refer figure

A conducting cross bar of mass m and length L can slide without friction up or down on two conducting rails. The rails are connected on the bottom via a variable resistance R and battery E, so that the rails and the bar form a rectangular conducting loop. A uniform magnetic field B points into the plane of the loop. Initially the bar is at rest. What is the total force acting on the bar, moving up at speed v?


I am not able to get the book answer, so please solve it.

To analyze the situation you've described, we need to consider the forces acting on the crossbar as it moves through the magnetic field. The setup involves a conducting bar sliding on rails in a magnetic field, which induces an electromotive force (emf) due to its motion. Let's break this down step by step to find the total force acting on the bar when it moves upward at speed v.

Understanding the Forces at Play

When the bar moves upward with speed v, it cuts through the magnetic field lines. This motion induces an emf in the circuit, which can be calculated using Faraday's law of electromagnetic induction. The induced emf (ε) is given by:

ε = B * L * v

where:

• B is the magnetic field strength,
• L is the length of the bar, and
• v is the velocity of the bar.

Current Flow and Resistance

With the induced emf, a current (I) will flow through the circuit, which can be calculated using Ohm's law:

I = ε / R

Substituting the expression for ε, we get:

I = (B * L * v) / R

Magnetic Force on the Bar

As the current flows through the bar in the magnetic field, it experiences a magnetic force (F) given by the Lorentz force law:

F = I * L * B

Substituting the expression for I, we find:

F = (B * L * v / R) * L * B

This simplifies to:

F = (B^2 * L^2 * v) / R

Net Force on the Bar

Now, let's consider the total force acting on the bar. The magnetic force acts upward, opposing the weight of the bar. The weight (W) of the bar is given by:

W = m * g

where:

• m is the mass of the bar, and
• g is the acceleration due to gravity.

Thus, the net force (F_net) acting on the bar as it moves upward can be expressed as:

F_net = F - W

Substituting the expressions we derived earlier, we have:

F_net = (B^2 * L^2 * v) / R - m * g

Final Thoughts

This equation gives you the total force acting on the bar as it moves upward at speed v. If the magnetic force exceeds the weight of the bar, the bar will accelerate upward; if not, it will decelerate or remain at a constant speed. Understanding these relationships is crucial in electromagnetic systems and helps illustrate the interplay between motion, magnetic fields, and electrical currents.