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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. |
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Answer» Solution :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^2dB)/( 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` |
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