if a charged particle enters a uniform magnetic field with a velocity v,...

the particle should not experience any force according to a person who is in a frame of reference which is moving with the particle, as for him the particle is at rest

according to the book H.C. Verma, for that case an electric field would exert that very force, an electric field which is present due to the magnetic field.......

but then....., how can we describe the motion of that particle, and if it is circular, how can we define the circle's radius, as it depends on its velocity, which is not even there when we've chosen that frame.

To tackle this question, we need to delve into the concepts of electromagnetic forces and reference frames. When a charged particle moves through a magnetic field, it experiences a magnetic force that is perpendicular to both its velocity and the magnetic field direction. However, when we switch to a frame of reference that moves with the particle, things can get a bit tricky. Let's break this down step by step.

Understanding Forces in Different Frames

In a stationary frame, the force on a charged particle moving with velocity v in a magnetic field B is given by the Lorentz force equation:

• F = q(v × B)

Here, q is the charge of the particle, and × denotes the cross product. This force is responsible for the circular motion of the particle in the magnetic field.

Switching Frames of Reference

Now, when you switch to a frame of reference that moves with the particle, it appears at rest. In this frame, the magnetic field still exists, but the particle does not experience a magnetic force because its velocity is zero. However, the magnetic field can induce an electric field due to the motion of the charged particle through the magnetic field. This induced electric field can exert a force on the particle.

According to the theory of electromagnetism, the relationship between electric and magnetic fields is described by Maxwell's equations. When a charged particle moves through a magnetic field, it can create an electric field in a co-moving frame. This electric field can be calculated using the relation:

• E = -v × B

In this case, the electric field E will exert a force on the particle, which can be described by:

• F = qE

Describing the Motion

Even though the particle is at rest in the moving frame, the induced electric field will still exert a force on it. This force will cause the particle to accelerate, and if we consider the effects of the magnetic field, the particle will still follow a circular path. The radius of this circular motion can be determined using the centripetal force equation.

Calculating the Radius of Circular Motion

The radius r of the circular path in the magnetic field can be derived from the balance of forces. The magnetic force provides the necessary centripetal force for circular motion:

• F = mv²/r

Setting the magnetic force equal to the centripetal force gives:

• qvB = mv²/r

Rearranging this equation allows us to solve for the radius:

• r = mv/(qB)

Here, m is the mass of the particle, v is its velocity, q is its charge, and B is the magnetic field strength. This equation shows that the radius of the circular path depends on the velocity of the particle, which is a crucial point.

Conclusion

In summary, even in a frame where the particle is at rest, the presence of an induced electric field due to the magnetic field allows us to describe the motion of the particle. The particle will still experience a force and can move in a circular path. The radius of that path can be calculated using the relationship between the particle's mass, charge, velocity, and the magnetic field strength. This interplay between electric and magnetic fields is a fundamental aspect of electromagnetism and illustrates the beauty of physics in different reference frames.