Step 1 : Understanding the Question:
The topic of this question is the Magnetic Effects of Current, specifically the magnetic field produced by a circular conductor. When an electric current flows through a circular loop, it generates a magnetic field in the surrounding space, with its maximum intensity being at the center of the loop. This question asks us to identify which physical variables determine the strength of this central magnetic field.
Step 2 : Key Formulas and approach:
The approach is to use the formula derived from the Biot-Savart Law for the magnetic field at the center of a circular loop:
1. Formula: $B = \frac{\mu_0 I}{2r}$ (for a single turn)
2. Formula for $N$ turns: $B = \frac{\mu_0 N I}{2r}$
Where $B$ is the magnetic field, $I$ is the current, $r$ is the radius, and $\mu_0$ is the permeability of free space.
Step 3 : Detailed Explanation:
According to the mathematical expression $B = \frac{\mu_0 I}{2r}$, there are two primary physical variables that can change (assuming we stay in the same medium like air or vacuum).
Dependency 1: The magnetic field ($B$) is directly proportional to the current ($I$). If you increase the current flowing through the wire, the magnetic field strength at the center will increase linearly.
Dependency 2: The magnetic field ($B$) is inversely proportional to the radius ($r$). If you make the loop larger (increase radius), the field at the center becomes weaker because the current-carrying wire is physically further away from the center point.
Resistance (Option D) is not a direct factor in the field formula, though it affects the current if the voltage is fixed. However, the field itself is governed by the resulting current and geometry.
Therefore, to know the magnetic field at the center, one must know both how much current is flowing and how large the circle is.
Step 4 : Final Answer:
The magnetic field strength depends on both the current ($I$) and the radius of the loop ($r$). Therefore, the correct option is (C).