What is the magnetic field at the center of a square loop of side a carrying cur

What is the magnetic field at the center of a square loop of side a carrying current I?

What is the magnetic field at the center of a square loop of side a carrying current I?

Step 1: Understand that a square loop is a shape with four equal sides, and we are interested in the magnetic field at the center of this loop.
Step 2: Recall that when current flows through a wire, it creates a magnetic field around it.
Step 3: Recognize that the magnetic field at the center of the loop is influenced by all four sides of the square.
Step 4: Use the formula for the magnetic field due to a straight current-carrying wire, which is B = (μ₀I)/(2πr), where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire to the point where we measure the field.
Step 5: For each side of the square loop, calculate the contribution to the magnetic field at the center. The distance from the center to each side is a/2 (half the side length).
Step 6: Since there are four sides, and they all contribute equally to the magnetic field at the center, multiply the contribution from one side by 4.
Step 7: Combine the contributions from all four sides to find the total magnetic field at the center of the square loop.
Step 8: Simplify the expression to arrive at the final result: B = μ₀I/4a.

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