Equation of resistance from given graph

  • #1
Aurelius120
192
20
Homework Statement
Given the graph of ##lnR## vs ##T^{-2}##, predict the relationship between resistance and temperature
Relevant Equations
NA
1000016494.jpg

From the graph:
$$lnR(T)=\frac{-lnR(0)T^2_○}{T^2}+lnR(0)$$
I have assumed ##R(0)## to be the value of ##R## at ##1/T^2=0## and ##T_○## to be the value of ##T## at ##lnR(T)=0##
From this I get,
$$R(T)=e^{lnR(0)×\left(1-\frac{T_○^2}{T^2}\right)}$$
$$R(T)=R(0)^{\left(1-\frac{T_○^2}{T^2}\right)}$$
This does not match with any of the options.
I couldn't reduce it any of the options either.

Maybe my choice of ##R(0)## and the one in the options refer to different quantities. Even so I could not get to the right answer(given option-c in the book).

Please help
 
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  • #2
Aurelius120 said:
I have assumed ##R(0)## to be the value of ##R## at ##1/T^2=0## and ##T_○## to be the value of ##T## at ##lnR(T)=0##
I would assume R0 is the value of R when T=T0.
 
  • #3
It is unclear to me why you have included ##\ln R_0## in the linear term. There is no reason to.

I suggest you write the linear equation on a more agnostic form ##\ln R = -k/T^2 + m##, solve for ##R##, and only then try to identify ##R_0## and ##T_0##.
 
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  • #4
haruspex said:
I would assume R0 is the value of R when T=T0.
I would not assume anything. I would wait with introducing the constants ##R_0## and ##T_0## until I can introduce them to get the result on one of the given forms. As should be clear from the given forms, ##R_0## is the value of ##R## in particular limits depending on the option. (##T = 0## for a and b, ##T \to \infty## for c, and ##T = 1## for d)
 
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  • #5
Orodruin said:
It is unclear to me why you have included ##\ln R_0## in the linear term. There is no reason to.

I suggest you write the linear equation on a more agnostic form ##\ln R = -k/T^2 + m##, solve for ##R##, and only then try to identify ##R_0## and ##T_0##.
Done
$$ln(R(T))=\frac{-k}{T^2}+m$$
$$R(T)=e^{(-k/T^2+m)}=e^m×e^{-k/T^2}=R_○e^{-T_○^2/T^2}$$
 
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