When a charged particle is moving with a velocity relative to an observer in frame S, the observer will measure the electric and magnetic fields produced by the moving charge. However, when the same moving charge is observed by an observer in a different reference frame S', which is moving with a relative velocity with respect to the first frame, the measured electric and magnetic fields will be different due to the effects of relativistic motion.

Suppose that a charge q is moving uniformly with velocity v along x axis with respect to frame S.

Consider a frame of reference fixed on moving charge. Consider a point P in y axis of S frame so that OP = b.

Since S frame appears to be moving along x axis, y and z coordinates doesn't change the coordinate that changes is only x coordinate

As S frame appears to be moving along x axis with respect to S'

The instantaneous distance from O' is

r' ² = OP² + OO'²

The electric field produced at P due to the charge q is

E' = q/ r²

which is along O'P.

The electric field has only x and y components.

In △ OPO'

cos ( π- θ ) = - vt' / r'

To transform t' to t, consider L- transformation

t' = γ ( t- vx/c² )

For P, x=0

t'= γt

Hence,

This gives the magnitude of the electric field generated by a charged particle moving at a constant velocity, taking into account the relativistic effects of the particle's motion.

The expression for the electric field generated by a charged particle moving at a constant velocity has several advantages. Some are as follow:

Understanding the behaviour of charged particles: The expression can help us understand how charged particles behave in electric fields. For example, we can use it to calculate the electric field strength at any point in space, and determine how the field changes as the particle moves.
Designing particle accelerators: The expression can be useful in designing particle accelerators, which use electric fields to accelerate charged particles to high speeds. By knowing the strength of the electric field required to accelerate a particle to a certain velocity, we can design accelerators that meet specific energy requirements.
Studying cosmic rays: Cosmic rays are high-energy particles that originate from outside the solar system. By studying the electric fields generated by cosmic rays, we can gain insight into the physical processes that generate them.
Understanding the principles of electromagnetism: The expression is derived from the principles of electromagnetism, which is a fundamental branch of physics. By studying the expression and its derivation, we can gain a deeper understanding of the underlying principles of electromagnetism.

Overall, the expression for the electric field generated by a charged particle moving at a constant velocity is a useful tool for understanding and studying the behaviour of charged particles in electric fields, and has applications in a variety of fields, including particle physics, astrophysics, and engineering.

This note is a part of the Physics Repository.

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