Series & Parallel Circuits

• Kirchhoff's First Law (Current): Based on the conservation of charge, it states that the sum of currents entering a junction must equal the sum of currents leaving it ($I_{in} = \sum I_{out}$).

• Kirchhoff's Second Law (Voltage): Based on the conservation of energy, it states that the sum of the e.m.f.s in any closed loop is equal to the sum of the potential differences across the components in that loop ($\sum \epsilon = \sum V$).

• Energy Dissipation: Electrical power ($P$) is the rate at which energy is transferred. It can be calculated using $P = IV$, $P = I^2R$, or $P = \frac{V^2}{R}$, depending on which variables are known.

Calculating Total Resistance

• Series Resistance: Simply sum the individual resistances: $R_{total} = R_1 + R_2 + R_3 + ...$. The total resistance is always greater than the largest individual resistor.
• Parallel Resistance: Use the reciprocal sum formula: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...$. The total resistance is always smaller than the smallest individual resistor.

Analyzing Combined Circuits

• Step 1: Identify blocks of components that are purely in series or purely in parallel.
• Step 2: Simplify these blocks into 'equivalent resistors' using the formulas above.
• Step 3: Redraw the circuit with equivalent resistors and repeat the process until a single loop remains.
• Step 4: Apply Ohm's Law ($V = IR$) to the simplified circuit to find the total current, then work backwards to find individual values.

• The 'Sanity Check' for Resistance: In a parallel circuit, if your calculated $R_{total}$ is larger than any of the individual resistors, you have likely forgotten to take the final reciprocal of your sum.

• Power Comparisons: When comparing brightness of bulbs, remember that in series, the bulb with the highest resistance dissipates the most power ($P = I^2R$), whereas in parallel, the bulb with the lowest resistance dissipates the most power ($P = \frac{V^2}{R}$).

• Ideal vs. Real Sources: Unless stated otherwise, assume wires have zero resistance and batteries have no internal resistance. If internal resistance is mentioned, treat it as a resistor in series with the e.m.f. source.

• Current Direction: Always mark the direction of current flow from the positive terminal to the negative terminal to avoid sign errors when applying Kirchhoff's Second Law.

• The 'Current Consumption' Myth: A common error is thinking current is 'used up' by components. Current is the flow of charge; the same number of electrons entering a component must leave it. It is the energy (potential) that is transferred, not the charge itself.

• Reciprocal Math Errors: Students often calculate $\frac{1}{R_1} + \frac{1}{R_2}$ and stop there. You must perform the final step of $R_{total} = 1 / (\text{your sum})$ to get the actual resistance value.

• Voltage in Parallel: Do not sum the voltages of branches in parallel. The potential difference across every parallel branch is identical to the potential difference across the entire parallel combination.