Question

A rectangular, a square, a circular and an elliptical loop, all in the $( x - y )$ plane, are moving out of a uniform magnetic field with a constant velocity, $\overrightarrow{ V }= v \cdot \hat{ i }$. The magnetic field is directed along the negative z-axis direction. The induced emf, during the passage of these loops, come out of the field region, will not remain constant for:

• A. the circular and the elliptical loops
• B. only the elliptical loop
• C. any of the four loops
• D. the rectangular, circular and elliptical loops

Answer 1

The Correct Option is A

To solve this problem, let's analyze the situation with respect to electromagnetic induction: 

1. According to Faraday's law of electromagnetic induction, the induced electromotive force (emf) in a closed loop is equal to the rate of change of the magnetic flux through the loop. Mathematically, this is represented as: \(\text{emf} = -\frac{d\Phi_B}{dt}\), where \(\Phi_B\) is the magnetic flux.
2. Here, the magnetic field is directed along the negative z-axis and the loops are moving with velocity \(\overrightarrow{V} = v \cdot \hat{i}\). As the loops move out of the magnetic field region, the area enclosed by the loop within the field changes over time, thus changing the magnetic flux.
3. Let's consider each loop:
   • Rectangular and Square Loops:
     • The change in magnetic flux will be linear and uniform as these loops have uniform dimensions along both axes. Thus, the rate of change of the area exposed to the magnetic field is constant, leading to a constant induced emf as they exit the field.
   • Circular and Elliptical Loops:
     • As these loops move out of the field, the rate of change of flux is not constant due to the varying lengths of the segments of these loops crossing the field boundary. The area exposed to the magnetic field does not change uniformly, resulting in an induced emf that varies with time.
4. Accordingly, the induced emf will not remain constant for the circular and elliptical loops. Their shape causes a non-uniform change in the area enclosed by the magnetic field, leading to a varying emf.

The correct choice is: the circular and the elliptical loops.