Two straight wires carry currents perpendicular to the xy-plane. One carries a 50 A current and passes through the point (5.0 cm, 0) on the x-axis. The other carries an 80 A current and passes through the point (0, 4.0 cm) on the y-axis. What is the magnitude of the resultant magnetic field at the origin?
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To find the magnitude of the resultant magnetic field at the origin due to two perpendicular wires carrying current, you'll need to use the Biot-Savart Law, which states that the magnetic field (B) due to a current (I) at a distance (r) away in free space is given by:
B = (μ₀I)/(2πr)
where μ₀ is the permeability of free space (μ₀ = 4π × 10⁻⁷ T·m/A).
For each wire, calculate the magnetic field at the origin and then use vector addition to find the resultant magnetic field.
Wire 1 carries a current of 50 A and passes through the point (5.0 cm, 0). Its distance from the origin is 5.0 cm or 0.05 m.
B₁ = (μ₀I₁)/(2πr₁) B₁ = (4π × 10⁻⁷ T·m/A × 50 A) / (2π × 0.05 m) B₁ = (2 × 10⁻⁵ T·m/A) / (0.05 m) B₁ = 4 × 10⁻⁴ T
The direction of the magnetic field due to wire 1 is into the page (or screen) because of the right-hand rule (thumb in direction of current, fingers curl towards the direction of the magnetic field lines).
Wire 2 carries a current of 80 A and passes through the point (0, 4.0 cm). Its distance from the origin is 4.0 cm or 0.04 m.
B₂ = (μ₀I₂)/(2πr₂) B₂ = (4π × 10⁻⁷ T·m/A × 80 A) / (2π × 0.04 m) B₂ = (3.2 × 10⁻⁵ T·m/A) / (0.04 m) B₂ = 8 × 10⁻⁴ T
The direction of the magnetic field due to wire 2 is out of the page (or screen), perpendicular to the direction of B₁.
Since the two magnetic fields are perpendicular, the magnitude of the resultant magnetic field (Bₜ) can be found by using the Pythagorean theorem (since they compose a right triangle in terms of vector addition):
Bₜ = √(B₁² + B₂²) Bₜ = √((4 × 10⁻⁴ T)² + (8 × 10⁻⁴ T)²) Bₜ = √(16 × 10⁻⁸ T² + 64 × 10⁻⁸ T²) Bₜ = √(80 × 10⁻⁸ T²) Bₜ = √(8 × 10⁻⁷ T²) Bₜ = 2.83 × 10⁻⁴ T
So, the magnitude of the resultant magnetic field at the origin is approximately 2.83 × 10⁻⁴ T.