If the radius of a circular loop carrying current is halved, how does the magnetic field at the center change?

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Q: If the radius of a circular loop carrying current is halved, how does the magnetic field at the center change?

Solution: The magnetic field at the center is inversely proportional to the radius, so it quadruples.

Steps: 7

Step 1: Understand that a circular loop carrying current creates a magnetic field at its center.
Step 2: Know that the strength of the magnetic field (B) at the center of the loop is related to the radius (r) of the loop.
Step 3: Remember the formula for the magnetic field at the center of a circular loop: B = (μ₀ * I) / (2 * r), where μ₀ is a constant and I is the current.
Step 4: Notice that in this formula, the radius (r) is in the denominator, which means that as the radius decreases, the magnetic field increases.
Step 5: If the radius is halved (r becomes r/2), substitute this into the formula: B = (μ₀ * I) / (2 * (r/2)) = (μ₀ * I) / (r) = 2 * (μ₀ * I) / (2 * r).
Step 6: This shows that the magnetic field at the center becomes 2 times stronger when the radius is halved.
Step 7: Since the magnetic field is inversely proportional to the radius, halving the radius results in quadrupling the magnetic field strength.