To tackle the problem of a magnetic monopole placed at the center of a current-carrying loop, we need to consider the interaction between the magnetic field produced by the loop and the monopole itself. Let's break this down step by step to find the force experienced by the loop due to the monopole.
Understanding the Magnetic Field of the Loop
When a current \( I \) flows through a circular loop of radius \( R \), it generates a magnetic field around it. The magnetic field at the center of the loop can be calculated using the formula:
B = \frac{\mu_0 I}{2R}
Here, \( \mu_0 \) is the permeability of free space, and \( R \) is the radius of the loop. This magnetic field is directed along the axis of the loop, and its strength depends on the current flowing through the loop and the radius.
Force on the Magnetic Monopole
A magnetic monopole, which we are considering to have a pole strength of \( +m \), will experience a force when placed in a magnetic field. The force \( F \) on a magnetic monopole in a magnetic field \( B \) is given by:
F = mB
Substituting the expression for the magnetic field \( B \) at the center of the loop, we get:
F = m \cdot \frac{\mu_0 I}{2R}
Final Expression for the Force
Now, substituting \( m \) with the pole strength of the monopole, we can express the force experienced by the loop due to the monopole as:
F = \frac{\mu_0 m I}{2R}
Identifying the Correct Option
From the options provided, we need to match our derived expression with one of them. The expression we obtained is:
F = \frac{\mu_0 m I}{2R}
Looking at the options:
- (1) \( \frac{\mu_0 n I}{R} \)
- (2) \( \frac{\mu_0 n I}{2R} \)
- (3) \( \frac{\mu_0 n I}{4R} \)
- (4) \( \frac{\mu_0 m I}{4\pi R} \)
None of the options directly match our derived expression. However, if we consider the possibility of \( n \) being related to \( m \) in some way, we can see that option (2) \( \frac{\mu_0 n I}{2R} \) is the closest match if we assume \( n \) is equivalent to \( m \). Thus, if \( n \) represents the pole strength of the monopole, then:
The correct answer is (2) \( \frac{\mu_0 n I}{2R} \)
Conclusion
In summary, the force experienced by the current-carrying loop due to the magnetic monopole at its center can be derived from the magnetic field generated by the loop and the interaction with the monopole. This problem illustrates the fascinating interplay between electricity and magnetism, showcasing how magnetic fields can exert forces on magnetic charges, even in hypothetical scenarios like this one.
