Question

The Richardson equation is given by $I = A{T^2}{e^{\dfrac{{ - B}}{{kT}}}}$ . The dimensional formula for $A{B^2}$ is same as that for:
A) $I{T^{ - 2}}$
B) $kT$
C) $I{k^2}$
D) $\dfrac{{I{k^2}}}{T}$

Answer 1

To solve the question, you need to realize that the constant $e$ is dimensionless and thus, there can be a direct relation between $I,A$ and $T$ . Similarly, the term in the power of $e$ is also supposed to be dimensionless since powers cannot have dimensions otherwise the numeric value of the term would mean nothing. Hence, a direct relation between the dimensions of $B,k$ and $T$ can also be developed rather easily. Once you’ve found out the dimensional relation of $A$ and $B$ individually, just find the dimensional formula of the term $A{B^2}$ to reach at the answer.

Complete step by step answer:
We will try to solve the question exactly as explained in the hint section of the solution to the question. For the given equation to be valid, the term as the power of the dimensionless constant $e$ should also be dimensionless, hence, we will find the dimensional formula of $B$ in terms of $k$ and $T$ from there. As for the dimensional formula of $A$ , we have already seen that the constant $e$ is dimensionless, hence, there is a direct relation between the dimensions of $A$ with the dimensions of $I$ and $T$ .
Let us find the dimensions of $A$ first:
Let us talk about the dimensions of the given equation. For it to be valid, the dimensions on both the sides of the equation should be exactly the same. We can see that $e$ is dimensionless, hence, the only terms with dimensions can be written as:
$I = A{T^2}$
From this equation, we can easily find the dimensions of $A$ in terms of $I$ and $T$ as:
$A = \dfrac{I}{{{T^2}}}$
Now, let us find the dimensions of $B$ :
We have already talked about the fact that the powers are always supposed to be dimensionless, hence, if we talk about the term as the power to the constant $e$ , we can safely say that this term should be dimensionless, hence, we can write the equation, in terms of dimensions as:
$\dfrac{B}{{kT}} = 1$
So, we can write the dimensions of $B$ in terms of $k$ and $T$ as:
$B = kT$
Now, the question has asked us about the dimensional formula of $A{B^2}$ . To find this, we simply have to substitute the dimensional relations of $A$ and $B$ as:
$A{B^2} = \left( {\dfrac{I}{{{T^2}}}} \right){\left( {kT} \right)^2}$
Upon solving it, we will get:
$A{B^2} = I{k^2}$

Hence, we can see that the option (C) is the correct option as the value matches with the one that we found out.

Note: Many students try to manipulate the given equation in hopes of finding a direct relation between $A{B^2}$ and the remaining other terms, which would never work in such questions. Also remember that in such questions, any symbol’s dimensions cannot be presumed to be something even if you know the symbols, this would only confuse you and make you lose marks.