Part 3: Mathematics for Maintenance Techs

# 3.2 Ohm’s Law, Joules Law, and Series/Parallel Formulas

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## Ohm’s Law

The current that flows through most substances is directly proportional to the voltage V applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is *directly proportional to the voltage applied*: I ∝ V.

This important relationship is known as Ohm’s law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn’t always occur.

### Resistance and Simple Circuits

If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance R. Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or I ∝ \(\frac{1}{R}\).

Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives I = \(\frac{V}{R}\).

This relationship is also called Ohm’s law. Ohm’s law in this form really defines resistance for certain materials. Ohm’s law (like Hooke’s law) is not universally valid. The many substances for which Ohm’s law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance R that is independent of voltage V and current I. An object that has simple resistance is called a *resistor*, even if its resistance is small. The unit for resistance is an ohm and is given the symbol Ω (upper case Greek omega). Rearranging I = *\(\frac{V}{R}\)* gives R=*\(\frac{V}{I}\)*, and so the units of resistance are 1 ohm = 1 volt per ampere: 1 Ω = 1 \(\frac{V}{A}\).

Figure shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in R.

*A simple electric circuit in which a closed path for current to flow is supplied by conductors (usually metal wires) connecting a load to the terminals of a battery, represented by the red parallel lines. The zigzag symbol represents the single resistor and includes any resistance in the connections to the voltage source.*

### Making Connections: Real World Connections

Ohm’s law (V=IR) is a fundamental relationship that could be presented by a linear function with the slope of the line being the resistance. The resistance represents the voltage that needs to be applied to the resistor to create a current of 1 A through the circuit. The graph (in the figure below) shows this representation for two simple circuits with resistors that have different resistances and thus different slopes.

*The figure illustrates the relationship between current and voltage for two different resistors. The slope of the graph represents the resistance value, which is 2Ω and 4Ω for the two lines shown.*

The materials which follow Ohm’s law by having a linear relationship between voltage and current are known as ohmic materials. On the other hand, some materials exhibit a nonlinear voltage-current relationship and hence are known as non-ohmic materials. The figure below shows current voltage relationships for the two types of materials.

Figure #. The relationship between voltage and current for ohmic and non-ohmic materials are shown.

Clearly the resistance of an ohmic material (shown in (a)) remains constant and can be calculated by finding the slope of the graph but that is not true for a non-ohmic material (shown in (b)).

Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of 1012Ω or more. A dry person may have a hand-to-foot resistance of 105Ω, whereas the resistance of the human heart is about 103Ω. A meter-long piece of large-diameter copper wire may have a resistance of 10−5Ω, and superconductors have no resistance at all (they are non-ohmic). Resistance is related to the shape of an object and the material of which it is composed, as will be seen in Resistance and Resistivity.

Additional insight is gained by solving I= *\(\frac{V}{R}\) *for V, yielding V=IR.

This expression for V can be interpreted as the *voltage drop across a resistor produced by the current *I. The phrase IR *drop* is often used for this voltage. For instance, the headlight in Example has an IR drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since PE=qΔV, and the same q flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See Figure.)

*The voltage drop across a resistor in a simple circuit equals the voltage output of the battery.*

### Making Connections: Conservation of Energy

In a simple electrical circuit, the sole resistor converts energy supplied by the source into another form. Conservation of energy is evidenced here by the fact that all of the energy supplied by the source is converted to another form by the resistor alone. We will find that conservation of energy has other important applications in circuits and is a powerful tool in circuit analysis.

## Resistors in Series and Parallel

Most circuits have more than one component, called a resistor that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections illustrated in Figure. The total resistance of a combination of resistors depends on both their individual values and how they are connected.

*Image shows (a) A series connection of resistors. (b) A parallel connection of resistors.*

When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.

## Joules Law

Power is associated by many people with electricity. Knowing that power is the rate of energy use or energy conversion, what is the expression for electric power? Power transmission lines might come to mind. We also think of lightbulbs in terms of their power ratings in watts. Let us compare a 25-W bulb with a 60-W bulb. (See Figure(a).) Since both operate on the same voltage, the 60-W bulb must draw more current to have a greater power rating. Thus the 60-W bulb’s resistance must be lower than that of a 25-W bulb. If we increase voltage, we also increase power. For example, when a 25-W bulb that is designed to operate on 120 V is connected to 240 V, it briefly glows very brightly and then burns out. Precisely how are voltage, current, and resistance related to electric power?

Electric energy depends on both the voltage involved and the charge moved. This is expressed most simply as PE = *qV*, where* q *is the charge moved and* V *is the voltage (or more precisely, the potential difference the charge moves through). Power is the rate at which energy is moved, and so electric power is *P = \(\frac{PE}{t}\) = \(\frac{qV}{t}\).*

Recognizing that current is *I = \(\frac{q}{t}\) (*note that Δt=t here), the expression for power becomes *P = IV.*

Electric power (*P*) is simply the product of current times voltage. Power has familiar units of watts. Since the SI unit for potential energy (PE) is the joule, power has units of joules per second, or watts. Thus, 1 A⋅V = 1 W. For example, cars often have one or more auxiliary power outlets with which you can charge a cell phone or other electronic devices. These outlets may be rated at 20 A, so that the circuit can deliver a maximum power* P* = *IV =* (20 A) (12 V) = 240 W. In some applications, electric power may be expressed as volt-amperes or even kilovolt-amperes 1 kA⋅V = 1 kW.

To see the relationship of power to resistance, we combine Ohm’s law with P = IV. Substituting *I = V/R* gives* P = (\(\frac{V}{R}\))V = V ^{2}/R*. Similarly, substituting

*V = IR*gives

*P = I(IR) = I*. Three expressions for electric power are listed together here for convenience:

^{2}R*P = IV*

*P = V ^{2}/R*

*P = I ^{2}R.*

Note that the first equation is always valid, whereas the other two can be used only for resistors. In a simple circuit, with one voltage source and a single resistor, the power supplied by the voltage source and that dissipated by the resistor are identical. (In more complicated circuits, *P* can be the power dissipated by a single device and not the total power in the circuit.)

Different insights can be gained from the three different expressions for electric power. For example,*P = V ^{2}/R*implies that the lower the resistance connected to a given voltage source, the greater the power delivered. Furthermore, since voltage is squared in

*P = V*, the effect of applying a higher voltage is perhaps greater than expected. Thus, when the voltage is doubled to a 25-W bulb, its power nearly quadruples to about 100 W, burning it out. If the bulb’s resistance remained constant, its power would be exactly 100 W, but at the higher temperature its resistance is higher, too.

^{2}/R

#### Joule’s Law