Section 9.5: Ohm's Law and Circuits

Ohm's Law relates the voltage (V), current (I), and resistance (R) in a circuit:

\[ V = I R \] where:
\( V \) = voltage (V)
\( I \) = current (A)
\( R \) = resistance (Ω)

Circuits can be classified as series, parallel, or combination circuits. In a series circuit, resistances add: \( R_\text{eq} = R_1 + R_2 + \dots \) In a parallel circuit, reciprocals add: \( \frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots \)

Example 1: Series Circuit

Two resistors, 4 Ω and 6 Ω, are connected in series with a 12 V battery. Find the current.

\( R_\text{eq} = 4 + 6 = 10 \, \Omega \)
\( I = V / R_\text{eq} = 12 / 10 = 1.2 \, \text{A} \)

Example 2: Parallel Circuit

Two resistors, 3 Ω and 6 Ω, are connected in parallel across a 12 V battery. Find the current through each resistor.

\( \frac{1}{R_\text{eq}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \Rightarrow R_\text{eq} = 2 \, \Omega \)
Total current: \( I = V / R_\text{eq} = 12 / 2 = 6 \, \text{A} \)
Current through 3 Ω: \( I_1 = V / R_1 = 12 / 3 = 4 \, \text{A} \)
Current through 6 Ω: \( I_2 = V / R_2 = 12 / 6 = 2 \, \text{A} \)

Practice Problems

A 10 Ω resistor is connected across a 20 V battery. Find the current.
Three resistors, 2 Ω, 4 Ω, and 6 Ω, are connected in series with 24 V. Find the total current.
Two resistors, 5 Ω and 10 Ω, are in parallel across 15 V. Find the current through each resistor.
Find the equivalent resistance of three resistors, 3 Ω, 6 Ω, and 2 Ω, in series.
A 12 V battery is connected to two parallel resistors, 8 Ω and 12 Ω. Find the total current.