What are dimensions of <math> <mi mathvariant="bold-italic">χ</mi> </math> , the magnetic susceptibility ? consider an hydrogen atom. Gives an expression for χ, upto a constant by constructing a quantity of dimensions of χ, out of parameters of the atom e,m,v,R and μ0?. 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 ?χ?&sim;10−5 for many solid materials.

This is a Long Answer Type Questions as classified in NCERT ExemplarExplanation- as is dimensionless quantity , it should have no involvement of charge Qin its dimensional formulaLet χ = μ 0 e 2 m a v b R c [ M 0 L 0 T 0 Q 0 ]= [ML Q - 2 ] × Q 2 M a × L T - 1 b L c = [ M 1 + a + L 1 + b + c T - b Q 0 ]After solving we get a=-1 , b= 0, c =-1Putting these values in equations χ = μ o e 2 m - 1 v 2 R - 1 = μ 0 e 2 m R By using their standard values we get χ =10 which proves it is dimensionless quantity

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Magnetism and Matter Physics Ncert Solutions Class 12th Physics Class 12th Physics NCERT Exemplar Solutions Class 12th Chapter Five

What are dimensions of $χ$ , the magnetic susceptibility ? consider an hydrogen atom. Gives an expression for χ, upto a constant by constructing a quantity of dimensions of χ, out of parameters of the atom e,m,v,R and μ0?. 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 ?χ?∼10−5 for many solid materials.

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alok kumar singh | Contributor-Level 10

4 months ago

This is a Long Answer Type Questions as classified in NCERT Exemplar

Explanation- as is dimensionless quantity , it should have no involvement of charge Qin its dimensional formula

Let $χ = μ_{0} e^{2} m^{a} v^{b} R^{c}$

[ $M^{0} L^{0} T^{0} Q^{0}$ ]= [ML $Q^{- 2}$ ] $\times [Q^{2}] [M^{a}] \times {[L T^{- 1}]}^{b} {[L]}^{c}$

= [ $M^{1 + a} + L^{1 + b + c} T^{- b} Q^{0}$ ]

After solving we get a=-1 , b= 0, c =-1

Putting these values in equations

$χ = μ_{o} e^{2} m^{- 1} v^{2} R^{- 1} = \frac{μ_{0} e^{2}}{m R}$

By using their standard values we get $χ$ =10 which proves it is dimensionless quantity

This is a Long Answer Type Questions as classified in NCERT ExemplarExplanation- as  is dimensionless quantity , it should have no involvement of charge Qin its dimensional formulaLet <math> <mi>χ</mi> <mo>=</mo> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>b</mi> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> </msup> </math> [ <math> <msup> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>Q</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msup> </math> ]= [ML <math> <msup> <mrow> <mrow> <mi>Q</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </mrow> </msup> </math> ] <math> <mo>×</mo> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <msup> <mrow> <mrow> <mi>Q</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfenced> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <msup> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> </msup> </mrow> </mrow> </mfenced> <mo>×</mo> <msup> <mrow> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mi>L</mi> <msup> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mi>b</mi> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> </msup> </math>    = [ <math> <msup> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>a</mi> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mi>b</mi> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>Q</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msup> </math> ]After solving we get a=-1 , b= 0, c =-1Putting these values in equations <math> <mi>χ</mi> <mo>=</mo> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>o</mi> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> <msup> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>m</mi> <mi>R</mi> </mrow> </mrow> </mfrac> </math> By using their standard values we get <math> <mi>χ</mi> </math> =10 which proves it is dimensionless quantity

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