Two identical wires with resistance,R and mass,m rest on two smooth parallel superconducting (zero resistance) rails. Initially, they are separated by a distance,D=70 cm. Suddenly, a uniform magnetic field B, perpendicular to the plane of the rails is switched on and the wires start rolling. What will be the final distance in centimeters between the wires? Assume that the wires are thin and that the process of switching on the magnetic field happens very fast.
To determine the final distance between the wires, we need to consider the forces acting on them.
When the magnetic field is switched on, a current is induced in the wires due to the changing magnetic flux. According to Faraday's law of electromagnetic induction, the induced emf is given by the equation:
ε = -dΦ/dt,
where ε is the induced emf, Φ is the magnetic flux, and dt is the change in time. Since the wires are rolling, the magnetic flux through each wire changes with time.
The induced current creates a magnetic field that interacts with the uniform magnetic field B. This interaction generates a force between the wires, causing them to move towards each other.
To analyze the forces, we can use the concept of magnetic force between two parallel conductors. The force per unit length between two parallel conductors carrying currents I1 and I2, separated by a distance r, is given by:
F = (μ0/2π) * (I1 * I2 / r),
where μ0 is the permeability of free space.
In this case, the wires have the same current (since they have the same resistance) and are separated by a distance D. Therefore, the force between the wires is:
F = (μ0/2π) * (I^2 / D).
Initially, the wires are at rest, so there is no force acting on them. When the magnetic field is switched on, the force between the wires causes them to move towards each other.
To find the final distance between the wires, we need to analyze the motion of the two wires. Since the rails are smooth and the wires are rolling, the frictional forces between the wires and the rails are negligible. Thus, the only force acting on the wires is the magnetic force.
The magnetic force will cause the wires to accelerate towards each other, and they will continue to move until the force of magnetic attraction is balanced by the upward gravitation force on each wire.
The acceleration of each wire can be calculated using Newton's second law:
F = m * a,
where F is the force acting on each wire, m is the mass of each wire, and a is the acceleration.
Substituting the expression for the force between the wires, we have:
(μ0/2π) * (I^2 / D) = m * a.
Since the wires have the same mass, the total mass is 2m.
Rearranging the equation, we can solve for the acceleration:
a = (μ0 * I^2) / (2π * D * m).
Now, we need to find the time taken for the wires to come to a stop. To do this, we can use the equations of motion. The wires accelerate towards each other until their final velocity becomes zero. The distance traveled by each wire can be calculated using the equation:
s = ut + (1/2) * a * t^2,
where s is the distance, u is the initial velocity, t is the time, and a is the acceleration.
Since the wires are initially at rest (u = 0), the equation becomes:
s = (1/2) * a * t^2.
At the moment the wires come to rest, the total distance they have traveled (2s) is equal to the initial separation between the wires (D). Therefore:
2s = D,
(1/2) * a * t^2 = D,
t^2 = (2 * D * m) / (μ0 * I^2).
Finally, we can solve for the time it takes for the wires to come to a stop:
t = √[(2 * D * m) / (μ0 * I^2)].
Once we have the time, we can determine the final distance between the wires using the equation:
s = (1/2) * a * t^2.
Substituting the values of a and t, we can calculate the final distance.
Note: To obtain a numerical answer, the values of the resistance R, mass m, and current I need to be specified.