Question

the refractive index μ of a transparent medium varies with the wavelength λ of light as μ = A + B/λ(SQUARE) . Where A and B ARE CONSTANT . find the dimensional formula and si unit of A and B

Answer 1

Given:

$\boxed{ \mu = A + \dfrac{B}{ {(\lambda)}^{2}} }$

To find:

Dimensional formula and SI UNIT of A and B ?

Calculation:

• In an equation, the dimensional formula of LHS should be equal to that in the RHS.

• Again, only physical quantities with same dimensional formula can be added.

Using the 1st rule, we can say that:

$\therefore \: \bigg \{\mu \bigg \} = \bigg \{A \bigg \}$

$\implies \: \bigg \{A \bigg \} = \bigg \{\dfrac{ velocity \: of \: light \: in \: air}{velocity \: of \: light \: in \: medium} \bigg \}$

$\implies \: \bigg \{A \bigg \} = \bigg \{\dfrac{L{T}^{ - 1} }{L{T}^{ - 1} } \bigg \}$

$\boxed{ \implies \: \bigg \{A \bigg \} = \bigg \{{ M }^{0} {L}^{0} {T}^{0} \bigg \}}$

Since A is dimension-less , it doesn't have any SI unit .

From the 2nd rule , we can say:

$\therefore \: \bigg \{ \dfrac{B}{ {(\lambda)}^{2}}\bigg \} = \bigg \{A \bigg \}$

$\implies \: \bigg \{ \dfrac{B}{ L^{2}}\bigg \} = \bigg \{ {(MLT)}^{0} \bigg \}$

$\boxed{ \implies \: \bigg \{ B\bigg \} = \bigg \{L^{2} \bigg \}}$

Since B has the dimension of (length)², its SI unit will be metre square (m²).