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Motional EMF in the presence of a time-varying magnetic field
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[QUOTE="TSny, post: 6851528, member: 229090"] [ATTACH type="full" alt="1675625992053.png"]321808[/ATTACH] I've marked some places that I believe need correction. Earlier, we defined ##\beta## as the constant in ##B(t) = B_0 - \beta t##. Thus, ##\beta = -B'(t)##. So, there is a sign error in your equation for ##\beta##. As pointed out in post #3, your expression for the area ##A = lx_0 + lx## implies that you are taking ##x## to be the displacement of the rod from its initial position. So, your initial condition should be x[0] = 0. Your symbol ##x_0## is the initial distance of the rod from the resistor, but this is not the initial value of your symbol ##x##. There appear to be two sign errors in the differential equation: (1)The right-hand side of your differential equation has the wrong overall sign. I think this is connected to the minus sign in ##\frac{dB(t)}{dt} = - \beta##. (2) For the sign error on the left-hand side of your differential equation, let's go back to your post #1 where you had [ATTACH type="full" alt="1675525693119-png.png"]321813[/ATTACH] which looks correct. You then rearranged this to [ATTACH type="full" alt="1675526052701-png.png"]321814[/ATTACH] I believe the minus signs on the left side of this equation should actually be plus signs. Since ##\frac {dB(t)}{dt} = - \beta##, we can write the corrected equation as $$\frac{Rm}{l} \ddot x + B(t)^2 l \dot x - \beta B(t) x = +\beta x_0 B(t_0)$$ Anyway, all of this leads to the need to make the sign correction for the second term on the left side of your Mathematica coded differential equation. The signs are a nightmare. I also made a sign error in one of the terms of the differential equation. Now, I find that Mathematica no longer gives a manageable analytic solution using the command DSolve. However, I do get a numerical solution using NDSolve. So you can still plot ##v(t)##. Note that with the sign corrections, your differential equation can be written as $$\ddot x + a(b-t)^2 \dot x - a(b-t)(x+x_0) = 0$$ Define a new x variable ##\tilde x= x + x_0##. ##\tilde x## represents the distance of the rod from the resistor. The area of the flux is then ##A = l\tilde x##. So, the initial value of ##\tilde x## is ##x_0##. The differential equation may then be written as $$\ddot {\tilde x} + a(b-t)^2 \dot {\tilde x} - a(b-t)\tilde x = 0$$ with initial conditions ##\tilde x(0) = x_0## and ##\dot {\tilde x} = 0##. Of course, at this point you can just drop the tilde with the understanding that ##x## now represents the redefined ##x##. [/QUOTE]
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Motional EMF in the presence of a time-varying magnetic field
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