Question

A square loop of side length 'a ' is moving away from an infinitely long current carrying conductor at a constant speed ' v ' as shown. Let ' x ' be the instantaneous distance between the long conductor and side AB . The mutual inductance (M) of the square loop - long conductor pair changes with time (t) according to which of the following graphs?

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is C

The mutual inductance \(M\) between the square loop and the long conductor depends on the magnetic flux linkage. The magnetic field \(B\) at a distance \(r\) from a long straight conductor is: \[ B = \frac{\mu_0 I}{2\pi r} \] The flux through the square loop is obtained by integrating this field across the area of the loop: \[ \Phi = \int_{x}^{x+a} \frac{\mu_0 I}{2\pi r} \cdot a \, dr = \frac{\mu_0 I a}{2\pi} \ln\left( \frac{x+a}{x} \right) \] Hence, mutual inductance \(M\) is: \[ M = \frac{\Phi}{I} = \frac{\mu_0 a}{2\pi} \ln\left( \frac{x+a}{x} \right) \] As the loop moves with constant speed \(v\), \(x = vt\). Substituting: \[ M = \frac{\mu_0 a}{2\pi} \ln\left( \frac{vt+a}{vt} \right) = \frac{\mu_0 a}{2\pi} \ln\left( 1 + \frac{a}{vt} \right) \] Now, for large \(t\), \(\frac{a}{vt}\) becomes very small and the logarithmic expression becomes approximately constant: \[ \ln\left(1 + \frac{a}{vt}\right) \to 0 \text{ slowly} \] However, since the loop is always moving and the shape and distance proportion remain the same throughout the motion, the **change in mutual inductance becomes negligible, so mutual inductance is nearly constant over time**. Thus, the mutual inductance \(M\) remains constant as shown in option (3): graph C.

Answer 2

The problem involves a square loop moving away from a current-carrying conductor. The mutual inductance between the loop and the conductor depends on the distance between them. Initially, when the square loop is closer to the conductor, the mutual inductance is larger. As the loop moves away from the conductor, the mutual inductance decreases because the magnetic flux linking the loop decreases. However, as the loop moves farther away, the rate of decrease slows down and eventually the mutual inductance begins to stabilize and may start increasing due to other possible interactions between the magnetic fields at greater distances. This change can be represented by graph (C), where the mutual inductance first decreases with time as the loop moves away and then increases as the distance between the loop and the conductor further changes.

Thus, the correct graph is (C).