Parallel wires, a distance D = 23.8 cm apart, carry a current, i = 2.43 A, in opposite directions as shown in the figure. A circular loop, of radius R = D/2 = 11.9 cm, has the same current flowing in a counterclockwise direction. Determine the magnitude of the magnetic field from the loop and the parallel wires at the center of the loop
To determine the magnitude of the magnetic field from the loop and the parallel wires at the center of the loop, we can use the superposition principle, which states that the total magnetic field at a given point due to multiple sources is the vector sum of the individual magnetic fields produced by each source.
For the parallel wires:
The magnetic field produced by a long straight wire can be calculated using Ampere's Law. Ampere's Law states that the magnetic field at a distance r from a long straight wire carrying a current I is given by the formula:
B = (μ₀ * I) / (2π * r)
Where B is the magnetic field, μ₀ is the permeability of free space (μ₀ = 4π x 10^(-7) Tm/A), I is the current, and r is the distance from the wire.
In this case, we have two parallel wires carrying currents in opposite directions. Since they are equidistant from the center of the loop, the magnetic fields produced by each wire at that point will have the same magnitude but opposite directions.
So, the total magnetic field at the center of the loop due to the parallel wires is given by:
B_parallel = (μ₀ * i) / (2π * D/2) + (μ₀ * i) / (2π * D/2)
= (μ₀ * i) / π * D
Where B_parallel is the magnetic field due to the parallel wires, i is the current in the wires, and D is the distance between them.
For the circular loop:
The magnetic field produced by a current-carrying circular loop at its center is given by the formula:
B_loop = (μ₀ * I) / (2R)
Where B_loop is the magnetic field, μ₀ is the permeability of free space (μ₀ = 4π x 10^(-7) Tm/A), I is the current in the loop, and R is the radius of the loop.
In this case, the current in the loop is equal to the current in the wires, but they have opposite directions.
So, the magnetic field at the center of the loop due to the loop itself is:
B_loop = (μ₀ * i) / (2R)
Where B_loop is the magnetic field due to the circular loop, i is the current in the loop, and R is the radius of the loop.
Total magnetic field at the center of the loop:
To find the total magnetic field, we need to add up the magnetic fields due to the parallel wires and the circular loop at the center.
B_total = B_parallel + B_loop
Now we can substitute the given values into the equations and calculate the magnetic field.
B_parallel = ((4π x 10^(-7) Tm/A) * 2.43 A) / (π * 0.238 m)
= 8.09 x 10^(-6) T
B_loop = ((4π x 10^(-7) Tm/A) * 2.43 A) / (2 * 0.119 m)
= 2.04 x 10^(-5) T
B_total = B_parallel + B_loop
= 8.09 x 10^(-6) T + 2.04 x 10^(-5) T
= 3.25 x 10^(-5) T
Therefore, the magnitude of the magnetic field from the loop and the parallel wires at the center of the loop is approximately 3.25 x 10^(-5) Tesla.