A rod of mass m and resistance R slides smoothly over two parallel perfectly conducting wires kept sloping at an angle with respect to the horizontal (figure). The circuit is closed through a perfect conductor at the top. There is a constant magnetic field B along the vertical direction. If the rod is initially at rest, find the velocity of the rod as a function of time.
The component of magnetic field along the inclined plane will be and other will be perpendicular i.e. . The conductor is moving perpendicular to . It is the vertical component of the magnetic field. The movement will cause motional emf across the two ends of the rod.
given by
This makes flow of induced current
where R is the resistance of rod. Now, current-carrying rod experience a magnetic force which is given by
(horizontally in backward direction).
Now, the component of magnetic force parallels to the inclined plane along the upward direction.
Where,
Also, the component of weight (mg) parallel to the inclined plane along downward direction =
Now, by Newton's second law of motion
But, this is the linear differential equation.
On solving, we get
A is a constant to be determined by initial conditions.
The required expression of velocity as a function of time is given by