Advertisements
Advertisements
Question
Find the current in the sliding rod AB (resistance = R) for the arrangement shown in figure. B is constant and is out of the paper. Parallel wires have no resistance. v is constant. Switch S is closed at time t = 0.
Solution
This is a similar problem as we discussed above. Here, a conductor of length d moves with speed v, perpendicular to the magnetic field B as shown in figure.
Due to this a motional emf is induced across two ends of rod (e = vBd). Since switch S is closed at time t = 0, capacitor is charged by this potential difference.
Let Q(t) be the charge on the capacitor and current flows from A to B.
Now, the induced current
`I = (dQ)/(dt) = (Bvd) - Q/(RC)`
`Q/(RC) + (dQ)/(dt) = (Bvd)/R`
`Q + RC (dQ)/(dt) = vBC` .....(Let `vBdC = A`)
`Q + RC Q/(dt) A`
`(dQ)/(A - Q) = 1/(RC) dt`
By integrating we have
`int_0^Q (dQ)/(A - Q) = 1/(RC) int_0^t dt - [In (A -Q) - In A] = t/(RC)`
In ``(A / Q)/A = - t/(RC)`
`(A - Q)/A = e^(-t/(RC))`
`Q = A(1 - e^(-t/(RC)))`
Current in the rod,
`I = (dQ)/(dt) = d/(dt) [A(1 - e^(-t/(RC)))]`
= `- A(e^-t/(RC))(- 1/(RC))`
`I = (vBd)/R e^(-t/(RC)`
APPEARS IN
RELATED QUESTIONS
An aircraft of wing span of 50 m flies horizontally in the Earth's magnetic field of 6 x 10-5 T at a speed of 400 m/s. Calculate the emf generated between the tips of the wings of the aircraft.
A wire of length 50 cm moves with a velocity of 300 m/min, perpendicular to a magnetic field. If the e.m.f. induced in the wire is 2 V, the magnitude of the field in tesla is ______.
The emf induced across the ends of a conductor due to its motion in a magnetic field is called motional emf. It is produced due to magnetic Lorentz force acting on the free electrons of the conductor. For a circuit shown in the figure, if a conductor of length l moves with velocity v in a magnetic field B perpendicular to both its length and the direction of the magnetic field, then all the induced parameters are possible in the circuit.
A conducting rod of length l is moving in a transverse magnetic field of strength B with velocity v. The resistance of the rod is R. The current in the rod is ______.
An e.m.f is produced in a coil, which is not connected to an external voltage source. This can be due to ______.
- the coil being in a time varying magnetic field.
- the coil moving in a time varying magnetic field.
- the coil moving in a constant magnetic field.
- the coil is stationary in external spatially varying magnetic field, which does not change with time.
A magnetic field B = Bo sin ( ωt )`hatk` wire AB slides smoothly over two parallel conductors separated by a distance d (Figure). The wires are in the x-y plane. The wire AB (of length d) has resistance R and the parallel wires have negligible resistance. If AB is moving with velocity v, what is the current in the circuit. What is the force needed to keep the wire moving at constant velocity?
A rectangular loop of wire ABCD is kept close to an infinitely long wire carrying a current I(t) = Io (1 – t/T) for 0 ≤ t ≤ T and I(0) = 0 for t > T (Figure). Find the total charge passing through a given point in the loop, in time T. The resistance of the loop is R.
An aeroplane, with its wings spread 10 m, is flying at a speed of 180 km/h in a horizontal direction. The total intensity of earth's field at that part is 2.5 × 10-4 Wb/m2 and the angle of dip is 60°. The emf induced between the tips of the plane wings will be ______.
A simple pendulum with a bob of mass m and conducting wire of length L swings under gravity through an angle θ. The component of the earth's magnetic field in the direction perpendicular to the swing is B. Maximum emf induced across the pendulum is ______.
(g = acceleration due to gravity)
A magnetic flux associated with a coil changes by 0.04 Wb in 0.2 second. The induced emf with coil is ______.
