Units and Measurements · Question 5

A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space 0, Planck's constant h, and speed of light c. If the dimensionless quantity is written as e 0 h c and n is a non-zero integer, then ( , , , ) is given by

1. (2n,-n,-n,-n)
2. (n,-n,-2n,-n)
3. (n,-n,-n,-2n)
4. (2n,-n,-2n,-2n)

Answer

(1) (2n,-n,-n,-n)

Solution

Key idea: the fine-structure constant f= e24 0 hc/(2 ) is dimensionless, so e2 0 hc is dimensionless.

Check: e2 0 has dimensions of (force) (length)2 = energy length, and hc is also energy length.

Any power of a dimensionless quantity is dimensionless: ( e2 0 hc )n=e2n 0-nh-nc-n.

So ( , , , )=(2n,-n,-n,-n).

Answer: (A)