First slide

Dimensions of physical quantities

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?

Moderate

A

$F = \frac{h c}{d^{2}}$
B

$F = \frac{h c}{d^{4}}$
C

$F = \frac{h d^{2}}{c}$
D

$F = \frac{d^{4}}{h c}$

Solution

$\begin{array}{l} F = \frac{Newton}{m^{2}} \\ \begin{aligned} (a) \frac{h c}{d^{2}} & = joule-sec. (\frac{metre}{sec.}) \frac{1}{(metre)^{2}} \\ = \frac{joule}{metre} = Newton \\ (b) \frac{h c}{d^{4}} & = (\frac{h c}{d^{2}}) \frac{1}{d^{2}} = \frac{Newton}{m^{2}} . Hence F = h c / d^{4} \\ (c) \frac{h d^{2}}{c} & = \frac{Joule.sec(metre)^{2}}{(\frac{metre}{\sec})} = {Newton-metre}^{2} \sec^{2} \\ (d) \frac{d^{4}}{h c} & = \frac{1}{(\frac{h c}{d^{4}})} = \frac{m^{2}}{Newton} \end{aligned} \end{array}$

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