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There is another useful system of units, besides the SI/mks A system, called the cgs (centimeter-gram-second) system. In this system Coloumb’s law is given by

F=Qq/r2 vector r

where the distance r is measured in cm (= 10–2 m), F in dynes (=10–5 N) and the charges in electrostatic units (es units), where

1es unit of charge=1/[3]x10-9C

The number [3] actually arises from the speed of light in vaccum which is now taken to be exactly given by c = 2.99792458 × 108m/s. An approximate value of c then is c = [3] × 108 m/s.

(i) Show that the coloumb law in cgs units yields

1 esu of charge = 1 (dyne)1/2 cm.

Obtain the dimensions of units of charge in terms of mass M, length L and time T. Show that it is given in terms of fractional powers of M and L.

(ii) Write 1 esu of charge = x C, where x is a dimensionless number. Show that this gives

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Or,

1 esu of charge = 1 (dyne)1/2 (cm)

Hence, [1 esu of charge] = [F]1/2 L = [MLT–2]1/2 L = M1/2 L3/2 T–1

[1 esu of charge] = M1/2 L3/2 T–1

Thus charge in cgs unit is expressed as fractional powers (1/2) of M and (3/2) of L.

(ii) Consider the coloumb force on two charges, each of magnitude

1 esu of charge separated by a distance of 1 cm:

The force is then 1 dyne = 10–5 N.

This situation is equivalent to two charges of magnitude x C separated by 10–2m.

This gives:

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