The Basic Forces and Messenger Particles — Question 2

Question 2

The Yukawa force has an additional radial factor

A potential is $V(r)=A e^{-r/\lambda}/r$, with positive A. Define outward radial force $F_r=-dV/dr$.

1. Differentiate V to find the force.

2. Find the force ratio to A/r squared at r=lambda and at r=2lambda.

Question 2 - Solution

The Yukawa force has an additional radial factor

(a) Method and calculation

$$\frac{dV}{dr}=-Ae^{-r/\lambda}\left(\frac1{r^2}+\frac1{\lambda r}\right).$$$$\boxed{F_r=\frac{A}{r^2}(1+r/\lambda)e^{-r/\lambda}}.$$

(b) Interpretation and further calculation

$$\boxed{F(\lambda)/(A/\lambda^2)=2e^{-1}=.735759}.$$$$\boxed{F(2\lambda)/(A/(2\lambda)^2)=3e^{-2}=.406006}.$$

Physical check

The sign reverses for negative A. Screening a potential is not equivalent to multiplying its force by the same exponential alone.