Charge Is Conserved — Question 5

Question 5

Annihilation changes particle counts while preserving total electric charge

An isolated model collection initially contains eight electrons and three positrons. Each electron has charge $-e$, each positron $+e$. Three electron–positron pairs annihilate, with each pair producing two neutral photons. Assume no other charged particles and no particles crossing the system boundary.

1. Find initial and final total charge, and the remaining charged-particle and photon counts.

2. For a general initial pair of counts $N_-,N_+$ and $a$ annihilations, prove the total charge is independent of $a$. State the allowed range of $a$.

Question 5 - Solution

Annihilation changes particle counts while preserving total electric charge

(a) Method and calculation

$$Q_i=(-8+3)e=-5e.$$$$N_{-,f}=8-3=5,\quad N_{+,f}=3-3=0,\quad N_{\gamma,f}=2(3)=6.$$$$Q_f=-5e+6(0)=-5e.$$

(b) Interpretation and further calculation

$$Q_f=-(N_--a)e+(N_+-a)e=(N_+-N_-)e=Q_i.$$$$0\le a\le\min(N_-,N_+),\qquad a\in\mathbb Z.$$Particle number is not the conserved quantity here. Equal positive and negative charges disappear together, leaving their signed sum unchanged.

Physical check

The neutral photons carry energy and momentum even though they contribute zero electric charge.