Parallel Circuits

Kirchhoff’s Laws

• Current Law (Junction Rule):   - The sum of currents entering a junction equals the sum of currents leaving it.   - In series circuits: $I_{battery} = I_1 = I_2 = I_3$   - In parallel circuits: $I_{total} = I_1 + I_2 + I_3 + …$

• Voltage Law (Loop Rule):   - The sum of voltage changes around a closed loop must be zero.   - In series circuits: $\Delta V_{battery} = \Delta V_1 + \Delta V_2 + \Delta V_3 + …$   - In parallel circuits: $\Delta V_{battery} = \Delta V_1 = \Delta V_2 = \Delta V_3 = …$

Parallel Circuits Resistance

• Total Resistance (RT or Req):   - Calculated as:     - $R_{eq} = \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + … \right)^{-1}$

• Branch Resistance:   - Use Ohm's Law: $V_1 = I_1R_1$ for individual branches.

Parallel Circuits Voltage

• Total Voltage:   - Total voltage equals the battery voltage: $V_T = I_T R_T$

• Branch Voltage:   - Voltage across each branch is equal to the total voltage.   - Use Ohm's Law: $V_1 = I_1R_1$ for individual branches.   - Note: Voltage affects the brightness of light bulbs; bulbs in parallel are brighter than in series.

Parallel Circuits Current

• Total Current:   - Calculated similarly to voltage: $V_T = I_T R_T$   - Total current is the sum of currents in each branch: $I_{total} = I_1 + I_2 + I_3 + …$

• Branch Current:   - Current splits at branches and combines when branches meet again.   - Use Ohm's Law: $V_1 = I_1R_1$ for individual branches.

Kirchhoff’s Laws

• Current Law (Junction Rule): The total current entering a junction equals the total current leaving it. In series, $I_{battery} = I_1 = I_2 = I_3$; in parallel, $I_{total} = I_1 + I_2 + I_3 + …$

• Voltage Law (Loop Rule): The sum of voltage changes in a closed loop is zero. In series, $ext{V}<em>{battery} = ext{V}_1 + ext{V}_2 + ext{V}_3 + …$; in parallel, $ext{V}</em>{battery} = ext{V}_1 = ext{V}_2 = ext{V}_3 = …$

Parallel Circuits Resistance

• Total Resistance: $R_{eq} = \bigg( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + … \bigg)^{-1}$

• Branch Resistance: Use $V_1 = I_1R_1$ for individual branches.

Parallel Circuits Voltage

• Total Voltage: Equals battery voltage: $V_T = I_T R_T$.

• Branch Voltage: Each branch has voltage equal to battery voltage. $V_1 = I_1R_1$ applies for individual branches.

Parallel Circuits Current

• Total Current: $V_T = I_T R_T$ and $I_{total} = I_1 + I_2 + I_3 + …$.

• Branch Current: Current splits at branches and combines when rejoining. Use $V_1 = I_1R_1$ for branches.