In the figure, the rolling axle, 1.60 m long, is pushed along horizontal rails a

Question

In the figure, the rolling axle, 1.60 m long, is pushed along horizontal rails at a constant speed v = 2.84 m/s.

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A resistor R = 0.335 ? is connected to the rails at points a and b, directly opposite each other. The wheels make good electrical contact with the rails, and so the axle, rails, and R form a closed-loop circuit. The only significant resistance in the circuit is R. There is a uniform magnetic field B = 7.55

Explanation / Answer

sol: Axle rolling on rails lets reference all voltage drops to the negative terminal of the battery, call it circuit ground. Axle rolling on rails 1.) With the switch open there is a total circuit current flowing out of the battery through the circuit and back into the battery of 8 amps. Both parallel branches of resistors add up to 6 ohms in each branch. Which means that each branch of resistors is drawing 24 v / 6 ohms = 4 amps, exactly half the total circuit current of 8 amps. Axle rolling on rails So, the voltage at "a" with reference to ground is (3 ohms) x ( 4 amps) = 12 volts. Now, we know that the other 4 amps of the 8 amp total circuit current has to be flowing through the remaining resistance branch. Thus, (1 ohm) x (4 amps) = 4 volts at "b". Now if you took a volt meter and connect it such that the postivie lead is at "a" and the negative lead is at "b", your meter will read (12v - 4v) or 8v volts difference Axle rolling on rails 2.) With the switch closed, do you see that you have place the top two resistors of the two branches in parallel, in series with the botton two resistors, of the two branches, which are now also in parallel. In otherwords, with the switch closed you have top rsistors ( 3 ohm in parallel with 5 ohm) in series with the bottom resisors (3 ohm in parallel with 1 ohm). Reduce this to two series resistors across 24 volts. 3 ohms in parallel with 5 ohms is 1.875 ohms; that in series with 3 ohms in parallel with 1 ohm, or 0.75 ohms. So, that makes 1.875 + 0.75 ohms across 24 volts. The total current is 24 volts / (1.875 + 0.75 ohms) = 9.143 amps. Axle rolling on rails 3. with the switch closed, point "a" and "b" are short circuited. There can be no potential difference with the switch closed. Axle rolling on rails (a)induced current=induced emf / resistance induced emf =magnetic field x axle length xspeed induced current=0.12 x 1.5 x 8 / 0.402 = 3,58 ampere (b)force=magnetic field x current x length of axle=0.12x3.58x1.5=0.64 newton

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