Question

The ratio of coefficient of thermal conductivity of two different materials is 5:3. If the thermal resistance of rods of the same area of these materials is the same, then what is the ratio of the length of these rods?
A. 3:5
B. 5:3
C. 25:9
D. 9:25

Answer 1

Hint: Recall that thermal conductivity quantifies the ease with which a material conducts heat through its volume, when there is a temperature gradient across its ends. We know that conductivity is the inverse of resistivity (which depends on the resistance, length and area of the conductor). Using this, determine the relationship between conductivity and the length of the conductor, which will subsequently lead you to the required ratio.

Formula used:
Thermal conductivity: $k={\displaystyle \frac{l}{R_{\theta }A}}$.

Complete step-by-step answer:
Let us begin by establishing what thermal conductivity means and how it can be quantified numerically.
Thermal conductivity of a material is a measure of its ability to conduct heat. This is in accordance with the second law of thermodynamics which suggests that heat flows from a hot system to a cold system in an attempt to equalize the temperature difference between the two. The extent of this
Now, thermal resistance is the impedance offered by the conducting material to the flow of heat (flux) across it as a result of a temperature gradient across its ends. Thermal resistivity is the thermal resistance of a conductor across a unit cross sectional area and unit thickness of the conductor and is an inherent property of the material of the conductor. Thermal conductivity is a measure of the ease with which heat flows through a conductor subjected to a temperature gradient and is given as the inverse of thermal resistivity, i.e.,
If $R_{\theta }K.W^{-1}$ is the thermal resistance at temperature $\theta$, then thermal resistivity
$r={\displaystyle \frac{R_{\theta }A}{l}}K.m.W^{-1}$, where A is the cross sectional area perpendicular to the path of heat flow, and l is the thickness (length) of the conductor parallel to the heat flow.
Then thermal conductivity is given as: $k={\displaystyle \frac{l}{R_{\theta }A}}$.
Now, we are given that the ratio of thermal conductivity of two different materials is 5:3.
We are required to find the ratio of the lengths of these two rods.
We are given that $R_1=R_2$ and $A_1=A_2$.
This means that, from the expression for conductivity, we will have $k\propto l$
${\displaystyle \frac{k_1}{k_2}}={\displaystyle \frac{l_1}{l_2}}$
$\Rightarrow {\displaystyle \frac{5}{3}}={\displaystyle \frac{l_1}{l_2}}$
$\Rightarrow l_1:l_2=5:3$

So, the correct answer is “Option B”.

Note: This whole concept of thermal conductivity can also be understood analogously with electrical conductivity. We know that electrical resistance is the impedance offered by the material of the wire to the flow of current. Resistivity is the electrical resistance of a conductor of a unit cross sectional area and unit length and is a characteristic property of each material. Conductivity is a measure of the ease with current passes through the conductor and is given as the inverse of resistivity, i.e.,
If $R\mathrm{\Omega }$ is the electrical resistance, then resistivity
$\rho ={\displaystyle \frac{RA}{l}}\mathrm{\Omega }m$, where A is the cross-sectional area and l is the length of the conductor.
Then conductivity is given as $\sigma ={\displaystyle \frac{1}{\rho }}{\mathrm{\Omega }}^{-1}{m}^{-1}$.