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Series and Parallel Circuits

Question: What is the difference between series and parallel circuits

In electrical circuits, series and parallel refer to two fundamental ways of connecting components, each with distinct characteristics affecting current, voltage, and resistance.

Series Circuits

  • Connection: Components are connected end-to-end in a single path for the current to flow.
  • Current: The same current flows through all components.
  • Voltage: The total voltage is divided among the components.
  • Resistance: Total resistance is the sum of individual resistances: $$ R_{\text{total}} = R_1 + R_2 + \dots + R_n $$

Parallel Circuits

  • Connection: Components are connected across the same two points, creating multiple paths for the current.
  • Voltage: The same voltage is applied across each component.
  • Current: The total current is the sum of the currents through each parallel branch.
  • Resistance: Total resistance is less than any individual resistance and is calculated using the reciprocal formula:
\[ R_{\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} \]

Example 1: Series Circuit

Imagine three resistors with resistances of 5 Ω, 10 Ω, and 15 Ω connected in series to a 12 V battery.

Total Resistance

\[ R_{\text{total}}=5 Ω+10 Ω+15 Ω=30 Ω \]

Current Through Circuit

\[ I = \frac{V}{R_{\text{total}}} = \frac{12\,V}{30\,\Omega} = 0.4\,A \]

Voltage Across Each Resistor

\[ V_1=I×R_1=0.4 A×5Ω = 2V \]
\[ V_2=I×R_2=0.4 A×10Ω = 4V \]
\[ V_3=I×R_3=0.4 A×15Ω = 6V \]

Example 2: Parallel Circuit Example

Now, connect the same resistors (5 Ω, 10 Ω, and 15 Ω) in parallel to a 12 V battery.

Total Resistance of Parallel Resistors

\[ \frac{1}{R_{\text{total}}}=\frac{1}{5 Ω} + \frac{1}{10 Ω} +\frac{1}{15 Ω} = \frac{6 + 3 + 2}{30 Ω} = \frac{11}{30 Ω} \]

So, $$ R_{\text{total}} = \frac{30 Ω}{11}≈2.73 Ω $$

Total Current

\[ I_{\text{total}}=V \times R_{\text{total}}=12 V \times 2.73 Ω ≈ 4.4 A \]

Current Through Each Resistor

\[ I_1 = \frac{V}{R_1} = \frac{12 V}{5Ω} = 2.4A \]
\[ I_2 = \frac{V}{R_2} = \frac{12 V}{10Ω} = 1.2A \]
\[ I_3 = \frac{V}{R_3} = \frac{12 V}{15Ω} = 0.8A \]

The sum of branch currents equals the total current:

\[ 2.4 A + 1.2 A + 0.8 A = 4.4A \]

These examples illustrate how series circuits have a single path with shared current and divided voltage, while parallel circuits have multiple paths with shared voltage and divided current.