Consider a set of conducting metal tracks (two parallel beams with a flat top su
ID: 1316945 • Letter: C
Question
Consider a set of conducting metal tracks (two parallel beams with a flat top surface) 0.3 m apart. Across the tracks, perpendicular to them, is a conducting rod of mass 0.65 kg. There is a uniform magnetic field of 0.80 T [Tesla, the SI unit of magnetic field strength, 1T = 1 N / (A m),] pointing down. A circuit consisting of a 47 Ohm resistor and a 15 V power supply is placed across the rails. After some time in motion, the velocity is constant. What is that velocity? Hint: The velocity is constant when the induced electromotive force is equal and opposite to the voltage applied to the rails. Please show work.
Explanation / Answer
Force due to current I following through rod where I = V/R is
F1 = ILB = VLB/R
Force due to induced current i:
E = induced voltage = -d(BA)/dt
E = -B d(Lx)/dt = -BL dx/dt
where x = distance along tracks and
L = length of rod
so force is
F2 = ELB/R = -(BL)^2/R * dx/dt
And net force is
F = F1 + F2
F = VLB/R - (BL)^2/R dx/dt
When dx/dt = constant,
net force is zero
since F = md^2x/dt^2 = m d(dx/dt)/dt = 0 if dx/dt = constant
so F1 = -F2
---> VLB/R = (BL)^2/R dx/dt
?--> dx/dt, v = V/BL = 15 / 0.8 * 0.3
v = 62.5 m/s