Question

The wavelength associated with a moving particle depends upon $p^{th}$ power of its mass $m,q^{th}$ power of its velocity $v$ and $r^{th}$ power of Planck’s constant $h$. Then the correct set of values of $p,q$ and $r$ is

• A. $p=1,q=-1,r=1$
• B. $p=1,q=-1,r=1$
• C. $p=1,q=1,r=1$
• D. $p=-1,q=-1,r=-1$

Answer 1

The correct option is A $p=1,q=-1,r=1$

Step 1: Write the dimensions of each quantity.
According to question $\lambda =k{m}^p{v}^q{h}^k\dots \left(i\right)$
From the principle of homogeneity, the dimensions of $RHS$ and $RHS$ should be equal.
So
$\left[M^{0}L{T}^0\right]=\left[M{]}^p\right[L{T}^{-1}{]}^q\left[M{L}^2{T}^{-1}{]}^r\text{or}\right[M^{0}L{T}^0]=[M^{p+r}\left]\right[L^{q+2r}\left]\right[T^{-q-r}]$

Step 2: Compare the powers of $M,L$ and $T$. Now, comparing the powers of $M,L$ and $T$, we get
$p+r=0q+2r=1-q-r=0$

After solving $p=-1,q=-1$ and $r=1$ Substituting the values in equation $\left(i\right)$
we get: $\lambda =k\frac{h}{mv}$