To calculate the equivalent resistance in series and parallel circuits, we need to understand how resistors behave when connected in different configurations:

1. Resistors in Series:

• When resistors are connected in series, the current through each resistor is the same, but the voltage across each resistor may differ.

• The total or equivalent resistance $R_{\text{eq}}$ in a series circuit is the sum of the individual resistances: $R_{\text{eq}} = R_1 + R_2 + R_3 + \dots + R_n$

  Example:

  • If three resistors of $2 \, \Omega$, $3 \, \Omega$, and $4 \, \Omega$ are connected in series, the equivalent resistance is: $R_{\text{eq}} = 2 \, \Omega + 3 \, \Omega + 4 \, \Omega = 9 \, \Omega$

2. Resistors in Parallel:

• When resistors are connected in parallel, the voltage across each resistor is the same, but the current through each resistor may differ.

• The total or equivalent resistance $R_{\text{eq}}$ in a parallel circuit is given by the reciprocal of the sum of the reciprocals of the individual resistances: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$

  • For two resistors: $R_{\text{eq}} = \frac{R_1 \times R_2}{R_1 + R_2}$

  Example:

  • If three resistors of $2 \, \Omega$, $3 \, \Omega$, and $6 \, \Omega$ are connected in parallel, the equivalent resistance is: $\frac{1}{R_{\text{eq}}} = \frac{1}{2 \, \Omega} + \frac{1}{3 \, \Omega} + \frac{1}{6 \, \Omega} = \frac{3 + 2 + 1}{6 \, \Omega} = \frac{6}{6 \, \Omega} = 1 \, \Omega$ Therefore, $R_{\text{eq}} = 1 \, \Omega$.

Summary:

• Series: $R_{\text{eq}} = R_1 + R_2 + R_3 + \dots$
• Parallel: $R_{\text{eq}} = \left(\sum \frac{1}{R_i}\right)^{-1}$

Would you like to explore this further, or do you have any more specific questions? Here are some related questions:

1. How do resistors affect the current in series vs. parallel circuits?
2. What are some real-world applications of series and parallel resistor combinations?
3. How does the concept of equivalent resistance apply to complex networks like Wheatstone bridges?
4. What are the effects of adding more resistors in parallel on the total resistance?
5. How can you determine the equivalent resistance of a mixed (series-parallel) resistor network?

Tip: Always simplify complex resistor networks step by step, combining resistors in series and parallel until you reach the simplest form.