
The resistance of a wire at temperatures \[{t^ \circ }C\] and \[{0^ \circ }C\] are related by
A. ${R_t} = {R_0}\left( {1 + \alpha t} \right)$
B. ${R_t} = {R_0}\left( {1 - \alpha t} \right)$
C. ${R_t} = {R_0}^2\left( {1 + \alpha t} \right)$
D. ${R_t} = {R_0}^2\left( {1 - \alpha t} \right)$
Answer
164.1k+ views
Hint: To get an expression relating the resistance at two different temperatures we use the relationship between resistance and temperature. The change in resistance with temperature is directly proportional to the resistance at that temperature and the constant of proportionality obtained is known as temperature coefficient of resistance.
Complete step by step solution:
The resistance is related to temperature as the rate of change of resistance with change in temperature is directly proportional to the resistance at that temperature and the proportionality constant is known as the temperature coefficient of resistance denoted by the symbol $\alpha $. That is,
$\dfrac{{dR}}{{dT}} = \alpha R$
If the change in resistance with temperature is small then this relation can be written as,
$\dfrac{{\vartriangle R}}{{\vartriangle T}} = \alpha R$
Given two temperatures are ${0^ \circ }C\,\& \,{t^ \circ }C$. Here, the change in temperature, that is, $\vartriangle T$ is ${\left( {t - 0} \right)^ \circ }C$. So the equation becomes,
$\vartriangle R = \alpha {R_0}\left( {t - 0} \right) \\ $
That is,
$\vartriangle R = \alpha {R_0}t \\ $ ...(1)
Also,
$\vartriangle R = {R_t} - {R_0} \\ $
This implies that,
${R_t} = {R_0} + \vartriangle R \\ $ ...(2)
Substituting equation (1) in equation (2), we get,
\[{R_t} = {R_0} + \alpha {R_0}t \\ \]
Taking ${R_0}$ common on right hand side of the equation, we get,
\[{R_t} = {R_0}\left( {1 + \alpha t} \right)\]
Hence, option A is the correct answer.
Note: Consider a basic model of resistance to better understand the temperature dependency. Atoms and molecules in a conductor obstruct the flow of electrons. It becomes more difficult for the electrons to pass by as these atoms and molecules bounce about more. As a result, resistance typically rises with temperature.
Complete step by step solution:
The resistance is related to temperature as the rate of change of resistance with change in temperature is directly proportional to the resistance at that temperature and the proportionality constant is known as the temperature coefficient of resistance denoted by the symbol $\alpha $. That is,
$\dfrac{{dR}}{{dT}} = \alpha R$
If the change in resistance with temperature is small then this relation can be written as,
$\dfrac{{\vartriangle R}}{{\vartriangle T}} = \alpha R$
Given two temperatures are ${0^ \circ }C\,\& \,{t^ \circ }C$. Here, the change in temperature, that is, $\vartriangle T$ is ${\left( {t - 0} \right)^ \circ }C$. So the equation becomes,
$\vartriangle R = \alpha {R_0}\left( {t - 0} \right) \\ $
That is,
$\vartriangle R = \alpha {R_0}t \\ $ ...(1)
Also,
$\vartriangle R = {R_t} - {R_0} \\ $
This implies that,
${R_t} = {R_0} + \vartriangle R \\ $ ...(2)
Substituting equation (1) in equation (2), we get,
\[{R_t} = {R_0} + \alpha {R_0}t \\ \]
Taking ${R_0}$ common on right hand side of the equation, we get,
\[{R_t} = {R_0}\left( {1 + \alpha t} \right)\]
Hence, option A is the correct answer.
Note: Consider a basic model of resistance to better understand the temperature dependency. Atoms and molecules in a conductor obstruct the flow of electrons. It becomes more difficult for the electrons to pass by as these atoms and molecules bounce about more. As a result, resistance typically rises with temperature.
Recently Updated Pages
Uniform Acceleration - Definition, Equation, Examples, and FAQs

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Electric field due to uniformly charged sphere class 12 physics JEE_Main

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Wheatstone Bridge for JEE Main Physics 2025

Charging and Discharging of Capacitor
