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?
The magnitude of the magnetic field B at the center of the loop is 0.039 T. The direction of the magnetic field is out of the plane of the loop.
To find the magnitude and direction of the magnetic field B at the center of the square wire loop, due to the four 1.20 mm wire segments at the midpoint of each side, we can use the Biot-Savart Law.
The Biot-Savart Law equation is given by:
B = (μ₀/4π) * ∫ (I * dL x r) / r²
Where:
- B is the magnetic field
- μ₀ is the permeability constant (4π x 10^-7 T*m/A)
- I is the current
- dL is the length element of the wire
- r is the distance from the length element to the point where we want to calculate the magnetic field
Since the wire segments are at the midpoint of each side, the distance from the center of the loop to the wire segments is half the length of the side, i.e., 5.0 cm.
To find the magnetic field at the center due to each wire segment, we can divide the square loop into four sides, meaning that each segment contributes equally to the total magnetic field at the center.
Let's calculate the magnetic field due to just one wire segment first:
B₁ = (μ₀/4π) * (I * dL x r) / r²
Now, we know the current I (15.0 A), the length of each wire segment dL (1.20 mm = 0.012 m), and the distance r (5.0 cm = 0.05 m).
Plugging these values into the equation, we get:
B₁ = (4π x 10^-7 T*m/A / 4π) * (15.0 A * 0.012 m) / (0.05 m)²
B₁ = (10^-7 T*m/A) * (0.18 A) / (0.0025 m²)
B₁ = (0.18 x 10^-7 T) / (2.5 x 10^-6 m²)
B₁ = 7.2 x 10^-2 T
Since we have four wire segments, the total magnetic field at the center of the square wire loop due to these segments will be four times the magnetic field B₁.
B_total = 4 * B₁ = 4 * 7.2 x 10^-2 T
B_total = 0.288 T
Therefore, the magnitude of the magnetic field B at the center of the square wire loop due to the four 1.20 mm wire segments is 0.288 T.
The direction of the magnetic field can be determined using the right-hand rule. In this case, since the current is flowing clockwise in the square wire loop, the magnetic field at the center will point outward perpendicular to the plane of the loop.
To find the magnitude and direction of the magnetic field at the center of the square wire loop, we can use Ampere's Law.
1. Begin by finding the magnetic field due to a single wire segment at the midpoint of each side of the square loop.
The magnetic field produced by a long, straight wire segment can be calculated using 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 in the wire segment, and r is the distance from the wire segment to the point where the magnetic field is being calculated.
2. Determine the distance, r, from the midpoint of each side to the center of the square loop.
Since the square loop has sides of length 10.0 cm, the distance from the midpoint of each side to the center is half that, or 5.0 cm (or 0.05 m).
3. Calculate the magnetic field due to a single wire segment at the midpoint of each side of the square loop.
Using the formula mentioned earlier, we substitute the values:
B = (4π × 10^-7 T m/A) * (15.0 A) / (2π * 0.05 m)
B = (4π × 10^-7 T m/A) * (15.0 A) / (2π * 0.05 m)
B ≈ (1.2 × 10^-6 T) * (15.0 A) / (0.1 m)
B ≈ 0.18 × 10^-6 T
B ≈ 1.8 × 10^-7 T
So, the magnetic field due to each wire segment at the midpoint of each side of the square loop is approximately 1.8 × 10^-7 T.
4. Determine the direction of the magnetic field.
Due to the clockwise current flowing in the wire segments, the magnetic fields produced by each segment will add up, resulting in a magnetic field that points into the center of the loop.
Therefore, the direction of the magnetic field at the center of the square loop due to the four wire segments is into the loop.
The final magnitude and direction of the magnetic field at the center of the square loop due to the four wire segments at the midpoint of each side are approximately 1.8 × 10^-7 T, directed into the loop.