WATT'S LAW: WHAT IS IT, EXAMPLES, APPLICATIONS - PHYSICAL - 2025

The law Watt is applied to electrical circuits and establishes that electric power P supplied by a circuit element, is directly proportional to the product between the supply voltage V of the circuit and the current I flowing through it.

Electric power is a very important concept, because it indicates how fast an element is to transform electrical energy into some other form of energy. Mathematically, the given definition of Watt's law is expressed like this:

Figure 1. Electrical power indicates how quickly electrical energy is transformed. Source: Pixabay

In the International System of Units (SI), the unit of power is called watt and abbreviated W, in honor of James Watt (1736-1819), Scottish engineer pioneer of the industrial revolution. Since power is energy per unit time, 1 W equals 1 joule / second.

We are all familiar with the concept of electrical power in one way or another. For example, household electrical devices in common use always have their power specified, including light bulbs, electric burners or refrigerators, among others.

Watt's law and circuit elements

Watt's law applies to circuit elements with different behavior. It can be a battery, a resistor or another. A potential difference V _B - V _A = V _AB is established between the ends of the element and the current flows from A to B, as indicated in the following figure:

Figure 2. A circuit element in which a potential difference has been established. Source: F. Zapata.

In a very short time dt, a certain amount of charge dq passes, so that the work done on it is given by:

Where dq is related to current as:

So:

And since power is work per unit of time:

-If V _AB > 0, the charges that pass through the element gain potential energy. The element supplies energy from some source. It could be a battery.

Figure 3. Power provided by a battery. Source: F. Zapata.

-If V _AB <0, the charges lose potential energy. The element dissipates energy, such as a resistor.

Figure 4. Resistance transforms energy into heat. Source: F. Zapata.

Note that the power supplied by a source does not depend solely on the voltage, but also on the current. This is important to explain why car batteries are so large, considering that they barely supply 12V.

What happens is that the starter motor needs a high current, for a short time, to provide the necessary power to start the car.

Watt's Law and Ohm's Law

If the circuit element is a resistor, Watt's law and Ohm's law can be combined. The latter states that:

Which by substituting in Watt's law leads to:

A version depending on voltage and resistance can also be obtained:

The possible combinations between the four quantities: power P, current I, voltage V and resistance R appear on the chart in figure 5. According to the data provided by a problem, the most convenient formulas are chosen.

For example, suppose that in a certain problem you are asked to find the resistance R, which is in the lower left quarter of the card.

Depending on the quantities whose value is known, one of the three related equations is chosen (in green). For example suppose that V and I are known, then:

If instead P and I are known, and the resistance is requested, use:

Finally, when P and V are known, the resistance is obtained by:

Figure 5. Formulas for Watt's law and Ohm's law. Source: F. Zapata.

Applications

Watt's law can be applied in electrical circuits to find the electrical power supplied or consumed by the element. Light bulbs are good examples of applying Watt's law.

Example 1

A special bulb to obtain several lightings in one, has two tungsten filaments, whose resistances are R _A = 48 ohm and R _B = 144 ohm. They are connected to three points, denoted 1, 2 and 3, as shown in the figure.

The device is controlled by switches to select the pairs of terminals and also connect it to the 120 V network. Find all the possible powers that can be obtained.

Figure 6. Scheme for the worked example 1. Source. D. Figueroa. Physics for Science and Engineering.

Solution

- When terminals 1 and 2 are connected, only resistor R _A remains activated. Since we have the voltage, which is 120 V and the resistance value, these values ​​are directly substituted in the equation:

- Connecting terminals 2 and 3, resistor R _B is activated, whose power is:

- Terminals 1 and 3 allow the resistors to be connected in series. The equivalent resistance is:

Thus:

- Finally, the remaining possibility is to connect the resistors in parallel, as shown in diagram d). The equivalent resistance in this case is:

Therefore the equivalent resistance is R _eq = 36 ohm. With this value, the power is:

Example 2

Besides the watt, another widely used unit for power is the kilowatt (or kilowatt), abbreviated as kW. 1 kW equals 1000 watts.

Companies that supply electricity to homes bill in terms of energy consumed, not power. The unit they use is the kilowatt-hour (kW-h), which despite having the name of watt, is a unit for energy.

a) Suppose that a household consumes 750 kWh during a given month. What will be the amount of the electricity bill for that month? The following consumption plan is followed:

- Base rate: $ 14.00.

- Price: 16 cents / kWh up to 100 kWh per month.

- The next 200 kWh per month are worth 10 cents / kWh.

- And above 300 kWh per month, 6 cents / kWh are charged.

b) Find the average cost of electrical energy.

Solution to

- The customer consumes 750 kW-h per month, therefore exceeds the costs indicated in each stage. For the first 100 kWh the money value is: 100 kWh x 16 cents / kWh = 1600 cents = $ 16.00

- The next 200 kWh have a cost of: 200 kWh x 10 cents / kWh = 2000 cents = $ 20.00.

- Above these 300 kW-h, the customer consumes 450 kW-h more, for a total of 750 kW-h. The cost in this case is: 450 kWh x 6 cents / kWh = 2,700 cents = $ 27.00.

- Finally, all the amounts obtained plus the base rate are added to obtain the price of the receipt for that month:

Solution b

The average cost is: $ 77/750 kWh = $ 0.103 / kW-h = 10.3 cents / kWh.

References

1. Alexander, C. 2006. Fundamentals of electrical circuits. 3rd. Edition. McGraw Hill.
2. Berdahl, E. Introduction to Electronics. Recovered from: ccrma.stanford.ed.
3. Boylestad, R. 2011. Introduction to circuit analysis. 13th. Edition. Pearson.
4. Electrical Rebuilder's Association. Ohm's Law & Watt's Law Calculator with Examples. Recovered from: electricalrebuilders.org
5. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 5. Electricity. Edited by Douglas Figueroa (USB).