In an experiment, the resistance of a material is plotted as a function of temperature (in some rang

Q: Why is the slope negative?

Because the line goes downwards as we move along the positive x-axis. This indicates an inverse relationship between the variables.

Q: What does the y-intercept represent in this context?

The y-intercept represents the value of $\ln R(T)$ when $1/T^2$ is zero, which corresponds to infinite temperature. We denote this constant as $\ln R_0$.

Q: How do we remove the natural logarithm from the equation?

By taking the exponential (base $e$) on both sides of the equation. The exponential function is the inverse of the natural logarithm.

Q: Why did we choose $T_0^2$ for the slope?

It's a convenient way to represent a positive constant with the same dimensions as $T^2$, ensuring the exponent is dimensionless. The negative sign is added explicitly to show the downward slope.

Q: Can this graph represent a semiconductor or a conductor?

The resistance decreases as temperature increases (since $1/T^2$ decreases, $T$ increases, and $\ln R(T)$ decreases). This behavior is characteristic of semiconductors, where resistance drops with rising temperature.

Question

In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line.
One may conclude that

Options:

(A) $R(T) = R_0 e^{-T^2/T_0^2}$

(B) $R(T) = R_0 e^{T^2/T_0^2}$

(C) $R(T) = R_0 e^{-T_0^2/T^2}$

(D) $R(T) = R_0 e^{T_0^2/T^2}$

Sigma Percentile · Accepted Answer

### Decoding the Graph

Imagine you are in a lab, carefully measuring the resistance of a mysterious material at various temperatures. You plot your data, and a beautiful straight line emerges. But wait, the axes aren't just resistance and temperature! The y-axis is the natural logarithm of resistance, $\ln R(T)$, and the x-axis is the inverse square of temperature, $\frac{1}{T^2}$.

This straight line is a visual goldmine. It tells us exactly how these two complex variables relate to each other. Our goal is to translate this geometric line back into a physical equation for $R(T)$.

### The Mathematical Translation

Every straight line follows the classic equation:
$$y = mx + c$$

Let's map our graph's variables onto this equation. Our "$y$" is $\ln R(T)$ and our "$x$" is $\frac{1}{T^2}$. Substituting these in, we get:
$$\ln R(T) = m \left(\frac{1}{T^2}\right) + c$$

Now, look at the slope. As we move to the right (increasing $\frac{1}{T^2}$), the line goes down. This means our slope, $m$, is **negative**. To represent this mathematically, let's define the slope as $m = -T_0^2$, where $T_0$ is just a positive constant.

What about the y-intercept, $c$? This is the value of $\ln R(T)$ when $\frac{1}{T^2} = 0$ (which physically means the temperature is infinite!). Let's call this constant intercept $\ln R_0$.

Plugging these constants back into our equation, we have:
$$\ln R(T) = -T_0^2 \left(\frac{1}{T^2}\right) + \ln R_0$$

### Unveiling the Final Function

We are almost there! Let's rearrange the equation to group the logarithmic terms:
$$\ln R(T) - \ln R_0 = -\frac{T_0^2}{T^2}$$

Using the fundamental property of logarithms, $\ln a - \ln b = \ln\left(\frac{a}{b}\right)$, we can combine the left side:
$$\ln\left(\frac{R(T)}{R_0}\right) = -\frac{T_0^2}{T^2}$$

To free $R(T)$ from the logarithm, we take the exponential (base $e$) of both sides. This is the crucial step that transforms our linear graph into an exponential physical law:
$$\frac{R(T)}{R_0} = e^{-\frac{T_0^2}{T^2}}$$

Finally, multiplying both sides by $R_0$, we arrive at the beautiful final expression:
$$R(T) = R_0 e^{-\frac{T_0^2}{T^2}}$$

This perfectly matches option (c). You've successfully decoded the graph and uncovered the hidden physics of the material!

Animated Solution for Physics - Current Electricity: In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line. One may conclude that

Select Answer:

Visualized Solution

Analyzing the Graph

Equation of the Straight Line

Determining the Slope and Intercept

Simplifying the Equation

Final Expression

The Way Forward

The Sigma Insight: Ohm's Law, Resistance and Electrical Power

Decoding the Graph

The Mathematical Translation

Unveiling the Final Function

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