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Question

Answers

$\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{3}}{{\theta }^{1}} \right]$

A. Coefficient of thermal conductivity

B. Coefficient of viscosity

C. Modulus of rigidity

D. Thermal resistance

Answer
Verified

Hint: For each of the options, from their basic formulae, find the dimensional formula for the terms.

Formula used: Formula for Coefficient of thermal conductivity, Coefficient of viscosity, Modulus of rigidity, Thermal resistance:

\[\begin{align}

& \text{coefficient of thermal conductivity}=\dfrac{\text{Rate of heat transfer per unit time}\times \text{distance}}{\text{Change in temperature in Kelvin}\times \text{Area of cross-section}} \\

& \text{coefficient of viscosity}\left( \eta \right)=\dfrac{Fdz}{Adv} \\

& \text{Modulus of rigidity }\left( \mu \right)=\dfrac{\text{stress}}{\text{strain}} \\

& \text{Thermal resistance }(R)=\dfrac{\text{Change in temperature in kelvin}}{\text{Rate of heat transfer per unit time}} \\

\end{align}\]

Complete step-by-step answer:

Every quantity can be expressed in the terms of the following seven dimensions

Dimension Symbol

Length L

Mass M

Time T

Electric charge Q

Luminous intensity C

Temperature $\theta $

Angle None

For option A. Coefficient of thermal conductivity.

Coefficient of thermal conductivity of a material is the rate of flow heat per unit area per unit change in temperature across a solid.

\[\begin{align}

& \text{coefficient of thermal conductivity}=\dfrac{\text{Rate of heat transfer per unit time}\times \text{distance}}{\text{Change in temperature in Kelvin}\times \text{Area of cross-section}} \\

& \Rightarrow k=\dfrac{Qd}{A\left( {{\theta }_{2}}-{{\theta }_{1}} \right)t} \\

\end{align}\]

Therefore, the dimensional formula equals $\left[ {{M}^{1}}{{L}^{1}}{{T}^{-3}}{{\theta }^{-1}} \right]$

Hence, this option is incorrect.

For option B. Coefficient of viscosity.

Coefficient of viscosity is defined as the tangential force required to maintain a unit velocity gradient in the depth of a unit area of a liquid.

$\text{coefficient of viscosity}\left( \eta \right)=\dfrac{Fdz}{Adv}$

Therefore, the dimensional formula equals $\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-1}} \right]$.

For option C. Modulus of rigidity.

Modulus of rigidity is defined as the ratio of shear stress to shear strain.

$\text{Modulus of rigidity }\left( \mu \right)=\dfrac{\text{stress}}{\text{strain}}$

Where, the unit of stress is the same as that pressure and strain is a dimensionless quantity.

Therefore, the dimensional formula equals $\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-2}} \right]$.

For option D. Thermal resistance.

Thermal resistance is defined as the resistance a body provides when heat is transferred. In simple words, it is the ratio of change in temperature to rate of heat transfer.

$\text{Thermal resistance }(R)=\dfrac{\text{Change in temperature in kelvin}}{\text{Rate of heat transfer per unit time}}$

Therefore, the dimensional formula equals $\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{3}}{{\theta }^{1}} \right]$.

Thus, the answer to this question option D. Thermal resistance.

Note: Firstly, taking change for a quantity does not change its dimension because it is essentially the difference of the quantity. Secondly, since there is a dimension of $\theta $ in the question, the term should have a term for temperature. Therefore, option B and C can be neglected.

Formula used: Formula for Coefficient of thermal conductivity, Coefficient of viscosity, Modulus of rigidity, Thermal resistance:

\[\begin{align}

& \text{coefficient of thermal conductivity}=\dfrac{\text{Rate of heat transfer per unit time}\times \text{distance}}{\text{Change in temperature in Kelvin}\times \text{Area of cross-section}} \\

& \text{coefficient of viscosity}\left( \eta \right)=\dfrac{Fdz}{Adv} \\

& \text{Modulus of rigidity }\left( \mu \right)=\dfrac{\text{stress}}{\text{strain}} \\

& \text{Thermal resistance }(R)=\dfrac{\text{Change in temperature in kelvin}}{\text{Rate of heat transfer per unit time}} \\

\end{align}\]

Complete step-by-step answer:

Every quantity can be expressed in the terms of the following seven dimensions

Dimension Symbol

Length L

Mass M

Time T

Electric charge Q

Luminous intensity C

Temperature $\theta $

Angle None

For option A. Coefficient of thermal conductivity.

Coefficient of thermal conductivity of a material is the rate of flow heat per unit area per unit change in temperature across a solid.

\[\begin{align}

& \text{coefficient of thermal conductivity}=\dfrac{\text{Rate of heat transfer per unit time}\times \text{distance}}{\text{Change in temperature in Kelvin}\times \text{Area of cross-section}} \\

& \Rightarrow k=\dfrac{Qd}{A\left( {{\theta }_{2}}-{{\theta }_{1}} \right)t} \\

\end{align}\]

Therefore, the dimensional formula equals $\left[ {{M}^{1}}{{L}^{1}}{{T}^{-3}}{{\theta }^{-1}} \right]$

Hence, this option is incorrect.

For option B. Coefficient of viscosity.

Coefficient of viscosity is defined as the tangential force required to maintain a unit velocity gradient in the depth of a unit area of a liquid.

$\text{coefficient of viscosity}\left( \eta \right)=\dfrac{Fdz}{Adv}$

Therefore, the dimensional formula equals $\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-1}} \right]$.

For option C. Modulus of rigidity.

Modulus of rigidity is defined as the ratio of shear stress to shear strain.

$\text{Modulus of rigidity }\left( \mu \right)=\dfrac{\text{stress}}{\text{strain}}$

Where, the unit of stress is the same as that pressure and strain is a dimensionless quantity.

Therefore, the dimensional formula equals $\left[ {{M}^{1}}{{L}^{-1}}{{T}^{-2}} \right]$.

For option D. Thermal resistance.

Thermal resistance is defined as the resistance a body provides when heat is transferred. In simple words, it is the ratio of change in temperature to rate of heat transfer.

$\text{Thermal resistance }(R)=\dfrac{\text{Change in temperature in kelvin}}{\text{Rate of heat transfer per unit time}}$

Therefore, the dimensional formula equals $\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{3}}{{\theta }^{1}} \right]$.

Thus, the answer to this question option D. Thermal resistance.

Note: Firstly, taking change for a quantity does not change its dimension because it is essentially the difference of the quantity. Secondly, since there is a dimension of $\theta $ in the question, the term should have a term for temperature. Therefore, option B and C can be neglected.

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