Is My Calculation for Terminal Speed of a Falling Conducting Loop Correct?

[edtman]

Please look for flaws in my reasoning, any help would be appreciated.
Question:
HRW CH31 #28P(if you have the book handy)
A long rectangular conducting loop of width L, resistance R, and mass m, is hung in a horizontal, uniform magnetic field B that is directed into the page and exists only above line aa. The loop is then dropped; during its fall, it accelerates until it reaches a certain terminal speed. Ignoring air resistance, what is this terminal speed?
Answer:
For the falling loop to reach a constant terminal speed the force of gravity pulling it downward must be canceled by a Lorentz Force pullint it upward

F(Lorentz)=mg eqn 1

To find F(L) we must first find the induce EMF, then the induced current and finally use this to determine F(L).

Magnetic Flux:(MF)=Integral[B*dot*dA] Let x equal vertical length of loop in B field.
(MF)=B*L*x

EMF=d/dt*(MF)=d/dt*BLX=BL*dx/dt=BLv

Now taking the derivative of both sides with respect to t:
d/dt*EMF=dv/dt*BL=abL

intergrate both sides with respect to t:
Integral[d/dt*EMF*dt=aBL*Integral[dt]
EMF=aBLt
This sounds reasonable because EMF should be increasing with time since change of flux is increasing with time.

Now in terms of current:
i=EMF/R=abLt/R

and Force:
F=aB^2*L^2*t/R

Subbing g for a and rewriting eqn 1 from way above:
mg=gB^2*L^2*t/R

solving for t:
t=MR/(M^2*L^2)

Using kinematics equation:
V(terminal)=g*t

Sound Plausible? Thanks for your time.

 

[gnome]

Here, I think, is a problem:

EMF=d/dt*(MF)=d/dt*BLX=BL*dx/dt=BLv

Now taking the derivative of both sides with respect to t:
d/dt*EMF=dv/dt*BL=abL

intergrate both sides with respect to t:
Integral[d/dt*EMF*dt=aBL*Integral[dt]
EMF=aBLt

(You could have saved yourself some time by simply claiming that v = at, which in effect is what you have done.)

Problem is you are assuming that a is constant. I think you will agree that this is the case here. The magnetic force exerted on the coil is increasing as v increases.

In fact, if you just remember that you are looking for a terminal velocity, it will be obvious that a_i=g, and a_f=0.

 

[edtman]

Ahhh , thanks for pointing that out; I sensed a bit of redundancy(sp?), but wasn't sure enough to simplify. Then again, I can use the practice showing relationships via Calculus.