Question

Two wires of resistances \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) have temperature coefficient of resistances \(\alpha_{1}\) and \(\alpha_{2}\) respectively they are joined in series the effective temperature coefficient of resistance is .... (A) \(\left[\left(\alpha_{1}+\alpha_{2}\right) / 2\right]\) (B) \(\sqrt{\alpha_{1} \alpha_{2}}\) (C) \(\left[\left(\alpha_{1} R_{1}+\alpha_{2} R_{2}\right) /\left(R_{1}+R_{2}\right)\right]\) (D) \(\left[\sqrt{\left(R_{1}\right.} R_{2} \alpha_{1} \alpha_{2}\right) / \sqrt{ \left.\left(R_{1}^{2}+R_{2}^{2}\right)\right]}\)

Short answer

The effective temperature coefficient of resistance for two wires connected in series is given by: \[\alpha_{total} = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2}\]

Step-by-step solution

Find the total resistance of the circuit in series

When two resistors are connected in series, their total resistance, denoted by R_total, can be found using the formula: \[R_{total} = R_1 + R_2\]

Find the resistance change for both resistors

The change in resistance for each wire due to a change in temperature can be found using the formula: \[\Delta R = \alpha R \Delta T\] For wire 1, this would be: \[\Delta R_1 = \alpha_1 R_1 \Delta T\] And for wire 2, this would be: \[\Delta R_2 = \alpha_2 R_2 \Delta T\]

Find the total change in resistance due to temperature

Since the wires are connected in series, their resistance changes are added together. So the total change in resistance is the sum of ΔR1 and ΔR2: \[\Delta R_{total} = \Delta R_1 + \Delta R_2\]

Substitute the resistance change formulas

Now, substitute the expressions for ΔR1 and ΔR2 from Step 2: \[\Delta R_{total} = \alpha_1 R_1 \Delta T + \alpha_2 R_2 \Delta T\]

Simplify the expression and find the effective temperature coefficient

We can simplify the expression by factoring out the temperature change ΔT: \[\Delta R_{total} = (\alpha_1 R_1 + \alpha_2 R_2) \Delta T\] Now, comparing with the standard formula for resistance change: \[\Delta R_{total} = \alpha_{total} R_{total} \Delta T\] And substituting the expression for total resistance (Step 1): \[\Delta R_{total} = \alpha_{total} (R_1 + R_2) \Delta T\] Now, we can equate the expressions for ΔR_total obtained in steps 4 and 5: \[(\alpha_1 R_1 + \alpha_2 R_2) \Delta T = \alpha_{total} (R_1 + R_2) \Delta T\] Since ΔT is non-zero, we can divide both sides by ΔT and by R_1 + R_2: \[\alpha_{total} = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2}\] This corresponds to answer choice C: \[\left[\left(\alpha_{1} R_{1}+\alpha_{2} R_{2}\right) /\left(R_{1}+R_{2}\right)\right]\]

Key concepts

Series Resistors

When resistors are arranged one after another in a circuit, they are said to be in series. This type of arrangement means that the same amount of current flows through each resistor. Here, the total resistance of a series circuit is simply the sum of the individual resistances.
Imagine a simple circuit with just two resistors, having resistances \( R_1 \) and \( R_2 \). To find the total resistance, we use the formula:

• \[ R_{total} = R_1 + R_2 \]

Regardless of how many resistors are added in series, the concept remains the same: total resistance is additive. This is very different from resistors in parallel, where the total resistance is calculated differently.
In summary, for resistors in series, just add up all their resistance values to get the total resistance.

Temperature Coefficient

The temperature coefficient of resistance is an important property of materials used in resistors. It explains how the resistance changes as the temperature changes. Think of it as a measure of how "sensitive" a resistor is to temperature changes.
For resistors, when temperature changes, their resistance doesn't stay the same. The resistance alters and this change is governed by the temperature coefficient, often denoted by \( \alpha \). It is given in per degree Celsius (or Kelvin) change.
We express the change in resistance as:

• \[ \Delta R = \alpha R \Delta T \]
• Where \( \Delta R \) is the change in resistance, \( R \) is the original resistance, and \( \Delta T \) is the change in temperature.

A positive temperature coefficient means resistance increases with temperature. Conversely, a negative one means resistance decreases when temperature rises.
This is crucial for understanding and predicting circuit behavior in different temperatures.

Resistance Change

Consider what happens when a resistor's temperature changes. The change in resistance (\( \Delta R \)) is determined by the product of the initial resistance, temperature coefficient, and temperature change.
For a resistor with initial resistance \( R \), temperature coefficient \( \alpha \), and temperature change \( \Delta T \), this is given by:

• \[ \Delta R = \alpha R \Delta T \]

Take a resistor with \( R_1 \) and coefficient \( \alpha_1 \). If the temperature changes, the formula tells us the new resistance value will be altered by \( \Delta R_1 = \alpha_1 R_1 \Delta T \).
Do this for each resistor, then add up all changes for those in series to know how the total resistance has been adjusted.
Understanding resistance change is critical for ensuring that circuits function correctly under various environmental conditions.

Resistors in Series

When multiple resistors are joined end to end, they form a series. The combined effect is an increase in overall resistance because current must pass through each resistor sequentially.
This impacts how circuits are designed and understood because resistors in series don't share the same current flow as those in parallel configurations.
The formula for the total resistance remains simple as noted:

• \[ R_{total} = R_1 + R_2 + ... + R_n \]

For temperature considerations, when two resistors with different temperature coefficients \( \alpha_1 \) and \( \alpha_2 \) are in series, the effective temperature coefficient \( \alpha_{total} \) can be calculated as:

• \[ \alpha_{total} = \frac{\alpha_1 R_1 + \alpha_2 R_2}{R_1 + R_2} \]

This helps predict how their total resistance changes with temperature.
Such calculations are required in design while modifying the properties of resistors joined sequence-wise in electronics.