Question

What is the dimensional formula for thermal resistance?
\[
  A.{\text{ }}\left[ {{M^{ - 1}}{L^{ - 2}}{T^{ - 1}}K} \right] \\
  B.{\text{ }}\left[ {M{L^2}{T^{ - 2}}{K^{ - 1}}} \right] \\
  C.{\text{ }}\left[ {M{L^{ - 3}}{T^2}{K^{ - 1}}} \right] \\
  D.{\text{ }}\left[ {{M^{ - 1}}{L^{ - 2}}{T^3}K} \right] \\
 \]

Answer 1

Hint: In order to solve this question first we will define the term thermal resistance, further then we will write its formula and with the help of this formula we will evaluate the required answer by manipulating the formula and using the dimensional formula for other units.

Formula used- ${\text{Thermal Resistance}} = \dfrac{{{\text{Temperature Difference}}}}{{{\text{Thermal Current}}}},{\text{Flow rate of heat}} = \dfrac{{\Delta Q}}{{\Delta t}}$

Complete step-by-step solution -

Thermal resistance is the property of heat and is the calculation of temperature variations in which the heat flow is resisted by substance or material.
Thermal resistance is inversely proportional to thermal Conductance.
Dimensional Formula is the representation of physical quantities with the aid of a simple unit in the appropriate dimensions.
As we know the formula for thermal resistance is:
$ \Rightarrow {\text{Thermal Resistance}} = \dfrac{{{\text{Temperature Difference}}}}{{{\text{Thermal Current}}}}$
As we know that the thermal current is the flow rate of heat so the thermal resistance can be written as:
$ \Rightarrow {\text{Thermal Resistance}} = \dfrac{{{\text{Temperature Difference}}}}{{{\text{Flow rate of heat}}}}$------ (1)
Now we know that temperature difference is $\Delta T$ and its dimensional formula is $\left[ K \right]$
And rate of flow of heat is
$ \Rightarrow {\text{Flow rate of heat}} = \dfrac{{\Delta Q}}{{\Delta t}}$----- (2)
From equation (1) and equation (2), we have:
$ \Rightarrow {\text{Thermal Resistance}} = \dfrac{{\Delta {\text{T}}}}{{\dfrac{{\Delta Q}}{{\Delta t}}}}.......(3)$
In the above equation we know that $\Delta Q$ or change in heat is a type of energy so its dimensional formula is: $\left[ {M{L^2}{T^{ - 2}}} \right]$
Also, $\Delta t$ is time so its dimensional formula is $\left[ T \right]$
So from the above values and equation (3), we have:
$
  \because {\text{Thermal Resistance}} = \dfrac{{\Delta {\text{T}}}}{{\left( {\dfrac{{\Delta Q}}{{\Delta t}}} \right)}} \\
  \because {\text{Dimensional formula of Thermal Resistance}} = \\
  \dfrac{{\left[ K \right]}}{{\left( {\dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ T \right]}}} \right)}} \\
 $
Solving the above equation we get:
Dimensional formula for thermal resistance:
$
   = \dfrac{{\left[ K \right]\left[ T \right]}}{{\left[ {M{L^2}{T^{ - 2}}} \right]}} \\
   = \left[ {{M^{ - 1}}{L^{ - 2}}{T^3}K} \right] \\
 $
Hence, the dimensional formula for thermal resistance is \[\left[ {{M^{ - 1}}{L^{ - 2}}{T^3}K} \right]\]
So, the correct answer is option D.

Note- The measure of a physical substance can be represented as a sum of the basic physical dimensions such as length, space, and time, each elevated to a logical power. A physical quantity's dimension is more fundamental than some unit of scale used to express the quantity of that physical amount. The dimensional equations have three uses: to test a physical equation's correctness. To derive the relation between a physical phenomenon involving different physical quantities. Move from one control configuration to another.