As shown in the figure, a rectangular loop with a length l of 20.0 cm and a width – [Free] B21
As shown in the figure, a rectangular loop with a length l of 20.0 cm and a width w of 15.0 cm has 20 turns and carries a current of 0.390 A counterclockwise around the loop when viewed from the positive x axis. A horizontal (parallel to the x-z plane) magnetic field of magnitude 0.0330 T is oriented at an angle of 65.0° relative to the perpendicular to the loop (the positive x axis). (Assume the length and width are measured along the z and y axes, respectively.)
![As shown in the figure, a rectangular loop with a length l of 20.0 cm and a width - [Free] B21](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20640%20262'%3E%3C/svg%3E)
Answer
🧲 Magnetic Force and Torque on a Current-Carrying Rectangular Loop
🔹 Given Parameters
- Number of Turns: N = 20
- Length of Loop: L = 20 cm = 0.20 m
- Width of Loop: W = 15 cm = 0.15 m
- Current: I = 0.390 A
- Magnetic Field Magnitude: B = 0.0330 T
- Angle with Perpendicular: α = 65°
🟦 Vector Form of Magnetic Field
→B = (B cos 65°) î + 0 ĵ + (B cos 25°) k̂
→B = (0.0139 î + 0 ĵ + 0.0299 k̂) T
🟨 Magnetic Forces on Each Side
🔸 Top Side (ab)
→FTop = N × I × →L × →B = (0 î + 0.0217 ĵ + 0 k̂) N
Components:
Fx = 0,
Fy = 2.17 × 10⁻² N,
Fz = 0
🔸 Bottom Side (cd)
→FBottom = (0 î + −0.0217 ĵ + 0 k̂) N
Components:
Fx = 0,
Fy = −2.17 × 10⁻² N,
Fz = 0
🔸 Left Side (bc)
→FLeft = (−0.0350 î + 0 ĵ + 0.0163 k̂) N
Components:
Fx = −3.50 × 10⁻² N,
Fy = 0,
Fz = +1.63 × 10⁻² N
🔸 Right Side (da)
→FRight = (+0.0350 î + 0 ĵ − 0.0163 k̂) N
Components:
Fx = +3.50 × 10⁻² N,
Fy = 0,
Fz = −1.63 × 10⁻² N
🟩 Magnetic Torque on the Loop
🔸 Magnetic Moment
A = L × W = 0.20 × 0.15 = 0.030 m²
M = N × I × A = 20 × 0.390 × 0.030 = 0.234 A·m²
Direction: Along +x-axis (î)
🔸 Torque Calculation
→τ = →M × →B = (0.234 î) × (0.0139 î + 0 ĵ + 0.0299 k̂)
→τ = 0 î − 0.007 ĵ + 0 k̂ = −0.007 ĵ N·m
Magnitude of Torque: 7.00 × 10⁻³ N·m
Direction: Clockwise about the y-axis when viewed from +y