### Power Distribution in Parallel Circuits

When resistances are wired in parallel, they each consume power according to the same formula, P = I 2R. But the current is not the same in each resistance. An easier method to find the power Pn dissipated by each of the various resistances Rn is to use the formula Pn = E2/Rn, where E is the voltage of the supply or battery. This voltage is the same across every branch resistance in a parallel circuit.

#### Problem 1

Suppose a dc circuit contains three resistances R1 = 22 Ω, R2 = 47 Ω, and R3 = 68 Ω across a battery that supplies a voltage of E = 3.0 V. Find the power dissipated by each resistance.

Let’s find the square of the supply voltage, E2, first. We’ll be needing this figure often: E2 = 3.0 × 3.0 = 9.0. Then the wattages dissipated by resistances R1, R2, and R3 respectively are P1 = 9.0/22 = 0.4091 W, P2 = 9.0/47 = 0.1915 W, and P3 = 9.0/68 = 0.1324 W. These should be rounded off to P1 = 0.41 W, P2 = 0.19 W, and P3 = 0.13 W. (But let’s remember the values to four significant figures for the next problem!) In a parallel circuit, the total dissipated wattage is equal to the sum of the wattages dissipated by the individual resistances.

#### Problem 2

Find the total consumed power of the resistor circuit in Problem 1 using two different methods.

The first method involves adding P1, P2, and P3. Let’s use the four-significant-digit values to avoid the possibility of encountering the rounding-off bug. The total power thus calculated is P = 0.4091 + 0.1915 + 0.1324 = 0.7330 W. Now that we’ve finished the calculation, we should round it off to 0.73 W.

The second method involves finding the net resistance R of the parallel combination. You can do this calculation yourself. Determining it to four significant digits, you should get a net resistance of R = 12.28 Ω. Then P = E 2/R = 9.0/12.28 = 0.7329 W. Now that the calculation is done, this can be rounded to 0.73 W. It’s the Law!

In electricity and electronics, dc circuit analysis can be made easier if you are acquainted with certain axioms, or laws. Here they are:

• The current in a series circuit is the same at every point along the way.
• The voltage across any resistance in a parallel combination of resistances is the same as the voltage across any other resistance, or across the whole set of resistances.
• The voltages across resistances in a series circuit always add up to the supply voltage.
• The currents through resistances in a parallel circuit always add up to the total current drawn from the supply.
• The total wattage consumed in a series or parallel circuit is always equal to the sum of the wattages dissipated in each of the resistances.

Now, let’s get acquainted with two of the most famous laws that govern dc circuits. These rules are broad and sweeping, and they make it possible to analyze complicated series-parallel dc networks.