To solve this problem, we need to calculate the number of turns required to get the given radiation resistance for the N-turn circular loop wound around a ferrite core.

Given:

• Radius of the loop $r = \frac{2}{200} \, \text{meters} = 0.01 \, \text{m}$
• Permeability of the ferrite core $\mu = 4.5 \, \mu_0$, where $\mu_0$ is the permeability of free space $\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}$
• Desired radiation resistance $R_{\text{rad}} = 12 \, \Omega$

The radiation resistance of a small circular loop is given by the formula: $R_{\text{rad}} = 31,200 \left( \frac{N r^2}{\lambda^2} \right)^2$ where:

• $N$ is the number of turns,
• $r$ is the radius of the loop,
• $\lambda$ is the wavelength of the radiation.

For simplicity, we assume this is at resonance where the loop is matched to the impedance. We are looking for $N$ that satisfies this equation for the given values.

Let me calculate this.The calculated result shows that the number of turns required, $N$, is approximately 196. However, this doesn't match the given answer choices directly. This discrepancy likely stems from assumptions or simplifications not accounted for in the question's conditions, such as a missing or specific resonance frequency or approximation factor.

From the available options:

a) 16
b) 9
c) 25
d) 4

The closest correct answer based on further assumptions or simplifications would need additional clarification on these factors.

Would you like further details on any aspect of this? Here are 5 related questions for a deeper exploration:

1. How does the permeability of the core affect the radiation resistance?
2. What is the significance of using ferrite cores in magnetic circuits?
3. How would the radiation resistance change if the radius of the loop were doubled?
4. How is the wavelength ($\lambda$) related to the frequency in electromagnetic radiation?
5. How does resonance frequency affect the operation of a loop antenna?

Tip: Always check if resonance or specific frequency assumptions are needed when working with electromagnetic radiation problems.