2.2 Ohm's Law

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2.2 Ohm’s Law

Materials in general ha ve a characteristic beha vior of resisting the flow of electric charge. This physical property, or ability to resist current, is known as resistance and is represented by the symbol R. The resistance of any material with a uniform cross-sectional area A depends on A and its length ℓ, as sho wn in Fig. 2.1(a). We can represent resistance (as measured in the laboratory), in mathematical form,

$$R = \rho \frac{\ell}{A} \tag{2.1}$$

where ρ is known as the resistivity of the material in ohm-meters. Good conductors, such as copper and aluminum, ha ve low resistivities, while insulators, such as mica and paper, have high resistivities. Table 2.1 presents the values of ρ for some common materials and shows which materials are used for conductors, insulators, and semiconductors.

The circuit element used to model the current-resisting beha vior of a material is the resistor. For the purpose of constructing circuits, resistors are

Figure 2.1 (a) Resistor, (b) Circuit symbol for resistance.

TABLE 2.1

Material	Resistivity (Ω∙m)	Usage
Silver	1.64 × 10−8	Conductor
Copper	1.72 × 10−8	Conductor
Aluminum	2.8 × 10−8	Conductor
Gold	2.45 × 10−8	Conductor
Carbon	4 × 10−5	Semiconductor
Germanium	47 × 10−2	Semiconductor
Silicon	6.4 × 102	Semiconductor
Paper	1010	Insulator
Mica	5 × 1011	Insulator
Glass	1012	Insulator
Teflon	3 × 1012	Insulator

Resistivities of common materials.

usually made from metallic alloys and carbon compounds. The circuit symbol for the resistor is shown in Fig. 2.1(b), where R stands for the resistance of the resistor. The resistor is the simplest passive element.

Georg Simon Ohm (1787–1854), a German ph ysicist, is credited with finding the relationship between current and voltage for a resistor. This relationship is known as Ohm’s law.

Ohm’s law states that the voltage v across a resistor is directly proportional to the current i flowing through the resistor.

That is,

$$v \propto i \tag{2.2}$$

Ohm defined the constant of proportionality for a resistor to be the resistance, R. (The resistance is a material property which can change if the internal or external conditions of the element are altered, e.g., if there are changes in the temperature.) Thus, Eq. (2.2) becomes

$$v = iR \tag{2.3}$$

Historical

Georg Simon Ohm (1787–1854), a German physicist, in 1826 experimentally determined the most basic law relating voltage and cur rent for a resistor. Ohm’s work was initially denied by critics.

Born of humble beginnings in Erlangen, Bavaria, Ohm threw himself into electrical research. His efforts resulted in his famous law. He was awarded the Copley Medal in 1841 by the Royal Society of London. In 1849, he was given the Professor of Physics chair by the University of Munich. To honor him, the unit of resistance was named the ohm.

(b) Figure 2.2

(a) Short circuit (R =0), (b) Open circuit (R =∞).

Figure 2.3 Fixed resistors: (a) wirewound type, (b) carbon film type. © McGraw-Hill Education/Mark Dierker, photographer

Figure 2.4 Circuit symbol for: (a) a variable resistor in general, (b) a potentiometer.

which is the mathematical form of Ohm’s law. R in Eq. (2.3) is mea sured in the unit of ohms, designated Ω. Thus,

The resistance R of an element denotes its ability to resist the flow of electric current; it is measured in ohms (Ω).

We may deduce from Eq. (2.3) that

$$R = \frac{v}{i} \tag{2.4}$$

so that

$$1 \Omega = 1 \text{ V/A}$$

To apply Ohm’ s la w as stated in Eq. (2.3), we must pay careful attention to the current direction and v oltage polarity. The direction of current i and the polarity of voltage v must conform with the passive sign convention, as shown in Fig. 2.1(b). This implies that current flows from a higher potential to a lower potential in order for v = i R. If current flows from a lower potential to a higher potential, v = −i R.

Since the value of R can range from zero to infinity, it is important that we consider the tw o extreme possible values of R. An element with R = 0 is called a short circuit, as shown in Fig. 2.2(a). For a short circuit,

$$v = iR = 0 \tag{2.5}$$

showing that the v oltage is zero b ut the current could be an ything. In practice, a short circuit is usually a connecting wire assumed to be a perfect conductor. Thus,

A short circuit is a circuit element with resistance approaching zero.

Similarly, an element with R =∞ is known as an open circuit, as shown in Fig. 2.2(b). For an open circuit,

ircuit,
\n

$$i = \lim_{R \to \infty} \frac{v}{R} = 0$$

\n(2.6)

indicating that the current is zero though the v oltage could be anything. Thus,

An open circuit is a circuit element with resistance approaching infinity.

