Question

If the resistance of a conductor is \(R_{1}\) at temperature \(t_{1},\) the resistance will change to a value \(R\) when the temperature changes to \(t\), where $$R=R_{1}\left[1+\alpha\left(t-t_{1}\right)\right]$$ and \(\alpha\) is the temperature coefficient of resistance at temperature \(t_{1} .\) Solve this equation for \(t_{1}.\)

Short answer

The original temperature \(t_1\) is found using the equation \(t_1 = t - \frac{1}{\alpha}\left(\frac{R}{R_1} - 1\right)\).

Step-by-step solution

Isolate the temperature-dependent term

First, start by isolating the term containing the temperatures on one side of the equation. You can do this by dividing both sides by \(R_1\): \[ \frac{R}{R_1} = 1 + \alpha(t - t_1). \]

Eliminate the Constant

Next, eliminate the constant 1 from the right side of the equation by subtracting 1 from both sides: \[ \frac{R}{R_1} - 1 = \alpha(t - t_1). \]

Isolate the term containing \(t_1\)

After that, isolate the term \(\alpha(t_1)\) by moving all other terms not containing \(t_1\) to the left side: \[ \alpha t_1 = \alpha t - \left(\frac{R}{R_1} - 1\right). \]

Solve for \(t_1\)

Finally, to solve for \(t_1\), divide both sides by \(\alpha\): \[ t_1 = t - \frac{1}{\alpha}\left(\frac{R}{R_1} - 1\right). \] This gives you the original temperature \(t_1\) in terms of \(R\), \(R_1\), \(\alpha\), and the new temperature \(t\).

Key concepts

Resistance and Temperature Relationship

One of the fundamental principles in electronics is that resistance changes with temperature. A key concept to understand here is that for most conductors, as temperature increases, their resistance also increases. This happens because as the thermal energy within the material goes up, it causes the atoms to vibrate more vigorously, hindering the flow of electrons, which results in increased resistance.

To quantify this relationship, physicists and engineers use the temperature coefficient of resistance, commonly denoted as \( \alpha \). The coefficient represents how much the resistance of a material changes with a one-degree change in temperature. In general, if \( \alpha \) is positive, the resistance increases with an increase in temperature; conversely, a negative \( \alpha \) means the resistance decreases as temperature rises. Materials with a very low or near-zero temperature coefficient can maintain a nearly constant resistance over a wide range of temperatures. These characteristics are essential when designing circuits that must operate consistently across various temperatures.

Solving for Temperature in Physics

In the realm of physics, solving for temperature often involves understanding the direct correlation between temperature and various physical properties, one of which includes resistance. The equation introduced earlier bridges the gap between theoretical physics and practical application by providing a formula that relates resistance to temperature through the temperature coefficient of resistance. When given this equation, \( R = R_{1}\left[1 + \alpha\left(t - t_{1}\right)\right] \), our main goal is to isolate and solve for the desired temperature, whether it be the old temperature \( t_{1} \) or the new one \( t \).

The solution process typically includes rearranging the equation step-by-step, which means isolating terms, eliminating constants, and using algebraic manipulation until we express the temperature of interest in terms of the given variables. This approach is applied universally in physics when dealing with variables that are temperature-dependent, such as volume in gases (Charles’s Law) or pressure (Gay-Lussac’s Law).

Linear Temperature-Resistance Equation

The relationship between temperature and resistance for a given material, assuming the relationship remains linear, can be expressed by the linear temperature-resistance equation. This equation takes the form \( R = R_{1}\left[1 + \alpha\left(t - t_{1}\right)\right] \). Here, \( R \) represents the resistance at the new temperature, \( R_1 \) is the resistance at the initial temperature \( t_1 \), \( t \) is the new temperature, and \( \alpha \) is the temperature coefficient of resistance.

The simplicity of this linear model allows for ease of calculation and prediction of resistance changes due to temperature variations. However, it's crucial to note that this linear relationship holds accuraTexCoord2 within certain temperature ranges and may not be valid for extreme temperatures, where the material properties could change more drastically or non-linearly. In such cases, the temperature coefficient of resistance itself may vary with temperature, requiring more complex models.