Question

A loop is rotating about the y-axis in a magnetic field \( B = B_0 \sin \omega t \ \hat{a}_z \text{ Wb/m}^2 \). The voltage induced in the loop is due to (Note: \(\hat{a}_z\) means the B field is in the z-direction. The loop rotates about y-axis.)

• A. Current density
• B. Flux density
• C. Electric field intensity
• D. Combination of motional and transformer emf

Hint:

Faraday's Law of Induction: \( \mathcal{E} = -d\Phi_B/dt \).
Transformer EMF: Induced in a stationary loop by a time-varying magnetic field.
Motional EMF: Induced in a conductor moving through a magnetic field (or a loop changing area/orientation in a field).
If both the field is time-varying and the loop is moving/changing orientation such that flux linkage changes, both types of EMF can contribute.

Answer 1

The Correct Option is D

Voltage (emf) can be induced in a loop by two primary mechanisms according to Faraday's Law of Induction: 

• Transformer EMF (or Statically Induced EMF): This is due to a time-varying magnetic flux (\(\Phi_B\)) passing through a stationary loop. The induced voltage is given by: 
  \(\mathcal{E}_{transformer} = -\frac{d\Phi_B}{dt}\) where the change in flux is due to a time-varying magnetic field \(B(t)\) through a fixed area.
• Motional EMF (or Dynamically Induced EMF): This is due to the motion of a conductor (forming the loop or part of it) in a magnetic field. The magnetic flux linkage changes because the motion changes the effective area or orientation. For a conductor of length \(L\) moving with velocity \(\vec{v}\) in a magnetic field \(\vec{B}\), the induced voltage is: 
  \(\mathcal{E}_{motional} = \oint (\vec{v} \times \vec{B}) \cdot d\vec{l}\)

In this problem:

• The magnetic field itself is time-varying: \(B(t) = B_0 \sin(\omega t)\). This time-varying field passing through the loop (even if it were stationary but had some area perpendicular to \(B\)) would induce a transformer EMF.
• The loop is also rotating about the \(y\)-axis. If the loop has an area and its orientation with respect to the \(z\)-directed magnetic field changes due to this rotation, the magnetic flux through the loop will change, thus inducing a motional EMF. For example, if the loop is in the \(xy\)-plane initially, as it rotates about the \(y\)-axis, the angle between its area vector and the \(B\)-field (which is along the \(z\)-axis) changes.

Since both conditions are present (time-varying magnetic field AND motion of the loop that changes its orientation relative to the field, thus changing flux linkage due to motion), the total induced voltage will be a combination of transformer EMF and motional EMF.

The general form of Faraday's law encompassing both is:

\(\mathcal{E} = -\frac{d\Phi_B}{dt}\), where the change in flux \(d\Phi_B/dt\) can arise from a time-varying field, a time-varying area, or a time-varying orientation. The term \(d\Phi_B/dt\) can be expanded to show both contributions explicitly in some formulations.

Therefore, the induced voltage is due to a combination of motional and transformer EMF.

Final Answer:

Combination of motional and transformer emf