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 vecF =(Qq)/(r^(2)).hatr where the distance is measured in cm (=10^(-2)m), F in dynes (=10^(-5) N) and the charges in electrostatic units (es units), where 1 es unit of charge 1/(3) xx 10^(-9) C. The number [3] actually arises from the speed of light in vacuum which is now taken to be exactly given by c = 2.99792458 xx 10^8 m/s. An approximate value of c then is c = [3] xx 10^8 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: 1/(4piepsilon_(0)) = 10^(-9)/x^(2) (Nm^(2))/C^(2) with x=1/[3] xx 10^(-9), we have 1/(4pi epsilon_(0)) = [3]^(2) xx 10^(9) (Nm^(2))/C^(2) or 1/(4pi epsilon_(0)) = (2.99792458)^(2) xx 10^(9) (Nm^(2))/C^(2) (exactly).

There is another useful system of units, besides the SI/mKs. A system,

1.	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 vecF =(Qq)/(r^(2)).hatr where the distance is measured in cm (=10^(-2)m), F in dynes (=10^(-5) N) and the charges in electrostatic units (es units), where 1 es unit of charge 1/(3) xx 10^(-9) C. The number [3] actually arises from the speed of light in vacuum which is now taken to be exactly given by c = 2.99792458 xx 10^8 m/s. An approximate value of c then is c = [3] xx 10^8 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: 1/(4piepsilon_(0)) = 10^(-9)/x^(2) (Nm^(2))/C^(2) with x=1/[3] xx 10^(-9), we have 1/(4pi epsilon_(0)) = [3]^(2) xx 10^(9) (Nm^(2))/C^(2) or 1/(4pi epsilon_(0)) = (2.99792458)^(2) xx 10^(9) (Nm^(2))/C^(2) (exactly).
Answer» Solution :(i) `F = (Qq)/r^(2)` `therefore` 1 dyne `=("1 esu charge")^(2)/(1 cm)^(2)` `therefore 1 esu = (1 "dyne")^(1//2) xx 1 cm` `=F^(1//2) L` `therefore` Dimensional formula of 1 esu, `=[M^(1//2)L^(3//2)T^(-1)]` Hence, in dimensional formula of esu charge, power of M is `1/2`and of L is `3/2`, which is non-integer. (II) Suppose 1 esu = xC, where X is a dimensionless number. The force between two CHARGES of 1 esu magnitude is `10^(-5)` N ( = 1 dyne) when they are at distance `10^(-2)` m (= 1 cm). `therefore F = 1/(4pi epsilon_(0)).x^(2)/(10^(-2))^(2)` `= 10^(-9)/x^(2).(Nm^(2))/C^(2)` but `x = 1/(\|3\| xx 10^(9))` `1/(4pi epsilon_(0)) = 10^(-9) xx \|3\|^(2) xx 10^(18) (Nm^(2))/C^(2)` `=9 xx 10^(9) (Nm^(2))/C^(2)` If `\|3\| = 2.99792458`, then `=1/(4pi epsilon_(0)) = 10^(-9) xx (2.99792548)^(2) xx 10^(18)` `=8.98755 xx 10^(9)`

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