Question

\[{{M}^{-1}}{{L}^{-2}}{{T}^{3}}{{\theta }^{1}}\] are the dimensions of
A. Coefficient of thermal conductivity
B. Coefficient of viscosity
C. Modulus of rigidity
D. Thermal resistance

Answer 1

Hint: Through dimensional analysis we can find the dimension of the given physical quantity and can change it unit by keeping its dimension same and similarly we can find the physical quantity from its dimension. The dimension for mass is M, for length is L, for time is T and for temperature it is θ.

Complete step-by-step answer:
Thermal resistance is the resistance which opposes the flow of heat in the conductor and it is measured in Kelvin per watt.
Hence we can write the unit of thermal resistance as
\[\begin{align}
  & \Rightarrow \dfrac{K}{W} \\
 & \Rightarrow \dfrac{K}{J{{s}^{-1}}} \\
 & \Rightarrow \dfrac{K}{Nm{{s}^{-1}}} \\
 & \Rightarrow \dfrac{K}{(kgm{{s}^{-2}})m{{s}^{-1}}} \\
 & \Rightarrow \dfrac{K}{kg{{m}^{2}}{{s}^{-3}}} \\
\end{align}\]
Here K is kelvin, W is watt, J is joules, N is newton, m is meters, kg is kilograms and s is seconds.
Therefore dimension of thermal resistance is given as
\[\begin{align}
  & \Rightarrow \dfrac{{{\theta }^{1}}}{{{M}^{1}}{{L}^{2}}{{T}^{^{-3}}}} \\
 & \Rightarrow {{M}^{-1}}{{L}^{-2}}{{T}^{3}}{{\theta }^{1}} \\
\end{align}\]

So, the correct answer is “Option D”.

Additional Information: Dimensions of the physical quantity is very important because through this we can change the unit conventionally and even can find its dimension. According to the rule of dimensions, only to physical quantity having same dimensions can be added or subtracted and so the dimensions on both the sides of the equation should be same
The unit of coefficient of thermal conductivity is watt per meter kelvin and so the dimension of the quantity is given \[{{M}^{1}}{{L}^{2}}{{T}^{-3}}{{\theta }^{-1}}\]
The unit of coefficient of viscosity is Newton second per \[\text{mete}{{\text{r}}^{2}}\] and the dimension is given by \[{{M}^{1}}{{L}^{-1}}{{T}^{-1}}\].
The unit of modulus of rigidity is Pascal and its dimension is \[{{M}^{1}}{{L}^{-1}}{{T}^{-2}}\].

Note: Thermal resistance can be mistaken as thermal resistivity which is also known as specific thermal resistance. Unit of thermal resistivity is also different, which is kelvin meter per watt, inverse of coefficient of thermal conductivity or just thermal conductivity.