Name: Magnets and Magnetism
Uploaded: 2019-07-19
Duration: 3 min 46 s
Description: In this video, we will learn about the different types of magnets and the magnetic field created by different types of magnets. Learn more.

Question

What are the dimensions of

χ

, the magnetic susceptibility? Consider an H-atom. Guess 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

μ_{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

| χ | ~ 10^{- 5}

for many solid materials.

Accepted Answer

Since, $χ = \frac{M}{H}$ , here, magnetisation $(M)$ and magnetising field $(H)$ are of same unit.

So, $χ$ is dimensionless.

$H$ couples to electrons through its charge $e$ . The effect on $m$ is via current $v$ which involves another factor of $R$ . The combination $| χ | ~ 10^{- 5}$ does not depend on the charge.

Expression of $χ$ is $χ = μ_{0} e^{2} m^{α} v^{β} R^{γ}$

Replacing vby their dimension in RHS we get,

$[M^{0} L^{0} T^{0} Q^{0}] = [M^{1} L^{1} Q^{- 2}] [Q^{2}] {[M^{1}]}^{α} {[L T^{- 1}]}^{β} {[L]}^{γ}$

Equating power of $[M]$ : $0 = 1 + α \Rightarrow α = - 1$

Equating powers of $[T]$ : $0 = - β \Rightarrow β = 0$

Equating power of $[L]$ : $0 = 1 + β + γ \Rightarrow γ = - 1$

So, $α = - 1, β = 0, γ = - 1$

Since, $χ = \frac{μ_{0} e^{2}}{m R} = \frac{(4 π \times 10^{- 7} T m A^{- 1}) {(1.6 \times 10^{- 19} C)}^{2}}{(9.1 \times 10^{- 31 k g}) (10^{- 10} m)}$

$χ = \frac{4 \times 3.14 \times 1.6}{9.1} 10^{- 7 - 19 - 19 + 31 + 10} = 3.5 \times 10^{- 4}$

For a given material, $|χ'| ~ 10^{- 5}$

So, $\frac{χ}{|χ'|} = \frac{3.5 \times 10^{- 4}}{10^{- 5}} = 35 \Rightarrow χ = 35 |χ'|$

Hence, it will be $35$ times of $|χ| ~ 10^{- 5}$ .

NCERT Solutions for Chapter: Magnetism and Matter, Exercise 4: LA

NCERT Physics Solutions for Exercise - NCERT Solutions for Chapter: Magnetism and Matter, Exercise 4: LA

Questions from NCERT Solutions for Chapter: Magnetism and Matter, Exercise 4: LA with Hints & Solutions

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