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?
2.4*10^-3
2400*10^-7
2.88*10^-6 T
To find the magnetic field at the center of the square wire loop due to the four wire segments at the midpoint of each side, we can use the formula for the magnetic field due to a straight wire at a certain distance.
The magnetic field due to a straight wire at a point a distance r away is given by the formula:
B = (μ₀ * I) / (2 * π * r)
where B is the magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ T*m/A), I is the current, and r is the distance from the wire.
In this case, we need to find the magnetic field at the center of the square wire loop, which is equidistant from each of the four wire segments at the midpoint of each side. The length of each side of the square loop is 10.0 cm, so the distance from the center to each wire segment is half of that, or 5.0 cm.
However, the formula for the magnetic field due to a straight wire assumes the distance is measured in meters, so we need to convert the distance to meters.
1 cm = 0.01 m
Therefore, the distance to each wire segment from the center becomes:
r = 5.0 cm * 0.01 m/cm = 0.05 m
Now we can calculate the magnetic field due to each wire segment using the formula given above:
B₁ = (μ₀ * I) / (2 * π * r)
B₁ = (4π × 10⁻⁷ T*m/A * 15.0 A) / (2 * π * 0.05 m)
Simplifying the expression, we get:
B₁ = (2 * 10⁻⁶ T*m) / (0.05 m)
B₁ = 4 * 10⁻⁵ T
Since the wire segments are at the midpoint of each side and are arranged symmetrically, the magnetic field due to each wire segment will have the same magnitude and direction. Therefore, the total magnetic field at the center of the square wire loop is simply the sum of the magnetic fields due to each wire segment.
B_total = 4 * B₁ = 4 * 4 * 10⁻⁵ T = 16 * 10⁻⁵ T
Therefore, the magnitude of the magnetic field at the center of the square wire loop due to the four 1.20 mm wire segments is 16 * 10⁻⁵ T.
As for the direction, we can determine it using the right-hand rule. If we align our right-hand thumb in the direction of the current (clockwise), then our curled fingers will point in the direction of the magnetic field at the center of the loop.