1.3 Charge and Current

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1.3 Charge and Current

The concept of electric charge is the underlying principle for explaining all electrical phenomena. Also, the most basic quantity in an electric circuit is the electric charge. We all experience the effect of electric charge when we try to remove our wool sweater and have it stick to our body or walk across a carpet and receive a shock.

Charge is an electrical property of the atomic particles of which matter consists, measured in coulombs (C).

We know from elementary physics that all matter is made of fundamental building blocks known as atoms and that each atom consists of electrons, protons, and neutrons. We also know that the char ge e on an electron is negative and equal in magnitude to 1.602 × 10⁻19 C, while a proton carries a positi ve charge of the same magnitude as the electron. The presence of equal numbers of protons and electrons leaves an atom neutrally charged.

The following points should be noted about electric charge:

• 1. The coulomb is a large unit for charges. In 1 C of charge, there are 1∕(1.602 × 10⁻19) = 6.24 × 1018 electrons. Thus realistic or laboratory values of charges are on the order of pC, nC, or μC.1
• 2. According to e xperimental observ ations, the only char ges that occur in nature are inte gral multiples of the electronic char ge e = −1.602 × 10⁻19 C.
• 3. The law of conservation of charge states that charge can neither be created nor destroyed, only transferred. Thus, the algebraic sum of the electric charges in a system does not change.

We now consider the flow of electric char ges. A unique feature of electric charge or electricity is the f act that it is mobile; that is, it can be transferred from one place to another , where it can be con verted to another form of energy.

When a conducting wire (consisting of se veral atoms) is connected to a battery (a source of electromotive force), the charges are compelled to move; positive charges move in one direction while ne gative charges move in the opposite direction. This motion of char ges creates elec tric current. It is conventional to take the current flow as the movement of positive charges. That is, opposite to the flow of ne gative charges, as Fig. 1.3 illustrates. This con vention w as introduced by Benjamin Franklin (1706–1790), the American scientist and in ventor. Although we now know that current in metallic conductors is due to ne gatively charged electrons, we will follo w the uni versally accepted con vention that current is the net flow of positive charges. Thus,

Electric current is the time rate of change of charge, measured in amperes (A).

Mathematically, the relationship between current i, charge q, and time t is

$$i \triangleq \frac{dq}{dt} \tag{1.1}$$

I – + – –– –

Figure 1.3

Electric current due to flow of electronic charge in a conductor.

A convention is a standard way of describing something so that others in the profession can understand what we mean. We will be using IEEE conventions throughout this book.

1 However, a large power supply capacitor can store up to 0.5 C of charge.

Historical

Andre-Marie Ampere (1775–1836), a French mathematician and physicist, laid the foundation of electrodynamics. He defined the electric current and developed a way to measure it in the 1820s.

Born in Lyons, France, Ampere at age 12 mastered Latin in a few weeks, as he was intensely interested in mathematics and many of the best mathematical works were in Latin. He was a brilliant scientist and a prolific writer. He formulated the laws of electromagnetics. He in vented the electromagnet and the ammeter. The unit of electric current, the ampere, was named after him.

© Apic/Getty Images

where current is measured in amperes (A), and

1 ampere = 1 coulomb/second

The charge transferred between time t0 and t is obtained by inte grating both sides of Eq. (1.1). We obtain

$$Q \triangleq \int_{t_0}^t i \, dt \tag{1.2}$$

The way we define current as i in Eq. (1.1) suggests that current need not be a constant-valued function. As many of the examples and problems in this chapter and subsequent chapters suggest, there can be se veral types of current; that is, charge can vary with time in several ways.

There are different ways of looking at direct current and alternating current. The best definition is that there are two ways that current can flow: It can always flow in the same direction, where it does not reverse direction, in which case we have direct current (dc). These currents can be constant or time varying. If the current flows in both directions, then we have alternating current (ac).

A direct current (dc) flows only in one direction and can be constant or time varying.

By convention, we will use the symbol I to represent a constant current. If the current v aries with respect to time (either dc or ac) we will use the symbol i. A common use of this w ould be the output of a rectifier (dc) such as i(t) = ∣5 sin(377t)∣ amps or a sinusoidal current (ac) such as i(t) = 160 sin(377t) amps.

An alternating current (ac) is a current that changes direction with respect to time.

An example of alternating current (ac) is the current you use in your house to run the air conditioner , refrigerator , w ashing machine, and other electric appliances. Figure 1.4 depicts tw o common e xamples of

Figure 1.4

Two common types of current: (a) direct current (dc), (b) alternating current (ac).

Figure 1.5 Conventional current flow: (a) positive current flow, (b) negative current flow.

dc (coming from a battery) and ac (coming from your home outlets). We will consider other types later in the book.

Once we define current as the movement of charge, we expect current to have an associated direction of flow. As mentioned earlier, the direction of current flow is conventionally taken as the direction of positive charge movement. Based on this convention, a current of 5 A may be represented positively or negatively as shown in Fig. 1.5. In other w ords, a negative current of −5 A flowing in one direction as shown in Fig. 1.5(b) is the same as a current of +5 A flowing in the opposite direction.

Example 1.1	How much charge is represented by 4,600 electrons?
	Solution: −19 C. Hence 4,600 electrons will Each electron has −1.602 × 10 have −1.602 × 10−19 C/electron × 4,600 electrons = −7.369 × 10−16 C
Practice Problem 1.1	Calculate the amount of charge represented by 6.667 billion protons.
	Answer: 1.0681 × 10−9 C.
Example 1.2	The total char ge entering a terminal is gi ven by q = 5 t sin 4 πt mC. Calculate the current at t = 0.5 s.
	Solution:
	dq___ __d i = dt = dt (5t sin 4πt) mC/s = (5 sin 4πt + 20πt cos 4πt) mA
	At t = 0.5,
	i = 5 sin 2π + 10π cos 2π = 0 + 10π = 31.42 mA
Practice Problem 1.2	= (10 − 10e−2t If in Example 1.2, q ) mC, find the current at t = 1.0 s.
	Answer: 2.707 mA.
Example 1.3	Determine the total charge entering a terminal between t = 1 s and 2 − t = 2 s if the current passing the terminal is i = (3t t) A.

Solution:

$$Q = \int_{t=1}^{2} i \, dt = \int_{1}^{2} (3t^2 - t) \, dt$$ $$= \left(t^3 - \frac{t^2}{2}\right)\Big|_{1}^{2} = (8 - 2) - \left(1 - \frac{1}{2}\right) = 5.5 \text{ C}$$

ough an element is

\n

$$i = \begin{cases} 4 \text{ A}, & 0 < t < 1 \\ 4t^2 \text{ A}, & t > 1 \end{cases}$$

Calculate the charge entering the element from t = 0 to t = 2 s.

Answer: 13.333 C.