B Electric+Magnetic Force Between a & b Charges: Inertial Frames

[Tahmeed]

Let's assume that a and b charges are moving. now in our lab frame there will be a electric+magnetic force whereas in a rest frame of either of the charges, there will be only an electric force.
So, two inertial observers will measure different forces?

 

[Orodruin]

So, two inertial observers will measure different forces?

Relativistically, it is not really accurate to separate the electric and magnetic forces. What appears as an only an electric field in one frame will appear as a magnetic and an electric field in others. The electric and magnetic fields are two sides of the same coin.

Also, the concept of a regular 3-force is not invariant between frames. What you want to look at is the 4-force acting on an object and its corresponding proper acceleration.

 

[vanhees71]

In any inertial reference frame the force on a charged particle is given by Lorentz
$$m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} = K^{\mu}=\frac{q}{c} F^{\mu \nu} \frac{\mathrm{d} x_{\nu}}{\mathrm{d} \tau},$$
i.e., the spatial components are
$$\vec{K}^{\mu} = \gamma q \left (\vec{E}+\frac{\vec{v}}{c} \times \vec{B} \right ),$$
where $\vec{v}=\mathrm{d} \vec{x}/\mathrm{d} t$ is the usual three-velocity in the given inertial reference frame, $\gamma=(1-\vec{v}^2/c^2)^{-1/2}$ the Lorentz factor, $q$ the electric charge of the particle, and $(\vec{E},\vec{B})$ the electromagnetic field.