Wire loop falling in a magnetic field: The upper part of a rectangular wire loop

Question

Wire loop falling in a magnetic field:

The upper part of a rectangular wire loop is intercepted by a homogeneous magnetic 7ield, perpendicular to the plane of the paper, of strength B. The lower part of the loop experiences no 7ield. The wire loop, whose ohmic resistance is R, is found to fall with constant speed v. (a) Calculate the magnitude of the current 7low induced in the loop as a result of the fall with constant speed. Express I in terms of B, l, v, and R. (b) Calculate the Lorentz force on the current loop, and show that the force is upwards. Express the force in terms of B, l, v, and R. (c) The magnitude of v is determined by a balance between mg and the Lorentz force. Use this to obtain a formula for v in terms of m, g, B, l, and R. (d) Calculated numerically the speed of fall if l = 10 cm, m = 1.8 g, R = 0.68

Wire loop falling in a magnetic field: The upper part of a rectangular wire loop is intercepted by a homogeneous magnetic field, perpendicular to the plane of the paper, of strength B. The lower part of the loop experiences no 7ield. The wire loop, whose ohmic resistance is R, is found to fall with constant speed v. (a) Calculate the magnitude of the current flow induced in the loop as a result of the fall with constant speed. Express I in terms of B, l, v, and R. (b) Calculate the Lorentz force on the current loop, and show that the force is upwards. Express the force in terms of B, l, v, and R. (c) The magnitude of v is determined by a balance between mg and the Lorentz force. Use this to obtain a formula for v in terms of m, g, B, l, and R. (d) Calculated numerically the speed of fall if l = 10 cm, m = 1.8 g, R = 0.68 x10^-2 ohm. (The wire is #15 gauge aluminum.) The magnetic field is produced by an electromagnetic and B = 1 T. (e) How would the value of v be changed if: a. The wire was twice as thick. b. The loop contained two turns. c. The magnetic field was doubled. d. The linear dimensions of the loop were doubled

Explanation / Answer

part A:

induced emf is defiend as the rate of change of magnetic flux

in mathematical form,

induced emf e = NAdB/dt    or NABW   or   NA dB/dt cos theta

where A = area

N = no. of tunrs

W = angular frequency

dB/dt = rate of change of magnetic field

induced current i = e/R

Where R = resisatnce

so current i = lvB/R------------------------<<<<<<<<<<<<<Answer

------------------------------
part B :

lorentz Force F = magnetic force+ Electrical force

so

F = iLB + Eq

F = (LvB* LB)/R   + 0

F = v L^2 B^2/R

---------------------------------

part C:

apply gravitational force F = mg = iLB

so

mg = v L^2 B^2/R

so

speed v = mgR /(L^2B^2)

---------------------------------------

part D:

speed v = 1.8 e-3 * 9.8 * 0.68 e-2/(0.1*0.1 * 1*1)

v = 0.01199   m/s

--------------------------------------------

if d is doubles, v also doubles

---------------------------
in no. of tuens increases, speed reduces by two

------------

if B is doubled, V also doubles

----------------------

is length is Doubled, v also doubles

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