I have a doubt related to magnetic field

Question

I have three doubts regarding the magnetic field due to the charge:-

1. Is a magnetic field generated when a charged particle moves with constant velocity or it only generated when the charge particle is accelerated or both?
2. Why and how does movement of charge result in the creation of a magnetic field around the particle?
3. We also know that motion is relative so if we observe in frame of moving charged particle then its magnetic field should not be there, then what we do in this case?
   Note (related to third question): as we use pseudo force in non-inertial frame then what do we use in this case?

Answer 1

1. The magnetic field is induced if the charge's velocity is non-zero. It doesn't matter whether the acceleration is zero or not.
2. As far as I know, there is no explanation for the why question, just a mathematical description of the process (Ampère's law, or, more generally, the Maxwell equations).
3. Yes, in the comoving frame there is no magnetic field, only the electric field. So the magnetic field strength is indeed frame dependent.

Answer 2

1. Both.
2. We have observations of interactions between electric currents. We interpret these interactions using the "magnetic field" abstraction. A moving particle is a special case of this. There is no deeper "why": the phenomena demand certain things of a model, and "magnetic field" is the model that works.
3. The way our model captures this is that if I have a pure electric field in my frame of reference, an observer in motion relative to me sees a mix of magnetic and electric fields. Thus, while a stationary charge in my frame produces no magnetic field, it will produce a magnetic field in a frame moving relative to mine, just as you'd expect from a moving charge.

Answer 3

Your third question is the exact one that led Einstein and others to construct the Special Theory of Relativity. In traditional Maxwellian Electrodynamics, there are magnetic fields, electromotive forces, and other concepts that appear and disappear depending on the frame of motion used. Under Special Relativity, it turns out that a charged particle being motionless has a purely electric field. The same field in a moving frame has its electric field $\vec E$ Lorentz Transformed into an electric field $\vec E$ with a different value and a magnetic field $\vec B$. The $\vec E$ and $\vec B$ fields are motion-dependent components of a quantity called the EM Field Tensor $F_{\alpha \beta}$, which is not motion-dependent.

In the process of establishing this theoretical framework, they also made the discovery that other quantities such as elapsed time between events $t$ and spatial separation between objects $x$ are also motion-dependent. In the case of $t$ and $x$, the invariant quantity is the spacetime interval $s$ or equivalently the proper time $\tau$ of an observer. This ended up being the more Earth-shattering aspect of the theory, vis a vis the scientific community (not to mention everyday people).