Introduction:
In this scenario, we have a conductor of length l and mass m placed along the east-west line on a table. A certain amount of charge is passed through the conductor, causing it to jump to a height h. We also know the earth's magnetic induction, which is denoted as B. We need to determine the charge passed through the conductor.

Explanation:

To solve this problem, we will use the principles of electromagnetism, specifically the Lorentz force law and the concept of electromagnetic induction.

Lorentz Force Law:
The Lorentz force law states that a charged particle moving in a magnetic field experiences a force. The force acting on the conductor can be given as F = qvB, where q is the charge, v is the velocity of the conductor, and B is the magnetic induction.

Electromagnetic Induction:
When a conductor carrying a current is placed in a magnetic field, an electromotive force (emf) is induced across the conductor. This emf can be given by the equation emf = Blv, where B is the magnetic induction, l is the length of the conductor, and v is the velocity of the conductor.

Analysis:
1. The conductor jumps to a certain height h due to the force acting on it.
2. We can equate the gravitational potential energy to the work done by the magnetic force on the conductor.
3. The gravitational potential energy can be given as mgh, where m is the mass of the conductor and g is the acceleration due to gravity.
4. The work done by the magnetic force can be given as qvBd, where q is the charge, v is the velocity, B is the magnetic induction, and d is the distance traveled by the conductor.
5. Since the conductor is placed along the east-west line, the distance traveled is equal to the length of the conductor, d = l.
6. Equating the gravitational potential energy to the work done, we have: mgh = qvBl.
7. Solving for the charge q, we get: q = (mgh)/(vBl).

Conclusion:
The charge passed through the conductor can be calculated using the equation q = (mgh)/(vBl). By substituting the given values for the conductor's length, mass, jump height, and the earth's magnetic induction, we can determine the charge passed through the conductor.