What is the expression for terminal speed in this homework statement?

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SUMMARY

The discussion centers on the expression for terminal speed in the context of electromagnetic induction, specifically addressing the relationship between induced electromotive force (emf) and the velocity of a loop in a magnetic field. The correct expression for induced emf is established as ε = -dφ/dt, where φ represents magnetic flux. The solution clarifies that for a constant magnetic field B, the induced emf can be expressed as ε = Bvl, where l is the length of the loop and v is the velocity, confirming that ε = -Bd(v.t)/dt is incorrect due to the non-constant nature of v.

PREREQUISITES
  • Understanding of electromagnetic induction principles
  • Familiarity with the concept of magnetic flux
  • Knowledge of calculus, specifically differentiation
  • Basic physics concepts related to motion in magnetic fields
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  • Study Faraday's Law of Electromagnetic Induction
  • Explore the relationship between magnetic fields and induced emf
  • Learn about the applications of terminal velocity in physics
  • Investigate the effects of varying magnetic fields on induced currents
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Homework Statement


29-59_zpsca4d2d1f.png



Homework Equations


That is the solution:
29-59_zps10e24f3f.png



The Attempt at a Solution


I think ε≠Bvl because ε = - dø/dt=- Bd(v.t)/dt, v is not constant.
Why ε=Bvl?
 
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The induced emf in the loop is related to the change in flux through a surface whose boundary is the loop: $$\epsilon = -\frac{\text{d} \phi}{\text{d}t} = -\frac{\text{d}}{\text{d}t} \int_S \mathbf{B} \cdot \mathbf{dS}$$
In your case, the B field is of constant magnitude and direction and dS instantaneously is Ldx.
 
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dø/dt = d(B. Area)/dt = d(BLx)/dt = BL dx/dt = BLv.
 
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