A square-shaped loop carries current I. What is the magnitude and the direction of the magnetic field at point P, which is located on the bottom segment of the loop? Each side of the square loop has length d. The answer should be given in terms of d, I, pi, and the permittivity constant.
I know that the magnetic field at P points into the page and that it is equal to the sum of the magnetic fields from each of the segments of the loop, but I'm not sure how to calculate each of the segments' magnetic field contributions. Thanks!
Adapt this problem to your problem: I do not understand your picture, but this should guide you.
https://www.physics.wisc.edu/undergrads/courses/spring09/248/HWSolutions/HW12Solutions.pdf
To calculate the magnetic field at point P due to each segment of the square loop, we can apply the Biot-Savart law, which relates the magnetic field at a point to the current and displacement vector of a small element of the current-carrying wire.
Let's consider each segment of the square loop separately and calculate the magnetic field at point P due to each segment.
Segment 1: This is the bottom segment of the loop. The current is flowing from left to right. To calculate the magnetic field at point P, we need to find the contribution from this segment. First, let's denote the distance from the segment to point P as r1. The displacement vector can be written as r1 = d/2 î, where î is the unit vector along the x-axis. Now we can use the Biot-Savart law:
dB1 = (μ₀ I / 4π) * (dL₁ x r₁) / r₁³,
where dB1 is the magnetic field contribution from segment 1, I is the current, dL₁ is the differential length of segment 1, and μ₀ is the permeability constant (also known as the vacuum permeability).
Segment 2: This is the left segment of the loop. The current is flowing from bottom to top. To calculate the magnetic field at point P, we need to find the contribution from this segment. Denote the distance from the segment to point P as r2. The displacement vector can be written as r2 = d/2 ĵ, where ĵ is the unit vector along the y-axis. Applying the Biot-Savart law again, we have:
dB2 = (μ₀ I / 4π) * (dL₂ x r₂) / r₂³,
where dB2 is the magnetic field contribution from segment 2, I is the current, dL₂ is the differential length of segment 2, and μ₀ is the permeability constant.
Segments 3 and 4: The magnetic field contributions from segments 3 and 4 can be calculated in a similar manner to segments 1 and 2.
Finally, to find the total magnetic field at point P, we sum up the contributions from all segments:
B_P = dB1 + dB2 + dB3 + dB4.
Note that the direction of each segment's magnetic field contribution should be taken into account. Since the loop is square-shaped and current is flowing in the same direction in each segment, the magnetic field at point P will be directed into the plane of the page.
By calculating the individual contributions from each segment and summing them up, you can find the magnitude and direction of the magnetic field at point P in terms of the given variables (d, I, π, and the permeability constant μ₀).