The correct answer is: A. $${\left( {{\text{conductor diameter}}} \right)^{\frac{1}{2}}}$$
Skin effect is the tendency of an alternating electric current (AC) to flow predominantly near the surface of a conductor, rather than through the entire cross-section of the conductor. This is due to the interaction between the electric field and the magnetic field produced by the current. The skin depth is the depth at which the current density falls to 1/e (about 0.368) of its value at the surface. The skin depth is inversely proportional to the square root of the frequency of the AC current.
The skin effect is a significant factor in the design of high-frequency AC circuits. It can cause a significant increase in the resistance of the conductor, and it can also lead to losses due to heating. To minimize the effects of skin effect, conductors are often made of multiple thin layers, or are made of a material with a high conductivity.
The options B, C, and D are incorrect because they are not proportional to the square root of the conductor diameter.