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, \(\vec{V}=v\hat{i}\). The magnetic field is directed along the negative z-axis direction. The induced emf, during the passage of these loops, out of the field region, will not remain constant for:

The correct answer is option 1) i.e. the circular and the elliptical loops.

CONCEPT:

• Motional emf: The emf induced due to motion relative to a magnetic field is called the motional emf. 

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• Consider a straight conductor PQ moving perpendicular to a uniform magnetic field B.
  • Assume the motion of the rod to be uniform with a constant velocity of (v m/s).
  • The rectangle PQRS forms a closed circuit enclosing a changing area due to the motion of the rod PQ.

The magnetic flux ϕ enclosed by the loop PQRS can be given as:

ϕ = B × area = B × (l.x)

• Since the conductor is moving, there is a change in the rate of area moving. This causes a rate of change of flux which induces an emf.
• This induced emf is given by:

 \(\epsilon =\frac{-dϕ }{dt}=\frac{-d}{dt}(Blx) =-Bl\frac{dx}{dt}=Blv\)

EXPLANATION:

Consider a rectangular, a square, a circular, and an elliptical loop moving perpendicularly out from a magnetic field B with contact velocity.

From the given figure:

• Assume that the shaded region is the area that moves out of the magnetic field in a time interval of Δt.
• For the rectangular and square loop, the area moving out in Δt is constant, and therefore the rate of change of flux is constant. (∵ ϕ = B × area)
• For the circular and elliptical loop, the area moving out in Δt is not constant, and therefore the rate of change of flux is not constant.
• It is known that \(\epsilon = \frac{-dϕ}{dt}\) ⇒ induced emf ∝ rate of change of magnetic flux.
• Hence, the emf induced will be constant in the rectangular and square loop, and emf will not remain constant for circular and elliptical loops.