Mathematics

Problem

For a certain semiconductor material, the electrical resistance

R

(in ohms) varies with absolute temperature

T

(in kelvins) according to

R(T)=kT^{n},

where

k

is a positive constant and

n

is a real exponent (not necessarily an integer). (a) Write an expression for

R'(T)

in terms of

k

n

, and

T

. (b) Describe in words the physical meaning of

R'(T)

for this material. (c) Determine the units of

k

and of

R'(T)

, explaining your reasoning. (d) Suppose

0<n<1

. Explain qualitatively how the magnitude of

R'(T)

changes as

T

increases and discuss what this indicates about the material’s sensitivity to temperature at high versus low temperatures.

Intuitive explanation

The key is recognizing that this power-law relationship between resistance and temperature lets you extract both unknowns $k$ and $n$ from experimental data. If you have two or more $(T, R)$ pairs, you can set up a system of equations—or take logarithms to linearize the relationship into $\log R = \log k + n \log T$ , turning it into a straight line. This transforms a tricky nonlinear problem into something you can solve with algebra or graphing.

Intuitive explanation

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