Question

A long, horizontal wire AB rests on the surface of a table and carries a current I. Horizontal wire CD is vertically above wire AB and is free to slide up and down on the two vertical metal guides C and D (the figure ).
Wire CD is connected through the sliding contacts
to another wire that also carries a current I, opposite in direction to the current in wire AB. The mass per unit length of the wire CD is λ. To what
equilibrium height (h) will the wire CD rise, assuming that the magnetic force on it is due entirely to the current in the wire AB? (Express your answer in terms of the variables I, λ, and appropriate constants (μ0,π, g))

Answer 1

Concepts and reason

The concepts used to solve this problem are ampere’s law, force of gravity and Lorentz law.

Initially, use ampere’s law to find the magnetic field at CD due to the current in AB

Finally, equate force of gravity and Lorentz force to find the equilibrium height of the wire CD increase.

Fundamentals

Ampere’s law for any closed loop states,

The line integral of the product of the magnetic field and the infinitesimal element of the curve over a closed path is equal to permeability times the current surrounded in the loop.

The expression for ampere’s law is,

Here, magnetic field is , infinitesimal element is , permeability is , and current is .

The force acts on the passing charged particles in an electromagnetic field is defined by Lorentz law.

The expression for force on current carrying wire in a magnetic field is,

Here, length is , current is , magnetic field is , and force is .

The expression for force of gravity is,

Here, mass is , and gravitational constant is .

The expression for ampere’s law is,

The area of the wire CD to rise the equilibrium height is . Here, height is .

Substitute for in ampere’s law.

Rearrange the above equation,

The expression for the magnetic at CD due to current in AB is,

The expression for force on current in magnetic field is,

The wire CD is right angles to the field (i.e.).

So,

For wire CD the magnetic force on current in magnetic field is,

…… (3

The expression for force of gravity is,

…… (4)

The mass per unit length of the wire CD is .

Hence,

Rearrange the above expression.

Substitute for in equation (4).

Substitute for in the above expression.

Substitute for .

Ans:

Thus, the equilibrium height of the wire CD increases is .