Suppose a charged particle is moving with velocity v. It will thus constitute a current and produce magnetic field.

Suppose you are moving with velocity v in the direction same as that of the charged partice. So the charge will appear stationary and so wont produce any magnetic field. But it is not stationary w.r.t. ground. Hence an observer on the ground will say that it will produce magnetic field.

Explain this paradox.........

To understand this scenario, we need to delve into the principles of electromagnetism and the concept of reference frames. The situation you've described involves a charged particle moving with a certain velocity and how its motion is perceived differently by observers in different frames of reference. Let's break this down step by step.

The Basics of Charge and Current

A charged particle, such as an electron, moving with a velocity \( v \) constitutes an electric current. According to Ampère's law, a current produces a magnetic field around it. The strength and direction of this magnetic field depend on the velocity of the charge and its position.

Reference Frames and Observers

In physics, a reference frame is essentially a perspective from which an observer measures and observes phenomena. When you move alongside the charged particle at the same velocity \( v \), you enter a frame of reference where the particle appears stationary. In this frame, it seems like there is no current, and thus, no magnetic field is produced.

Ground Observer's Perspective

Now, consider an observer on the ground who is stationary relative to the charged particle's original motion. From this observer's perspective, the charged particle is indeed moving and producing a magnetic field due to its velocity. This creates a paradox: how can the same particle produce a magnetic field in one frame and not in another?

Resolving the Paradox

The key to resolving this paradox lies in understanding the principles of electromagnetism and the effects of relative motion. Here are some important points to consider:

• Magnetic Fields and Motion: The magnetic field produced by a moving charge is dependent on the velocity of the charge relative to the observer. In the ground observer's frame, the charge is moving, hence it generates a magnetic field.
• Relativity of Simultaneity: According to Einstein's theory of relativity, events that are simultaneous in one frame may not be simultaneous in another. This means that the perception of the magnetic field can differ based on the observer's motion.
• Transformation of Fields: When switching between reference frames, electric and magnetic fields transform according to specific rules. The Lorentz transformations show how electric fields can appear as magnetic fields when viewed from different velocities.

Example for Clarity

Imagine you are on a train moving at a constant speed alongside a charged particle. To you, the particle is at rest, and you might think it has no magnetic field. However, if you look out the window and see a stationary observer on the platform, they would see the particle moving and generating a magnetic field. This difference arises because the magnetic field is not an absolute quantity; it depends on the relative motion of the observer and the charge.

Conclusion

In essence, the apparent paradox is resolved by recognizing that magnetic fields are not absolute but are relative to the observer's frame of reference. The charged particle does produce a magnetic field when viewed from the ground, while it appears to be stationary and thus not producing a magnetic field from your moving perspective. This interplay of motion and electromagnetic fields is a fundamental aspect of modern physics, illustrating the beauty and complexity of our universe.