Q.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 $\chi \sim 10^{-5}$ for many solid materials.

Dimensions of $\chi$. Since $M = \chi H$ and both $M$ and $H$ are measured in $\text{A/m}$, the susceptibility $\chi$ is dimensionless.

Building $\chi$ from atomic parameters. Seek $\chi \sim \mu_0^{a} e^{b} m^{c} v^{d} R^{f}$ that is dimensionless. Using $[\mu_0]=MLT^{-2}I^{-2}$, $[e]=IT$, $[m]=M$, $[v]=LT^{-1}$, $[R]=L$ and demanding all four dimensions cancel gives $b=2a$, $c=-a$, $d=0$, $f=-a$. The velocity power is zero, so $v$ drops out and

Estimate (Bohr atom): with $\mu_0=4\pi\times10^{-7}$, $e=1.6\times10^{-19}\,\text{C}$, $m=9.1\times10^{-31}\,\text{kg}$, $R=5.3\times10^{-11}\,\text{m}$, …

Dimensions of $\chi$

Susceptibility is defined by $\vec{M} = \chi\,\vec{H}$. Magnetisation and magnetising field are both in $\text{A/m}$, so their ratio is a pure number:

Constructing $\chi$ from atomic parameters

Look for $\chi \sim \mu_0^{a} e^{b} m^{c} v^{d} R^{f}$. In terms of $M,L,T,I$:

Requiring the product to be dimensionless gives

From the $I$-equation $b = 2a$; substituting into the $T$-equation gives $d = 0$; then $c = -a$ and $f = -a$. The combination is therefore $\left(\dfrac{\mu_0 e^2}{m R}\right)^{a}$, and the fundamental dimensionless quantity is

Dimension check: $\dfrac{(MLT^{-2}I^{-2})(I^2T^2)}{(M)(L)} = 1.$

Numerical estimate

For the hydrogen atom (Bohr radius $R = a_0$):

• Numerator: $4\pi\times10^{-7} \times 2.56\times10^{-38} \approx 3.22\times10^{-44}$.
• Denominator: $9.1\times10^{-31} \times 5.3\times10^{-11} \approx 4.82\times10^{-41}$. …

Method: Dimensional Analysis to Guess an Unknown Physical Quantity from a Set of Parameters

Use this whenever you're asked to "construct" or "estimate" a quantity (up to a constant) from a given list of physical parameters, without deriving it from first principles.

Steps

Step 1: Find the target quantity's own dimensions first

Before guessing a formula, work out the dimensions of the quantity you're building directly from its definition. Here, susceptibility is defined by $\vec M=\chi\vec H$; since magnetisation $M$ and magnetising field $H$ are both measured in the same unit (A/m), their ratio $\chi$ is dimensionless:

Knowing the target dimension in advance tells you exactly what condition the guessed combination must satisfy.

Step 2: Propose a general power-law combination of the given parameters

Write the unknown quantity as a product of the given parameters, each raised to an unknown power:

Step 3: Express every parameter's dimensions in the base set $M,L,T,I$

Look up (or derive from a known formula) the dimensions of each quantity — e.g. $[\mu_0]=MLT^{-2}I^{-2}$, $[e]=IT$, $[m]=M$, $[v]=LT^{-1}$, $[R]=L$ — and substitute into Step 2's combination.

Step 4: Set up and solve the simultaneous equations

Collect the total power of each base dimension ($M$, $L$, $T$, $I$) in the combined expression, and set each equal to the target dimension's corresponding power found in Step 1 (all zero here, since $\chi$ is dimensionless). This gives one linear equation per base dimension; solve them together for the exponents $a,b,c,d,f$. …