Falling Loop in a uniform magnetic field As shown in the figure, a wire loop wit

Question

Falling Loop in a uniform magnetic field

As shown in the figure, a wire loop with length L, width w and mass m with one side connected with a resistor with resistance R is falling into a uniform magnetic field that points out of the page. Suppose at t = 0 the loop is at rest with the lower end align with the margin of the magnetic field. The loop is then released and starts to fall for t > 0. The distance between the lower end of the loop and the upper margin of the magnetic field is x(t) as shown in the figure. Assume the loop is long enough such that the loop can reach the terminal speed before totally immerse into the magnetic field. After the release and before the whole loop immerse into the magnetic field, what is the direction of the induced current, clockwise or counterclockwise? What is the acceleration of the loop a(t)? Express it in terms of m, g, B, w, R and v(t) Find the terminal speed v0 of the wire loop. Does the magnetic force do power (work done per unit time) throughout the process? In part(e) and part (f) v(t)

Explanation / Answer

a) loop wants to oppose motion into the field so current goes clockwise

b)

Fmagnetic = I L B = ( B L v/R) L B = B^2 L^2 v/R

mg - Fmagnetic = ma

g - B^2 L^2 v/(m R) = a

c) terminal velocity when a = 0

v = g m R/(B^2 L^2)

d) P = F v = B^2 L^2/R * (g m R/(B^2 L^2))^2 = g^2 m^2 R/(B^2 L^2)

e) P = m g v

f) P = I emf = B L v/R * B L v = B^2 L^2 v^2/R

no since the loop is still accelerating so it is gaining energy

g) at terminal speed

P = B^2 L^2/R * ( g m R/B^2 L^2)^2 = m^2 g^2 R/B^2 L^2

but this equal P gravity

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