What Produces a Magnetic Field? — Question 1

Question 1

Electrical neutrality does not determine whether a moving mixture carries current

Two overlapping uniform particle streams have number density n each and charges +e and -e. In case A both move with velocity $v\hat x$; in case B the positive stream moves with $v\hat x$ and the negative stream with $-v\hat x$. Ignore their intrinsic moments.

1. Find charge density and current density in each case.

2. Explain which case can act as a steady current source of magnetic field despite being electrically neutral.

Question 1 - Solution

Electrical neutrality does not determine whether a moving mixture carries current

(a) Method and calculation

$$\rho_A=\rho_B=ne-ne=0.$$$$\vec J_A=nev\hat x-nev\hat x=0,\quad\vec J_B=nev\hat x+nev\hat x=2nev\hat x.$$

(b) Interpretation and further calculation

$$I_A=0,\quad I_B=2nevA\quad\text{through a perpendicular area }A.$$$$\rho=0\ \not\Rightarrow\ \vec J=0.$$Case B has nonzero conventional current; neutrality alone does not remove its magnetic effect.

Physical check

Charge density and current density are different source properties: one counts charge, the other weights charge by velocity.