Question

If e, h, µ , and G denote electron charge, Plank's constant, permiability, and universal gravitational constant. Then write the product which gives speed of light in terms of dimensions.

Answer 1

e is denoted electron charge .
∴ dimension of e = dimension of { current × time} = [AT]
[∵ dimension of current = [A] and dimension of time = [T]

h is denoted Plank's constant .
∵ E = hν , here E is energy of photons , h is plank's constant and ν is frequency of light .
∴ dimension of h = dimension of { E/v} = dimension of {E}/dimension of v
= [ML²T⁻²]/[T⁻¹] = [ML²T⁻¹]

μ is denoted permeability of medium
we know, F = Bil and B = μidlsinΘ/4πr² [ both equations are General formula ]
after solving we get
dimension of μ = dimension of {F/i²} = dimension of F/dimension of i²
= [MLT⁻²]/[A²] = [MLT⁻²A⁻²]

G is denoted gravitational constant
∵ F = GmM/r² [ Gravitational force ]
∴ dimension of G = dimension of {Fr²/mM} = dimension of Fr²/dimension of {mM}
= [MLT⁻²][L²]/[M²]
= [M⁻¹L³T⁻²]

Now, speed of light in term of all above :
dimension of speed = [LT⁻¹]
Let, [LT⁻¹] = k[AT]^a [ML²T⁻¹]^b [MLT⁻²A⁻²]^c [M⁻¹L³T⁻²]^d
= [M]^(b+c -d) [L]^(2b+c+3d) [T]^(a-b-2c-2d) [A]^(a -2c)
Compare both sides,
b + c - d = 0⇒b + c = d
2b + c + 3d = 1
a - b - 2c - 2d = -1
a - 2c = 0 ⇒ a = 2c

after solving above equations ,
d = 0 , b = 1 , c = -1 and a = -2

Now, dimension of speed of light = h/(e² × μ )