Question

A, B, C, and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation $AD = C\ln (BD)$ holds true. Then which of the combination is not a meaningful quantity
A. $\dfrac{C}{{BD}} - \dfrac{{A{D^2}}}{C}$
B. ${A^2} - {B^2}{C^2}$
C. $\dfrac{A}{B} - C$
D. $\dfrac{{A - C}}{D}$

Answer 1

Hint: Dimensions play a key role in physics. Various quantities have various dimensions. Dimensions of the main fundamental quantities are M for mass while L for length, T for time, K for temperature, while A for the current. Dimensional analysis is useful in solving various problems and checking for the validity of the options. We use one of the properties of dimensions here to solve this question.

Complete step-by-step solution:
There are some rules which we use in case of problems involving dimension usage. Like in the case of any formula, dimensions of the left-hand part of the equation must be the same as dimensions of the right-hand part of the equation. We can add or subtract quantities of the same dimensions. Logarithm function doesn’t have any dimensions and all trigonometric functions don’t have any dimensions
and exponential functions also don’t have any dimensions.
If we have a product of two physical quantities and the dimension of one quantity is inverse of the dimension of the other quantity then that product will be dimensionless.
We use all these rules in dimensions to solve this.
The equation we have is
$AD = C\ln (BD)$
We know that the logarithm function should not have any dimension. So BD must not have any dimension.
That means the dimension of AD is the same as the dimension of C.
We can add or subtract only physical quantities of the same dimensions. If we do $A - C$ That means A and C have the same dimensions.
But from the point, we discussed the product of A and D i.e AD and C has the same dimensions, and moreover, D is not dimensionless. Which states that A and C don't have the same dimensions. Hence $A - C$ is not valid
Hence $\dfrac{{A - C}}{D}$ is not a meaningful quantity.
Hence option D will be the correct answer.

Note: There are some quantities which have units while don't have dimensions. Few instances for that are plane angle which has a unit or radian but no dimensions and solid angle which has a unit of steradian and no dimension and angular displacement also has the unit of radian but no dimension.