Final answer:
The self-inductance of the coil increases by a factor of 27 when each side of the equilateral triangular frame is increased by a factor of 3, with the number of turns per unit length remaining unchanged.
Step-by-step explanation:
The question pertains to the concept of self-inductance in physics, specifically in the context of a coil wound around a triangular frame. According to the principles of inductance, increasing the dimensions of the frame while keeping the number of turns per unit length constant will increase the total number of turns proportionally to the increase in length. Since the inductance of a coil is proportional to the square of the number of turns (N), and each side of the triangle is increased by a factor of 3, the overall factor of increase for the number of turns N is also 3.
Therefore, the self-inductance will increase by a factor of N squared, which is 3 squared, resulting in an increase by a factor of 9. However, because the inductance also depends on the area enclosed by the coil and the shape of the frame is an equilateral triangle, the area (A) increases by a factor of 9 (since Area ≈ (side length) squared). The total self inductance will thus increase by a factor of 3^2 * 9, that is, by a factor of 27. So, the correct answer is D. increases by a factor of 27.