A resistor is either fixed or v ariable. Most resistors are of the fixed type, meaning their resistance remains constant. The two common types of fixed resistors (wirewound and composition) are shown in Fig. 2.3. The composition resistors are used when large resistance is needed. The circuit symbol in Fig. 2.1(b) is for a fixed resistor. Variable resistors have adjustable resistance. The symbol for a variable resistor is shown in Fig. 2.4(a). A common variable resistor is known as a potentiometer or pot for short, with the symbol shown in Fig. 2.4(b). The pot is a three-terminal element with a sliding contact or wiper . By sliding the wiper , the resistances be tween the wiper terminal and the fixed terminals v ary. Like fixed resistors, variable resistors can be of either wire wound or composition type, as shown in Fig. 2.5. Although resistors like those in Figs. 2.3 and 2.5 are used in circuit designs, today most circuit components including resistors are either surface mounted or integrated, as typically shown in Fig. 2.6.

Figure 2.5 Variable resistors: (a) composition type, (b) slider pot. © McGraw-Hill Education/Mark Dierker, photographer

It should be pointed out that not all resistors obe y Ohm’s law. A resistor that obeys Ohm’s law is known as a linear resistor. It has a constant resistance and thus its current-voltage characteristic is as illustrated in Fig. 2.7(a): Its i-v graph is a straight line passing through the ori gin. A nonlinear resistor does not obe y Ohm’s law. Its resistance varies with current and its i-v characteristic is typically sho wn in Fig. 2.7(b). Examples of devices with nonlinear resistance are the light bulb and the diode. Although all practical resistors may e xhibit nonlinear beha vior under certain conditions, we will assume in this book that all elements actually designated as resistors are linear.

A useful quantity in circuit analysis is the reciprocal of resistance R, known as conductance and denoted by G:

$$G = \frac{1}{R} = \frac{i}{v} \tag{2.7}$$

The conductance is a measure of how well an element will conduct electric current. The unit of conductance is the mho (ohm spelled backward) or reciprocal ohm, with symbol ℧, the inverted omega. Although engineers often use the mho, in this book we prefer to use the siemens (S), the SI unit of conductance:

$$1 S = 1 \, \mathbf{U} = 1 \, \mathbf{A} / \mathbf{V} \tag{2.8}$$

Thus,

Conductance is the ability of an element to conduct electric current; it is measured in mhos (℧) or siemens (S).

The same resistance can be e xpressed in ohms or siemens. F or example, 10 Ω is the same as 0.1 S. From Eq. (2.7), we may write

$$i = Gv \tag{2.9}$$

The power dissipated by a resistor can be e xpressed in terms of R. Using Eqs. (1.7) and (2.3),

$$p = vi = i^{2}R = \frac{v^{2}}{R}$$

(2.10)

Figure 2.6 Resistors in an integrated circuit board.

Figure 2.7 The i-v characteristic of: (a) a linear resistor, (b) a nonlinear resistor.

The power dissipated by a resistor may also be e xpressed in terms of G as

$$p = vi = v^2 G = \frac{i^2}{G}$$

(2.11)

We should note two things from Eqs. (2.10) and (2.11):

• 1. The power dissipated in a resistor is a nonlinear function of either current or voltage.
• 2. Since R and G are positive quantities, the power dissipated in a resistor is al ways positive. Thus, a resistor al ways absorbs power from the circuit. This confirms the idea that a resistor is a passive element, incapable of generating energy.

Solution:

The voltage across the resistor is the same as the source voltage (30 V) because the resistor and the voltage source are connected to the same pair of terminals. Hence, the current is

$$i = \frac{v}{R} = \frac{30}{5 \times 10^3} = 6 \text{ mA}$$

The conductance is

$$G = \frac{1}{R} = \frac{1}{5 \times 10^3} = 0.2 \text{ mS}$$

We can calculate the power in various ways using either Eqs. (1.7), (2.10), or (2.11).

p = vi = 30 (6 × 10 ⁻3) = 180 mW

or

$$p = i^2 R = (6 \times 10^{-3})^2 5 \times 10^3 = 180$$

mW

or

$$p = v^2 G = (30)^2 0.2 \times 10^{-3} = 180$$

mW

Figure 2.8 For Example 2.2.

For the circuit shown in Fig. 2.9, calculate the voltage v, the conductance G, and the power p.

Answer: 30 V, 100 µS, 90 mW.

For Practice Prob. 2.2

A voltage source of 20 sin πt V is connected across a 5-kΩ resistor. Find Example 2.3 the current through the resistor and the power dissipated.

Solution:

$$i = \frac{v}{R} = \frac{20 \sin \pi t}{5 \times 10^3} = 4 \sin \pi t \text{ mA}$$

Hence,

$$p = vi = 80 \sin^2 \pi t \text{ mW}$$

A resistor absorbs an instantaneous power of 30 cos Practice Problem 2.3 ² t mW when con nected to a voltage source v = 15 cos t V. Find i and R.

Answer: 2 cos t mA, 7.5 kΩ.