Magnetic Field at a Point Above Two Opposite Current-Carrying Wires

Magnetic Field Due to Two Opposite Currents at a Point Above the Midpoint

Question:

Two long, parallel wires carry equal currents of 8.87 A in opposite directions. The wires are separated by a distance d = 5.62 cm. Find the magnitude of the magnetic field at point P, which is located a vertical distance h = 7.69 cm above the center of the wires.

Answer:

Step 1: Determine the Distance from Each Wire to Point P

The wires are separated by a horizontal distance d = 5.62 cm = 0.0562 m, so the horizontal distance from the midpoint to each wire is d/2 = 0.0281 m.
The vertical distance from the midpoint to point P is h = 7.69 cm = 0.0769 m.

Use the Pythagorean Theorem to find the distance from each wire to point P:

r = √[(0.0281)² + (0.0769)²] = √(0.00078961 + 0.005916) ≈ √0.00670561 ≈ 0.0819 m

Step 2: Magnetic Field Due to One Wire

The magnetic field due to a long straight wire at a perpendicular distance r is given by:

B = (μ₀ × I) / (2π × r)

Substitute the values:

μ₀ = 4π × 10⁻⁷ T·m/A
I = 8.87 A
r = 0.0819 m

B = (4π × 10⁻⁷ × 8.87) / (2π × 0.0819) = (4 × 10⁻⁷ × 8.87) / 0.0819 ≈ 3.548 × 10⁻⁶ / 0.0819 ≈ 4.33 × 10⁻⁵ T

Step 3: Find Vertical Component of Magnetic Field

The total magnetic field from each wire has both horizontal and vertical components. Due to symmetry, horizontal components cancel each other. Vertical components add.

Use trigonometry to find cos(θ), where:

cos(θ) = h / r = 0.0769 / 0.0819 ≈ 0.940

Vertical component from one wire:

B_y = B × cos(θ) = 4.33 × 10⁻⁵ × 0.940 ≈ 4.07 × 10⁻⁵ T

Step 4: Net Magnetic Field at Point P

Both wires contribute equally to the vertical magnetic field at point P. Since directions are the same, they add up:

B_net = 2 × B_y = 2 × 4.07 × 10⁻⁵ = 8.14 × 10⁻⁵ T

✅ Final Answer:
The magnitude of the magnetic field at point P is 8.14 × 10⁻⁵ Tesla.

learnlyfly