Electric Circuits

Series circuit. Components connected end-to-end in a single loop. Only one path for current.

Rules.

• Current: SAME at every point in the circuit.
• Voltage: total supply voltage = sum of voltages across each component.
• Resistance: $R_{\text{tot}} = R_{1} + R_{2} + R_{3} + \ldots$

Why current is the same. Charge has nowhere else to go — what flows in must flow out at every point.

Worked. A $9\,\text{V}$ battery in series with $R_{1} = 2\,\Omega$ and $R_{2} = 4\,\Omega$.

• $R_{\text{tot}} = 2 + 4 = 6\,\Omega$.
• $I = V/R_{\text{tot}} = 9/6 = 1.5\,\text{A}$.
• $V_{1} = IR_{1} = 1.5 \times 2 = 3\,\text{V}$.
• $V_{2} = IR_{2} = 1.5 \times 4 = 6\,\text{V}$. ($3 + 6 = 9\,\text{V}$ ✓.)

Disadvantage. If one bulb fails (open), the WHOLE circuit breaks — like Christmas lights in older designs.

Parallel circuit. Components on separate branches between the same two points. Multiple paths for current.

Rules.

• Voltage: SAME across each parallel branch (= supply voltage if branches connect directly to the cell).
• Current: total current from supply = sum of currents in each branch.
• Resistance: $\dfrac{1}{R_{\text{tot}}} = \dfrac{1}{R_{1}} + \dfrac{1}{R_{2}} + \ldots$ The total resistance is LESS than the smallest individual resistor.

Worked. Two resistors in parallel: $R_{1} = 4\,\Omega$, $R_{2} = 6\,\Omega$.

• $\dfrac{1}{R_{\text{tot}}} = \dfrac{1}{4} + \dfrac{1}{6} = \dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}$.
• $R_{\text{tot}} = 12/5 = 2.4\,\Omega$.
• Confirm: less than $4$ ✓.

Two-resistor shortcut. $R_{\text{tot}} = \dfrac{R_{1}R_{2}}{R_{1} + R_{2}}$.

Worked. Same as above: $R_{\text{tot}} = (4 \times 6)/(4 + 6) = 24/10 = 2.4\,\Omega$ ✓.

Advantage. If one bulb fails, the others stay on (each on its own branch). Used for household wiring.

Thermistor (NTC type).

• Resistance DECREASES as temperature increases.
• Used in: temperature-sensing circuits, fire alarms, oven controllers.
• Rule of thumb: at room temp $\sim$ kΩ, at boiling point $\sim$ tens of Ω.

LDR (Light-Dependent Resistor).

• Resistance DECREASES as light intensity increases.
• Used in: street lights (turn on at dusk), camera light meters, automatic doors (sometimes).

Both are non-linear, non-ohmic components. A graph of $R$ vs temperature (thermistor) or $R$ vs light intensity (LDR) is a curve, not a straight line.

Cambridge tip. When asked "describe the action of a thermistor" — say something like "as temperature rises, the resistance of the thermistor falls". Cambridge mark schemes look for this exact phrasing.

Potential divider. Two resistors $R_{1}$ and $R_{2}$ in series across a supply $V_{\text{in}}$. The output voltage taken across $R_{2}$ is: $V_{\text{out}} = V_{\text{in}} \times \dfrac{R_{2}}{R_{1} + R_{2}}.$

So the supply voltage divides between the two resistors in proportion to their values.

Worked. $V_{\text{in}} = 12\,\text{V}$, $R_{1} = 200\,\Omega$, $R_{2} = 400\,\Omega$. Find $V_{\text{out}}$ across $R_{2}$.

• $V_{\text{out}} = 12 \times 400/(200+400) = 12 \times 400/600 = 8\,\text{V}$.

Use with sensors. Replace one of the resistors with a thermistor or LDR. Then $V_{\text{out}}$ varies with temperature or light — useful for triggering alarms or switches.

Worked qualitative. Light-sensitive circuit: replace $R_{2}$ with an LDR. In darkness LDR has high $R$ → $V_{\text{out}}$ across it is large. In bright light LDR has low $R$ → $V_{\text{out}}$ small. Feed $V_{\text{out}}$ to a transistor that switches a streetlight: light goes ON in the dark.