Question

Experiment shows that two perfectly neutral parallel metal plates separated by a small distance $d$, attract each other via a very weak force, known as the Casimir force. The force per unit area of the plates, $F,$ depends only on the Planck constant $h$, on the speed of light $c$, and on $d$. Which of the following has the best chance of being correct for $F$

• A. $F=\frac{hc}{d^{2}}$
• B. $F=\frac{hc}{d^{4}}$
• C. $F=\frac{hd^{2}}{c}$
• D. $F=\frac{d^{4}}{hc}$

Answer 1

Answer: (B)

$F=\frac{newton}{m^{2}}$
$\frac{hc}{d^{2}}=$ joule-s.$\left(\frac{\text{metre}}{s.}\right)\frac{1}{\left(\text{metre}\right)^{2}}$
$=\frac{\text{joule}}{\text{metre}}=$newton
$\frac{hc}{d^{4}}=\left(\frac{hc}{d^{2}}\right)\frac{1}{d^{2}}=\frac{\text{newton}}{m^{2}}$. Hence $F=hc/ d^{4}$
$\frac{hd}{c}=\frac{\text{joule.s}\left(\text{metre}\right)^{2}}{\left(\frac{\text{metre}}{s.}\right)}=$newton-metre$^{2}s^{2}$
$\frac{d^{4}}{hc}=\frac{1}{\left(\frac{hc}{d^{4}}\right)}=\frac{m^{2}}{\text{newton}}$