A conducting wire XY of mass m and negligible resistance slides smoothly on two parallel conducting wires as shown in figure. The closed circuit has a resistance R due to AC. AB and CD are perfect conductors. There is a magnetic field
(i) Write down an equation for the acceleration of the wire XY
(ii) If B is independent of time, obtain v(t), assuming v(0) = u0
(iii) For (ii), show that the decrease in kinetic energy of XY equals the heat lost in.
Upon analyzing the situation:
The wire is parallel with y-axis at y=0 and y=L. It placed along the x-axis.
At t=0, the wire starts from x=0 and is moving with a velocity of v. At certain time wire is at is the displacement function_). The diagram is redrawn as:
The magnetic flux linked with the loop is given by
and as we know both (area vector) and (magnetic field vector) are directed along z-axis. So angle between them is 0.
So,
At any instant of time t,
Magnetic flux
Emf induced due to change in magnetic field
Emf induced due to motion
Total emf in the circuit = emf due to change in field (along XYAC) + the motional emf across XY
Electric current in clockwise direction (as shown in equivalent diagram) is given by
The force acting on the conductor is given by
Substituting the vlues, we have
Applying Newton's second law of motion,
which is the required equation.
If B is independent of time, i.e., B = Constant
or
substituting the above value in Eq (i), we have
or
Intergrating using variable separable form of differential equation, we have
Applying given conditions, at
This is the required equation.
Since the power consumption is given by
Here,
Now, energy consumed in time interval dt is given by energy consumed Therefore, toal energy consumed in time t.
= decrease in kinetic energy
This proves that the decrease in kinetic energy of XY equals the heat lost in R.