# [Physics] Magnetic field from different reference frame

electromagnetism

We know if a particle moves under a magnetic field then the force applied on it is given by
F=q(V×B)
Where V is the velocity vector and B is the magnetic field intensity.

Now consider a hypothetical situation that I throw a charged ball in a uniform magnetic field and very naturally it will get deflected under the magnetic force.

Now again this time,say,I am sitting in a car and have the ball in my hand so in my frame the magnetic field should be the same but the velocity of the ball is zero so there should be no magnetic force at all and the ball should not get deflected.
A physical phenomena if I am observing from an outside static frame or from a car frame should not be different,at least upto the extent of viewing a deflection in ball's path.

This is an absurd unrelaistic situation,so the only solution is that the magnetic field should change by some means.
So,does it really happen in practice?

Yes, the electric and magnetic fields change. When you move to a different reference frame with relative speed $\vec v$, the components of the fields which are perpendicular to your new velocity become:

$$\vec E'_\perp = \gamma\left(\vec E_\perp +\vec v \times \vec B\right)$$ $$\vec B'_\perp = \gamma\left(\vec B_\perp -\frac{1}{c^2}\vec v \times \vec E\right)$$ where $$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

The components of the fields which are parallel to your new velocity are unchanged.

As you can see, the electric field gains a component perpendicular to your new velocity which precisely compensates for the fact that the magnetic field does not act on the charged ball which is stationary in your new reference frame.

The result is that both observers ("stationary" and co-moving) would agree$^\dagger$ on the forces acting on the ball, but would disagree on whether those forces were due to the electric or the magnetic fields. In this sense, the electric and magnetic fields aren't separate entities, but are part of a unified electromagnetic field; how much of the force on a particle is due to the electric or magnetic fields depends on one's reference frame.

$^\dagger$Clarification: The observers would not actually agree on the magnititude of the forces because of the factor of $\gamma$ which is present in the new reference frame. This reflects the fact that force and acceleration are not 4-vector quantities - without going into too much detail, the difference can be chalked up to time dilation, and can be neglected if $v/c <<1$.