Final answer:
Ampere's law can be applied both at the center and on the axis of a circular loop to calculate the magnetic field; however, it's more directly applied at the center and may require additional considerations or be complemented by the Biot-Savart law when used on the axis.
Step-by-step explanation:
To apply Ampere's law for calculating the magnetic field (B) due to a current-carrying circular loop, we need to consider symmetry and the path over which we integrate.
(a) At the center of a circular loop, the magnetic field can be derived using Ampere's law combined with the Biot-Savart law, which states that B = μ_0I / (2R), where μ_0 is the permeability of free space, I is the current through the loop, and R is the radius of the loop. The field is directed perpendicular to the plane of the loop, and its magnitude is given by the formula.
(b) On the axis of a circular loop, Ampere's law alone is not typically used because the magnetic field on the axis varies with distance from the center of the loop. Here, we commonly use the Biot-Savart law or the formula for the magnetic field on the axis of a loop. However, Ampere's law can be used if combined with other symmetrical considerations to simplify the calculations, but for the axis of a loop, the Biot-Savart law is usually more practical.
Therefore, the answer to the student's question would be (c) Both (a) and (b) because Ampere's law can be applied in both scenarios, though it is more direct for (a) than for (b).