Magnetic Field Calculation at a Point Near a Square Wire Loop
Question:
A square loop of wire, measuring 0.14 m × 0.14 m, carries a current of I = 16.00 A. Find the magnitude of the magnetic field at point P, which is located 0.07 m (i.e., a/2) away from the center of one side of the square loop.
Answer:
Step 1: Given Data
- Side of the square loop, a = 0.14 m
- Current in the loop, I = 16.00 A
- Distance from the center to point P, r = a/2 = 0.07 m
Step 2: Concept Used
We use the **Biot-Savart Law** to calculate the magnetic field at a point along the perpendicular axis of a straight current-carrying conductor:
B = (μ₀I / 4πr) × [L / √(L² + 4r²)]
Where:
- μ₀ is the permeability of free space =
4π × 10⁻⁷ T·m/A - L is the length of one side of the square
- r is the perpendicular distance to point P from the wire
Step 3: Magnetic Field from One Side
Substitute values into the Biot-Savart Law:
B₁ = [(4π × 10⁻⁷) × 16] / [4π × 0.07] × [0.14 / √(0.14² + 4 × 0.07²)]
Simplify step-by-step:
B₁ = (2.286 × 10⁻⁶) × (0.14 / √0.0392)
⇒ B₁ ≈ 2.286 × 10⁻⁶ × 0.707
⇒ B₁ ≈ 1.617 × 10⁻⁶ T
⇒ B₁ ≈ 2.286 × 10⁻⁶ × 0.707
⇒ B₁ ≈ 1.617 × 10⁻⁶ T
Step 4: Total Magnetic Field
Due to symmetry, only the **vertical sides** contribute to the magnetic field at point P in the horizontal direction. The horizontal sides cancel each other out.
So, the total magnetic field is:
B_total = 2 × B₁ = 2 × 1.617 × 10⁻⁶ T = 3.23 × 10⁻⁶ T
✅ Final Answer:
The magnitude of the magnetic field at point P is 3.23 μT (microtesla).
The magnitude of the magnetic field at point P is 3.23 μT (microtesla).