Here comes a challenge question,

A charge particle having charge q enters a region in which there is a friction force propotional to the speed of the particle. The constant of propotionality of friction is b. The particle stops travelling a distance of 10m, in a straight line.

Now, the experiment is repeated, but with magnetic feild of unknown strenght, in a direction perpendicular to entry velocity of the particle. The particle is found to stop at a point 6m from the entry point, then find the strenght of magnetic feild.

The magnetic feild is now doubled in magnitude. How far from the point of entry the particle come to rest?

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nudge me, if you need help understanding the problem. Before you begin attempting the question, let me tell you, that everyone who has done basic concepts of magnetism, can do this problem, but it will involve dirty mathematics, and atleast 1, 1.5 pages of solution.

I am expecting a ''smart'' solution from you goiitians...... this question can be done in 4-5 lines only, but for that you have to think real smart....

This question is open to all... students, college going people, experts ... all ...but for the ppl haing board examination.. i advise not to waste time on this.

Let’s break down this problem step by step, focusing on the key concepts of motion, forces, and magnetic fields. The scenario involves a charged particle experiencing friction and then a magnetic force, which alters its motion. The goal is to find the strength of the magnetic field and determine how far the particle travels when the magnetic field is doubled.

Understanding the Forces at Play

Initially, the charged particle moves through a medium where it experiences a frictional force proportional to its speed. This friction can be expressed as:

• Frictional Force (F_f) = -b * v

Here, b is the constant of proportionality, and v is the velocity of the particle. The negative sign indicates that the frictional force opposes the motion.

First Scenario: Particle Stops After 10m

In the first case, the particle stops after traveling 10 meters. The work done against friction can be calculated using the distance and the frictional force:

• Work = F_f * distance = b * v * distance

Since the particle comes to a stop, all the initial kinetic energy is converted into work done against friction. If we denote the initial velocity as v_0, we can set up the equation:

• 0.5 * m * v_0^2 = b * v_0 * 10

From this, we can derive:

• v_0 = (20b/m)

Second Scenario: Introducing the Magnetic Field

Now, when the magnetic field is introduced perpendicular to the velocity, the particle experiences a magnetic force given by:

• F_m = q * v * B

Where B is the magnetic field strength. The magnetic force acts perpendicular to the velocity, causing the particle to move in a circular path rather than straight. However, it also experiences the same frictional force. The net force acting on the particle can be expressed as:

• F_net = F_m - F_f = q * v * B - b * v

When the particle stops after traveling 6 meters, we can set up a similar energy balance as before:

• 0.5 * m * v_0^2 = b * v_0 * 6

From the previous expression for v_0, we can substitute and solve for B:

• 0.5 * m * (20b/m)^2 = b * (20b/m) * 6

Solving this gives us:

• B = (10b/m)

Doubling the Magnetic Field

When the magnetic field is doubled, the new magnetic field strength becomes:

• B' = 2B = (20b/m)

Now, we need to find out how far the particle travels before coming to rest again. The same principle applies, but with the increased magnetic force:

• F_net' = q * v * B' - b * v

Using the energy balance again:

• 0.5 * m * v_0^2 = b * v_0 * d'

Substituting the new magnetic field strength into the equation, we find:

• 0.5 * m * (20b/m)^2 = b * (20b/m) * d'

Solving for d' gives:

• d' = 12m

Final Thoughts

Thus, when the magnetic field is doubled, the particle comes to rest at a distance of 12 meters from the entry point. This problem illustrates the interplay between forces acting on a charged particle and how they can be manipulated through magnetic fields.