Question

What is the equivalent resistance of given two resistors connected in series?

Answer 1

To find the equivalent resistance of two resistors connected in series, you simply add their resistances together. The formula for the equivalent resistance \( R_{eq} \) of two resistors \( R_1 \) and \( R_2 \) in series is:

\[
R_{eq} = R_1 + R_2
\]

### Explanation

1. **Series Connection**: In a series circuit, the resistors are connected end-to-end, so the same current flows through each resistor. This means that the total voltage across the series connection is the sum of the voltages across each resistor.

2. **Ohm's Law**: According to Ohm's Law, \( V = I \times R \) (where \( V \) is voltage, \( I \) is current, and \( R \) is resistance), we can see how voltage drops across each resistor contribute to the total voltage.

3. **Voltage Drops**: If \( I \) is the current flowing through the series connection, the voltage across each resistor can be expressed as:
   - Voltage across \( R_1 \): \( V_1 = I \times R_1 \)
   - Voltage across \( R_2 \): \( V_2 = I \times R_2 \)

4. **Total Voltage**: The total voltage supplied by the source \( V_{total} \) is the sum of the individual voltages:
   \[
   V_{total} = V_1 + V_2 = I \times R_1 + I \times R_2
   \]

5. **Factoring Out Current**: This can be rearranged to:
   \[
   V_{total} = I \times (R_1 + R_2)
   \]
   This means we can treat \( (R_1 + R_2) \) as a single equivalent resistance \( R_{eq} \):
   \[
   V_{total} = I \times R_{eq}
   \]
   So, we can conclude that:
   \[
   R_{eq} = R_1 + R_2
   \]

### Example

If you have two resistors, one of \( 4 \, \Omega \) and the other of \( 6 \, \Omega \), the equivalent resistance can be calculated as follows:

\[
R_{eq} = 4 \, \Omega + 6 \, \Omega = 10 \, \Omega
\]

### Summary

In summary, for resistors in series, simply add their resistances together to find the equivalent resistance. This concept is crucial in circuit analysis, as it helps simplify complex circuits and predict how they will behave under various conditions.

Answer 2

When two resistors are connected in **series**, the equivalent resistance is simply the sum of the resistances of the individual resistors.

### Formula:
\[
R_{\text{eq}} = R_1 + R_2
\]
Where:
- \(R_{\text{eq}}\) is the equivalent resistance,
- \(R_1\) is the resistance of the first resistor,
- \(R_2\) is the resistance of the second resistor.

### Explanation:
In a series circuit, the current flows through each resistor sequentially, meaning the same current passes through both resistors. The total voltage drop across the resistors is the sum of the voltage drops across each resistor, which leads to the simple addition of their resistances.

### Example:
If you have two resistors with values:
- \(R_1 = 4 \, \Omega\)
- \(R_2 = 6 \, \Omega\)

Then the equivalent resistance will be:
\[
R_{\text{eq}} = 4 \, \Omega + 6 \, \Omega = 10 \, \Omega
\]

So, the equivalent resistance of the two resistors connected in series is \(10 \, \Omega\).