• Several resistors may be connected in two ways.

• The equivalent resistance (R′) is the single resistance that has the same effect on the circuit as the whole group of resistors.

• Life application.

• Household appliances are connected in parallel to the mains, so that each can be switched on or off independently, and each gets the full mains voltage.
• If one lamp burns out, the others keep working, since each has its own path to the source.

Worked example 4: series

• Three identical resistors are connected in series to a 12 V source. The current is 2 A. Find the resistance of each.
• Total (equivalent) resistance R′ = V / I = 12 / 2 = 6 Ω.
• R′ = 3R, so R = 6/3 = 2 Ω.
• Check: each resistor has 4 V across it (2 A × 2 Ω), and 3 × 4 = 12 V.

Worked example 5: parallel

• Three identical 6 Ω resistors are connected in parallel to a 12 V battery.
• Find the equivalent resistance, the total current, and the current in each resistor.
• R′ = R/n = 6/3 = 2 Ω. Total current I = 12/2 = 6 A. Each branch: 12/6 = 2 A, and 3 × 2 = 6 A.

Worked example 6: two ways with the same resistors

• Two identical 10 Ω resistors are connected to a 10 V battery, first in series, then in parallel.
• In series R′ = 20 Ω, I = 0.5 A.
• In parallel R′ = 5 Ω, I = 2 A.
• The parallel arrangement gives a current four times as large.
• This is why, if you want the maximum reading on an ammeter with two identical resistors and two identical cells, you connect the cells in series (to double the emf) and the resistors in parallel (to halve the resistance).

• Pacemaker (integration with medicine). A pacemaker is a small electronic circuit implanted under the skin: a battery, a microchip, and wires to the heart muscle.
• If the heartbeat becomes irregular, the circuit sends tiny electric pulses to make the heart contract regularly.