To solve this problem, we need to understand the concept of controlling torque in a spring-controlled moving iron instrument.

In moving iron instruments, the controlling torque (T_C) is provided by a spring. The amount of deflection (\theta) in the pointer of the instrument is directly proportional to the controlling torque applied by the spring. The relation between the controlling torque, the spring constant, and the deflection is given by Hooke's Law for springs:

T_C = K \theta

where:

• T_C is the controlling torque.
• K is the spring constant, representing the stiffness of the spring.
• \theta is the deflection angle of the pointer.

This equation states that the controlling torque is proportional to the deflection angle, with the spring constant as the proportionality constant. This ensures that the pointer returns to its original position when the force causing the deflection is removed.

Therefore, the correct option for the controlling torque in a spring-controlled moving iron instrument is:

T_C = K\theta, which corresponds to the fourth option in the provided list.

Let's rule out the other options:

• T_C = \frac{K}{\theta}: This implies an inverse relation which does not apply to spring-controlled instruments.
• T_C = K + \theta: Suggests a linear relationship different from the proportionality required for spring torque.
• T_C = K - \theta: Implies subtractive interaction, which does not represent the physical relationship in these instruments.

Therefore, the only plausible equation is T_C = K\theta.