A square wire loop 10.0 cm on each side carries a clockwise current of 15.0 A. Find
the magnitude and direction of the magnetic field B at its center due to the four 1.20
mm wire segments at the midpoint of each side?
To find the magnitude and direction of the magnetic field at the center of the square wire loop, you can use the Biot-Savart law. The law states that the magnetic field at a point due to a small current element is directly proportional to the current, the length of the wire segment, and inversely proportional to the square of the distance from the wire segment to the point.
To apply the Biot-Savart law to this problem, you need to consider each of the four wire segments separately and then sum their contributions to find the total magnetic field.
The formula to calculate the magnetic field due to a wire segment at the center of a square loop is:
B = (μ₀ * I * L) / (4π * d)
Where:
B is the magnitude of the magnetic field,
μ₀ is the permeability of free space (a constant, approximately 4π x 10^-7 T m/A),
I is the current in the wire,
L is the length of the wire segment, and
d is the distance from the wire segment to the center of the loop.
In this case, each wire segment has a length of 1.20 mm, and their distance to the center of the loop is half of the side length, which is 5.0 cm or 0.05 m.
Now, let's calculate the magnetic field due to each of the four wire segments:
B₁ = (μ₀ * I * L) / (4π * d)
Substituting the given values:
B₁ = (4π x 10^-7 T m/A * 15.0 A * 1.20 x 10^-3 m) / (4π * 0.05 m)
Calculating this expression will give you the magnitude of the magnetic field due to one wire segment. Since all four segments are identical, you can then multiply this magnitude by 4 to find the total magnetic field at the center of the loop.
After calculating the magnitude, you can determine the direction of the magnetic field by using the right-hand rule. The magnetic field lines will circulate around the wire segments in a counterclockwise direction, so the direction of the magnetic field at the center of the loop will be out of the plane of the square loop.
By performing the necessary calculations and using the right-hand rule, you will be able to find both the magnitude and direction of the magnetic field at the center of the square wire loop.