Question

The V-I graph for a conductor at temperatures $T_1$ and $T_2$ are shown in the figure, Then ($T_2$ – $T_1$) is proportional to:
A. tan 2θ
B. sin 2θ
C. cos 2θ
D. cot 2θ

Answer 1

For this question we will use the fact that the resistance of a body is proportional to its temperature. We will find the slope of the graph which will be equal to the resistance of the conductor and then solve according to the linear relation of resistance and temperature.

Complete step-by-step answer:
According to Ohm’s law, the slope of the V-I graph will be equal to the resistance on the material at those temperatures. The difference between the resistances at two different temperatures is proportional to the difference in temperatures. Resistance of an object is linearly dependent on temperature.
So, $\cot \theta -\tan \theta \propto {{T}_{2}}-{{T}_{1}}$

& \cot \theta -\tan \theta =\dfrac{1}{\tan \theta }-\tan \theta =\dfrac{1-{{\tan }^{2}}\theta }{\tan \theta }=\dfrac{1-\dfrac{{{\sin }^{2}}\theta }{{{\cos }^{2}}\theta }}{\tan \theta }=\dfrac{\dfrac{{{\cos }^{2}}\theta -{{\sin }^{2}}\theta }{{{\cos }^{2}}\theta }}{\dfrac{\sin \theta }{\cos \theta }}=\dfrac{\cos 2\theta }{{{\cos }^{2}}\theta }\dfrac{\cos \theta }{\sin \theta }=\dfrac{\cos 2\theta }{\cos \theta \sin \theta } \\\ & =\dfrac{\cos 2\theta }{\dfrac{1}{2}\sin 2\theta }=2\cot 2\theta \\\ \end{aligned}$$ So, as we can see here that cotθ - tanθ is proportional to $T_2$ – $T_1$ and cotθ – tanθ can be written as 2cot2θ. So, $T_2$ – $T_1$ is proportional to cot2θ. **So, the correct answer is “Option D”.** **Additional Information:** The resistance of a body usually depends on its dimensions. The resistance of the body is directly proportional to the length along the direction in which current flows and it is inversely proportional to the cross-section area of the body in a plane perpendicular to which the current flows. But the resistance of a body also changes with change in temperature. When the temperature of a body increases, the vibrational motion of atoms of the body increases and due to this, the probability of collision between atoms of the body and flowing electrons increases. **Note:** Take care when doing trigonometric calculations as students can make a mistake there. Remembering trigonometric identities is also important to solve this question. The relation between the resistance and temperature is not exactly linear but higher powers are ignored as their contribution is very low.