Final answer:
The maximum magnetic moment for a current-carrying loop made from a wire of length l is achieved when n = 1, as a single loop maximizes both the area and the magnetic field inside the loop. Option A is the correct answer.
Step-by-step explanation:
The student is asking about the condition for maximizing the magnetic moment of a circular loop made from a wire of length l when carrying a current i and formed into n turns. The magnetic moment (μ) of a coil is given by the formula μ = nIA, where n is the number of turns, I is the current, and A is the area of the loop. Since the length of the wire is constant, forming the wire into a loop of one turn will maximize the area, and therefore, the magnetic moment will be largest for n = 1. So, the correct option is (a) n = 1.
Consider each portion of wire as being part of the circumference of a circle. With more turns (larger n), each circular loop would have a smaller radius to maintain the same total length. The area of a circle is proportional to the square of its radius (A = πr^2), so by increasing n, you decrease the radius exponentially which results in a smaller area and thus a smaller magnetic moment. When n = 1, the entire length of the wire forms the circumference of one single loop, yielding the largest possible area and magnetic moment.
When considering the factors contributing to the magnetic moment, it should also be noted that the strength of the magnetic field inside the coil is inversely proportional to n, and thus, the single-turn loop also ensures the maximum magnetic field.