If Planck’s constant (h ) and speed of light in vacuum (c ) are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities?

According to plank, the energy of photon is given by the expression,

Where,

f= frequency of light

h= Planck’s constant

Dimension of E=

Dimension of f=

So, dimension of h,

As we can see in the question, we have to relate M, L and T in terms of h and c as fundamental quantities.

There is no charge or ampere dimension so option (c) is eliminated.

We have to check for option a, b and d.

Dimensions of m_e, m_p and G are , and respectively.

For m_e and m_p,

For mass, let

m

By substituting the dimension for each quantity

Now using principal of homogeneity, which states that the dimensions on both sides of a dimensional equation is same, we get

=

Equating like terms and solving above we get,

a+b=1, 2b+c=0, -b-c=0

a=1, b=0, c=0

Substituting the values in equation,

m

m=m_e

Similarly for length,

l

=

Solving we get,

and for time,

This holds true for both m_e and m_p.

And similarly for G,

Also we know,

Dimension of G=

Let,

m ...........(1)

By substituting the dimension for each quantity

Now using principal of homogeneity, which states that the dimensions on both sides of a dimensional equation is same, we get

=

Equating like terms and solving above we get,

a-c=1, 2a+b+3c=0, -a-b-2c=0

a=1/2, b=1/2, c=-1/2

Substituting the values in equation,

m

Similarly for length and time,

and