What are the dimensions of `chi`, the magnetic susceptibility? Consider an H-atom. Guess an expression for `chi`, upto a constant by constructing a quantity of dimensions of `chi`, out of parameters of the atom: e, m, v, R and `mu_0`. Here, m is the electronic mass, v is electronic velocity, R is Bohr radius. Estimate the number so obtained and compare with the value of `|chi|10^(-5)` for many solid materials.

Explanation

Magnetic susceptibility of substance `chi_(m) = (M)/( H)`
`chi_(m) = (I)/( I) [ because` Units of M and H are same dimension formula of `chi_(m) [M^(0) L^(0) T^(0) ]`.
From Biot-Savart.s law,
`dB= (mu_0)/( 4pi )( I dl sin theta )/( r^2)`
`therefore mu_0 = (4pir^2 dB)/( Idl sin theta)`
`= (4pi r^2)/( I dl sin theta ) xx (F)/( qv sin theta) [ because dB= (F)/( dv sin theta)]`
The dimensional formula of `mu_0`
`therefore` Dimension of `mu_0 = (L^(2) xx [M^(1) L^(1) T^(-2) ] )/( (QT^(-1) ) (L) (Q) (L^(-1) T^(-1) ) )`
`= M^(1) L^(1) Q^(-2)" "` [Where Q is the dimension of charge]
An `chi` is dimensionless, it should have no involvement of charge Q in its dimension formula. It will be so if `mu_0 and e^(2)` together should have the value `mu_(0) e^(2)` as e has the dimension of charge.
Let `chi= mu_(0) e^(2) m^(a) v^(b) R^(c ) " "...(1)`
where a, b, c are the power of m, v and R respectively, such that relation (1) is satisfied
Dimension equation of (1) is,
`[ M^(0) L^(0) T^(0) Q^(0) ] = [ M^(1) L^(1) Q^(-2) ] [Q^(2) ] [Q^(2) ] [M]^(a) [LT^(-1) ]^(b) [L]^(c )`
`= [M^(1+a)] [L^(1+ b+ c)] [T^(-b) ] [Q^(0)]`
Equating the power of M, L, T
Equating the power of M,
`a+1 =0 therefore a= -1 " "...(2)`
Equating the power of T,
`-b=0 therefore b=0" "...(3)`
Equating the power of Q,
`1+b + c =0`
`therefore c=-1" "...(4)`
Putting value of a, b, c in equation (1)
`chi= mu_(0)e^(2) m^(-1) v^(0) R^(-1)`
`chi = (mu_(0) e^(2) ) /( mR) [ because v^(0) =1]" "...(5)`
Here `mu_0 = 4pi xx 10^(-7) Tm//A`
`e= 1.6 xx 10^(-19) C`
`m=9.1 xx 10^(-31) Kg`
`R= 10^(-10) m`
`therefore chi = ((4pi xx 10^(-7) ) (1.6 xx 10^(-19) )^(2) )/( (9.1 xx 10^(-3) ) (10^(-10) ) ) ~~ 10^(-4)`
`(chi )/( chi_(+) ) = (10^(-4) ) /( 10^(-5)) = 10`