A rectangular loop with width L and a slide wire with mass m, are shown in the f

Question

A rectangular loop with width L and a slide wire with mass m, are shown in the figure (Figure. 1). A uniform magnetic field B vector is directed perpendicular to the plane of the loop into the plane of the figure. The slide were given an initial speed of v_a and then released. There is no friction between the slide wire and loop, and the resistance of the loop is negligible in comparison to the resistance R of the slide wire. Obtain an expression for F, the magnitude of the force exerted on the wire while it is moving at speed v. Express your answer in terms of the variables B, V, R, L, and m. Find the distance x that the wire moves before coming to rest. Express your answer in terms of the variables B, V, R, L, and m.

Explanation / Answer

A.

For part A i'm getting the same answer.

try to use this

F = B2L2v/R

I mean small v in the place of capital V in your expression.

B.

Using F = ma

a = -F/m

a = -B^2L^2v/mR

a = dv/dt = -B^2L^2v/mR

dv/v = [-B^2*L^2/mR]*dt

B^2L^2/mR = K

dv/v = -K*dt

after integration

ln v - ln vo = -Kt

ln (v/vo) = -Kt

v = vo*e^(-Kt)

at t = 0, v = vo

and as v tends to 0, then t tends to infinity

Now,

v = dx/dt = vo*e^(-Kt)

dx = vo*e^(-Kt)*dt

after integration

x = -(vo/K)*[e^(-Kt)(from 0 to t)]

x = (vo/K)*[1 - e^(-Kt)]

Given that comes to rest means v = 0, this happens when t tends to infinity

using this

x = (vo/K)*[1 - e^(-infinity)]

x = vo/K

x = mRvo/(B^2L^2)

Let me know if you have any doubt.

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