Why is the Lorentz force always perpendicular to velocity?

[JiuBeixin]

TL;DR

Exploring the relativistic origin of Lorentz force. I'm deriving how a transverse force (from length contraction) and a longitudinal force (from relativistic momentum compensation) combine to stay perpendicular to velocity.

I represent the conductor as two parallel lines: stationary protons and electrons moving at velocity $v$. A test object with a positive charge also moves at velocity $v$ parallel to the electrons.
In frame $S'$ (the rest frame of the object), the object experiences a repulsive force (the origin of the Lorentz force) due to the length contraction of the moving proton line.

If I introduce an additional vertical velocity $u$ to the object in the frame $S$ the rest frame of the conductor), the entire conductor appears to move upward in frame $S'$. Since length contraction only occurs in the direction of relative motion, this vertical shift seemingly shouldn't create any additional charge density imbalance, so there isn't any extra force.

Yet, someone told me that because of a kind of Transformation of Forces, the force in frame $S$ would be vertical, but WHT and HOW it transform?

PS: I am a high school student and not a native English speaker, so please excuse any grammar issuee and awkward phrase :)​

 

[A.T.]

If I introduce an additional vertical velocity u to the object in the frame S the rest frame of the conductor), the entire conductor appears to move upward in frame S′.

Not just upward, but to the right (like it was moving before) and upward. So you have to apply the Lorentz transformation along that diagonal direction.

Since length contraction only occurs in the direction of relative motion, this vertical shift seemingly shouldn't create any additional charge density imbalance, so there isn't any extra force.

The Lorentz transformation is non-linear, so you cannot decompose it into independent dimensions like with Galilean transformations.

 

[TSny]

If I introduce an additional vertical velocity $u$ to the object in the frame $S$ the rest frame of the conductor), the entire conductor appears to move upward in frame $S'$. Since length contraction only occurs in the direction of relative motion, this vertical shift seemingly shouldn't create any additional charge density imbalance, so there isn't any extra force.

Consdider the case where the test charge moves perpendicularly away from the conductor ("wire") in the lab frame $S$ and suppose the charge has no motion parallel to the wire. Let $S'$ be the frame moving with the test charge. In both $S$ and $S'$, there is no imbalance of charge density in the wire. In frame $S$ there is no electric field produced by the conductor, only a magnetic field.

In frame $S'$ there are both an electric field and a magnetic field . See this thread for a discussion of how the electric field arises in $S'$ even though there is no charge imbalance. The electric field in $S'$ is parallel to the wire. Thus, in this frame there is an electric force on the test charge parallel to the conductor and no magnetic force (because the test charge is not moving in this frame).

Frame $S$ also finds a force on the test charge parallel to the wire. But, in this frame the force is interpreted as a magnetic force.

 

[PeterDonis]

perpendicular to velocity.

In general this is not the case. The Lorentz force is $F = q \left( \vec{E} + \vec{v} \times \vec{B} \right)$ (modulo some constant factors that depend on your choice of units). The $\vec{v} \times \vec{B}$ component will always be perpendicular to $\vec{v}$, but the $\vec{E}$ component does not have to be. For a simple example, consider an electron that starts out at rest between two oppositely charged plates, as in a capacitor: in the rest frame of the plates, the EM field is pure $\vec{E}$ and the force on the electron will be parallel to its velocity once it starts moving towards the positively charged plate.

 

[A.T.]

The Lorentz force is $F = q \left( \vec{E} + \vec{v} \times \vec{B} \right)$

The OP might be using a different naming convention:

Some sources refer to the Lorentz force as the sum of both components, while others use the term to refer to the magnetic part alone.