Combinations of Resistors, Electric Power and Heating Effect

Introduction — Building Real Circuits

Real appliances contain many resistors (bulbs, motors, heater coils) wired together. In this part we learn how resistors combine in series and in parallel, then turn to the heating effect of current (Joule's law) and to electric power. Finally we see how these ideas are applied in household wiring.

11.8 Combinations of Resistors

Two (or more) resistors can be wired in a circuit in two fundamental ways — series (end-to-end) or parallel (side-by-side).

11.8.1 Resistors in Series

Resistors are said to be connected in series when they are joined end-to-end so that the same current flows through each of them. The potential difference across the combination is the sum of the potential differences across each resistor.

If V is the total voltage and \(V_1, V_2, V_3\) are the voltages across \(R_1, R_2, R_3\), then \(V = V_1 + V_2 + V_3\). Applying \(V = IR\) to each:

\[ IR_s = IR_1 + IR_2 + IR_3 \quad\Rightarrow\quad \boxed{R_s = R_1 + R_2 + R_3} \]

11.8.2 Resistors in Parallel

Resistors are in parallel when they share the same two end-points, so the same potential difference acts across each. The total current is the sum of currents through each branch.

Total current: \(I = I_1 + I_2 + I_3\). Since each branch sees the same V:

\[ \dfrac{V}{R_p} = \dfrac{V}{R_1} + \dfrac{V}{R_2} + \dfrac{V}{R_3} \quad\Rightarrow\quad \boxed{\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}} \]

11.8.3 Why Household Appliances are Wired in Parallel

• Each appliance gets the full mains voltage (220 V in India), regardless of whether others are on.
• If one device fails, others keep working — the circuit is not broken.
• Each appliance can be switched on/off independently.
• Total resistance decreases → total current drawn from the mains is adequate for multiple devices.

11.9 Worked Examples — Series and Parallel

11.10 Heating Effect of Electric Current — Joule's Law

When current flows through a resistor, electrical energy is converted to heat. The drifting electrons collide with metal atoms, giving up kinetic energy as thermal vibrations. This is the heating effect of current, used in heaters, geysers, irons and bulbs.

11.10.1 Derivation of H = I²Rt

Work done by the battery to drive charge Q through potential difference V is \(W = VQ\). Since \(Q = It\), we have \(W = V I t\). For a resistor obeying Ohm's law, \(V = IR\). So work done appears as heat:

\[ H = W = VIt = (IR)(I)(t) = I^2 R t \]

11.11 Electric Power

The rate at which electrical energy is dissipated (or consumed) in a circuit is called electric power.

11.11.1 Commercial Unit of Electrical Energy — the kWh

The joule is too tiny to measure household consumption. Electric companies use a bigger unit — the kilowatt-hour (kWh), popularly called a "unit".

\[ 1\ \text{kWh} = 1\ \text{kW} \times 1\ \text{h} = 1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^{6}\ \text{J} \]

The electrical energy consumed by an appliance is: Energy (kWh) = Power (kW) × time (h).

11.12 Worked Numericals on Power and Heating

11.13 Applications of the Heating Effect

• Electric heaters, irons, geysers, toasters, kettles — coils of nichrome wound on ceramic, which glow red-hot.
• Incandescent bulbs — a tungsten filament (melting point 3380 °C) heats to ~2500 °C in an inert gas (argon + a little nitrogen) and emits white light. Most of the energy goes out as heat, which is why LEDs are far more efficient.
• Electric fuse — a short piece of wire made of a tin-lead alloy with low melting point. If the current exceeds a safe value, the fuse heats up and melts, breaking the circuit and protecting the appliance and wiring.

11.14 Household Electric Wiring (Brief)

Electricity reaches Indian homes at 220 V ± 10%, 50 Hz. Three wires enter the main switch: live (red or brown), neutral (black or blue) and earth (green or yellow-green). The live wire carries current; the neutral completes the circuit; the earth wire is connected to a metal plate buried in the ground and is connected to metal casings of appliances for safety.

11.14.1 Short-Circuit and Overloading

• Short circuit — if the live and neutral wires touch directly (due to damaged insulation), the resistance becomes almost zero and the current becomes very large. This melts the fuse (or trips the MCB) instantly.
• Overloading — if too many high-power appliances are plugged into the same socket, the wire carries more current than it is rated for. It heats up, possibly enough to cause a fire.