# Force on a current carrying conductor and Hall effect

If we consider a thin wire on which flows current, inside a magnetic field, we observe a force $\mathbf{dF}=i\mathbf{ds} \land \mathbf{B}$ on each $\mathbf{ds}$ of the wire. This force is caused by electrons, on which is acting Lorentz force, which bump into the Crystal structure of the metal. However after a while (assuming $\mathbf{B}$ constant and uniform and the wire firm) Lorentz force generates a separation of charges between two opposite side of the wire (Hall effect), and the force on each electron becomes zero, so they should stop bumping (all togheter and all in the same direction) into the Crystal structure of the metal. Then how is possible that we still observe a force on the conductor? Where I am wrong?

## 2 Answers

Then how is possible that we still observe a force on the conductor? Where I am wrong?

The net force due to the Hall electric field acting on the moving electrons in a small volume element of metal indeed does cancel the net force due to external magnetic field on those same moving electrons. But there are other particles in the wire and there are forces acting on them that are not cancelled in this way.

The Hall electric field is electrostatic so we can assume it is produced by stationary charge distribution. Since there cannot be non-zero charge density inside the metal, it is the surface of the wire where there has to be non-zero charge density. This means there are charged particles present on the surface of the wire.

These particles are distributed on the surface in such a way that their net electric force on the conduction electrons inside the metal cancels the external magnetic force on them. But if there is electric force from surface particle acting on the particle inside the wire, there has to be also the corresponding reaction - an electric force due to particle inside acting on the particle on the surface.

So when we consider some element of wire, there is net electric force acting on the charged particles on its surface. Since the surface charges are bound to the wire, this force is experienced also by the rest of the wire (non-conduction part) and since all other forces acting on the particles in the wire are cancelled, this is equal to total force acting on the wire.

Nitpick: The total force acting on the wire in magnetic field is often called magnetic force or incorrectly Lorentz force, because it is, in magnitude and direction, the same as the actual external magnetic force acting on the moving electrons. But now you know the total force that moves the wire is actually result of both external magnetic forces and internal electric forces between the inside and the surface particles. This total force is better called ponderomotive force (force acting on a piece of matter with mass) than the Lorentz force (this is best thought of as magnetic force acting on moving charged particles) or electromotive force (force acting on the conduction electrons that keeps them moving along the wire).

You are right that the force on the electrons is negated by the Hall effect. The short classical explanation is that the Hall-effect electric field also acts on the ionic lattice that makes up the rest of the metal.

In terms of classical electromagnetism, the explanation would go as follows:

When a current is flowing in the metal, there are two parts: the free, delocalised electrons, and the fixed lattice of positive ions through which they move. The electrons are, on average, moving at the drift velocity, while the ions are not able to drift, so only the electrons experience a Lorentz force. This causes them to experience a magnetic force perpendicular to the current, creating a small excess of positive surface charge on one side of the wire (where there are fewer electrons than ions), and a small excess of negative charge on the other (where there are more electrons than ions). This creates an electric field that balances the magnetic forces on the moving electrons, stopping further buildup of charge - the Hall effect.

The electric field does not only act on the electrons, however. It also acts on the positive ions, and because they are positive, the force on them is opposite the direction of the force on the electrons, i.e. in the same direction as the original magnetic forces! So, since the electric field exactly balances the magnetic field, the net force on the wire as a whole is in the same direction and (since the ions have the exactly opposite charge to the electrons) magnitude as the magnetic forces on the electrons.

This subject is discussed in much more detail in this paper, which presents a classical quantitative model for finding the force on the wire from the microscopic forces and the relevant Hall electric field. However, to fully describe these kinds of microscopic interactions, quantum mechanics is necessary. According to the linked paper, more advanced condensed matter textbooks develop the quantum version using the Fermi gas or Bloch functions. The above picture - where the Hall effect is all that is relevant - works in a metal where only one conduction band is present, but in semiconductors without a Hall effect, collisions between charge carriers and the lattice may be more important.