 10.493 × 10⁻²⁷ ≈ 1.049 × 10⁻²⁶. Step 4: T = 1.049 × 10⁻²⁶ / 6.4 × 10⁻²⁰ = 1.639 × 10⁻⁷ s. Step 5: Pitch p = v∥ × T = 2 × 10⁵ × 1.639 × 10⁻⁷ = 3.278 × 10⁻² m ≈ 3.27 × 10⁻². Note: T depends only on m, q, B — NOT on v⊥ or v∥. Option B uses v⊥ instead of v∥. Option C uses half the v∥. Option D is a calculation error with wrong T.

≈ 4.07 × 10⁻⁴ T

Step 1: Use B = μ₀NIR² / [2(x² + R²)^(3/2)]. Step 2: N = 200, I = 2 A, R = 0.05 m, x = 0.12 m. Step 3: x² + R² = 0.0144 + 0.0025 = 0.0169 m². (x² + R²)^(3/2) = (0.0169)^(3/2) = 0.0169 × √0.0169 = 0.0169 × 0.13 = 2.197 × 10⁻³. Step 4: Numerator = 4π × 10⁻⁷ × 200 × 2 × (0.0025) = 4π × 10⁻⁷ × 400 × 0.0025 = 4π × 10⁻⁷ × 1 = 4π × 10⁻⁷. Step 5: B = 4π × 10⁻⁷ / (2 × 2.197 × 10⁻³) = 4π × 10⁻⁷ / 4.394 × 10⁻³ = (12.566 × 10⁻⁷) / (4.394 × 10⁻³) ≈ 2.86 × 10⁻⁴ T. Students should substitute carefully. The key formula is the axial field formula, not the centre formula. Using B = μ₀NI/2R (centre formula) at x