Two identical thin wires are oriented along the x and y axes.
With current on x plane going North. Current on y plane going West.Both carry the same current I, and there is no electrical contact between the wires. The points A, B, C, D are each equidistant from both wires.
A: top left
B:top right
C: bottom left
D: bottom right.
the net magnetic field at each point are?
To determine the net magnetic field at each point (A, B, C, D) due to the two wires, we will use the principle of superposition. This principle states that the total magnetic field at a point due to multiple sources is the vector sum of the magnetic fields produced by each source individually.
Let's calculate the net magnetic field at each point step by step:
Step 1: Determine the magnetic field at each point due to the wire along the x-axis:
The magnetic field produced by a current-carrying wire can be calculated using Ampere's Law:
B_x = (μ₀ * I) / (2π * r_x),
where B_x is the magnetic field at point x (A or B), μ₀ is the permeability of free space (constant), I is the current in the wire, and r_x is the perpendicular distance from the wire to point x.
Since points A and B are equidistant from the wire along the x-axis, their distances r_Ax and r_Bx are the same.
Step 2: Determine the magnetic field at each point due to the wire along the y-axis:
Similar to Step 1, we calculate the magnetic field due to the wire along the y-axis using Ampere's Law:
B_y = (μ₀ * I) / (2π * r_y),
where B_y is the magnetic field at point y (C or D), I is the current in the wire, μ₀ is the permeability of free space (constant), and r_y is the perpendicular distance from the wire to point y.
Since points C and D are equidistant from the wire along the y-axis, their distances r_Cy and r_Dy are the same.
Step 3: Find the net magnetic field at each point:
The net magnetic field at each point will be the vector sum of the magnetic fields produced by the current-carrying wires along the x and y axes.
At point A:
The net magnetic field, B_A, at point A is equal to the vector sum of B_x and B_y:
B_A = √((B_x)^2 + (B_y)^2),
where B_x is the magnetic field at point A due to the x-axis wire, and B_y is the magnetic field at point A due to the y-axis wire.
Similarly, we can calculate the net magnetic field at points B, C, and D.
At point B:
B_B = √((B_x)^2 + (B_y)^2)
At point C:
B_C = √((B_x)^2 + (B_y)^2)
At point D:
B_D = √((B_x)^2 + (B_y)^2)
Note: Remember to consider the directions and signs of the magnetic fields when performing vector addition.
By following these steps and calculating the individual magnetic fields, you can find the net magnetic field at each point: A, B, C, and D.