# Is momentum of electrons conserved when wires form a junction?

Assume constant current flows through the wires. I feel that at the junction, electrons suddenly gain or lose velocity. This is a result of $$I=n_e\cdot e\cdot A \cdot v_d$$ with $$v_d$$ (drift velocity) and that the current is a scalar quantity.

I want to know what is the reason behind this, or where does the force which changes direction and magnitude of drift velocity come from, since battery's emf only accelerates them along the direction of wire, and thus whenever wire is curved or junction is present, I encounter this problem.

Think of pipes of varying diameter (resistance) with constant mass flow rate. Here the flow is faster for smaller diameter and vice versa. This is an effect of the continuity equation. So in some sense the velocity is faster in some places because momentarily there would be accumulation of charges over there. These accumulations would push charges away faster. So in the steady state there is an equilibrium.

• Is this accumulation just explained qualitatively or we also have some models to measure the stuff like charge density due to accumulation etc? – AbsoluteZero Mar 10 '20 at 5:18

In a perfectly ideal wire, you'll only see the drift velocity effects you describe. However, in any real wire, there are many more interactions going on. In particular there are interactions between the atoms of the wire and the electrons, and any themodynamic interactions therein. This provides ways for the electrons to affect the total momentum of the wire.

Indeed, if we look at ion thrusters, this ability to transfer momentum to other atoms (positively charged ions, in this case) is fundamental to how they propel a craft forward.

Suppose there is no electric field. Then, the net movement of the electron must be zero since the probability of it moving at any direction is equal. (No one direction is favored). Now we apply an electric field, and that electric field will always be parallel to the wire. The electrons will be accelerated opposite to the electric field vector but they will collide with obstacles, reducing their speeds back to 0. So this time, there is random thermal motion but it is not 0 because it is influenced by an electric field.

As the wire bends, say from vertical(down to top) to horizontal(right to left), the electron will continue exhibiting its random motion, except this time, the electric field is pointing along the wire to the left side rather than to the top side. This new electric field that is pointing to the left, will influence the electron to move to the left, gaining a new velocity in that direction. So to answer your question, yes momentum is conserved.

• Actually NO, momentum is not conserved here obviously since direction of velocity of electrons changes. There is some charge accumulating on surface of wire phenomenon that I just learned which causes this but I do not know how or why – AbsoluteZero Mar 9 '20 at 17:47
• I’m talking about collisions with the lattice ions since that’s the extent of my knowledge (the drude model) In those, when an electron collides with the lattice ion, the electron loses ALL of its speed and the lattice ions have more vibration.(transfer of momentum from electron to lattice ion). Realistically, the electron’s position isn’t even definite. It can only be described using a wave function. – Bandoo Mar 9 '20 at 18:49
• Please understand that the electron is in random thermal motion, and in the drude model when it does collide momentum is conserved. – Bandoo Mar 9 '20 at 18:53