To determine the magnetic field at the point (0, 0, -a) due to three infinitely long wires carrying the same current i along the positive x, y, and z directions, we can use the Biot-Savart law. Each wire contributes to the magnetic field at the specified point.
Magnetic Field Contributions
For a long straight wire, the magnetic field at a distance r from the wire is given by:
B = (μ₀i)/(2πr)
where μ₀ is the permeability of free space. The direction of the magnetic field is determined by the right-hand rule.
Calculating Contributions from Each Wire
- Wire along x-axis: At (0, 0, -a), the distance from the wire is a. The magnetic field direction is in the negative y-direction.
- Wire along y-axis: The distance is also a, and the magnetic field direction is in the positive x-direction.
- Wire along z-axis: The distance remains a, and the magnetic field direction is in the negative x-y plane, contributing to both x and y components.
Resultant Magnetic Field
Combining these contributions, we find:
The magnetic field due to the wire along the x-axis is:
B_x = -μ₀i/(2πa) ĵ
The magnetic field due to the wire along the y-axis is:
B_y = μ₀i/(2πa) î
Adding these vectors gives:
B = μ₀i/(2πa)(î - ĵ)
Final Answer
The correct option for the magnetic field at the point (0, 0, -a) is:
B = μ₀i/(2πa)(î - ĵ)
Thus, the answer is Option B.
