Effects of Couloumb’s law on electron beam In an electron beam there are about $10^{15}$ electrons. If they are all electrons, then they all have the same charge, so why don't they repel each other? Furthermore, the force would extend to infinity because it is inversely proportional to the distance.
Where is that force?
 A: In a beam of highly relativistic charged particles, their magnetic attraction is almost as large as their electric repulsion. The net repulsive force is reduced by the Lorentz factor $\gamma$ compared to the repulsion in their rest frame, and $\gamma$ can be very large.
A: The simple answer is that the Coulomb force exists, but in the high momentum environment of each individual electron in a particle beam, it plays a very small role in defocusing the beam. I could find this paper in a first search:
>
Focusing ofrelativistic electron (positron) bunches is considered in three different descriptions of cold overdense plasma-rigid electron bunch system. In all three models Coulomb component of field exists but for large values of the bunch Lorentz-factor it is negligible in comparison with the wake field component. Total charge and current densities in general are not compensated. For narrow bunches they are nearly proportional to each other. The resulting focusing force is a complex combination of magnetic and electric forces, whose relative strength depends on bunch parameters
As for the Pauli exclusion principle, it ony applies to bound states, and the electrons in a beam are not in a bound state.
A: Firstly, the exclusion principle does not say that electrons repel each other. It’s states that fermions can not occupy the same quantum state. The state depends on more than just charge and position. It also depends on spin. And, although the consequences of the exclusion principle mean that electrons don’t get too close to each other, they don’t actively repel each other. Also, because of how minuscule the effects of quantum mechanics on a macroscopic scale are, the electrons would not repel each other enough to cause visual change. These factors all are reasons an electron beam can stay together.
