ELECTRODYNAMICS

Electrostatic Discharge. In 1731, Stephen Gray communicated to the Royal Society, through one its members, a striking discovery that he had made while experimenting with a electrified glass tube. He had found, as others had before him, that if the glass tube is rubbed in the dark, tiny sparks are seen to pass between the glass tube and a finger held close to it. He reasoned that if the rubbed glass tube communicates sparks to a nearby object, perhaps it is at the same time communicating the "electric virtue" to the object. To test this conjecture, he procured a glass tube about 3 feet long and over an inch in diameter. He put a cork in each end of it "to keep the dust out." But being careful experimenter, he decided to check whether the corks would interfere with the operation of the tube. Upon rubbing the tube he found that it would attract a nearby feather just as well when stopped at both ends as when left open. Holding the feather near one end of the tube, he saw it approach and touch the cork and then be repelled, just as it had at the rubbed glass surface. Here was a new phenomenon; the cork had not been rubbed, but nevertheless it attracted the feather and hence was electrified. Seemingly, Gray concluded, the "attractive virtue" had been communicated to the cork, presumably because it was in contact with the electrified tube. He abandoned his original conjecture, and turned all his efforts to exploit his discovery. Continuing his experiments he replaced the cork with an ivory ball and suspended it from the glass tube by a string. To be able to use longer strings he stood on furniture and balconies. He succeeded in transmitting the "electric virtue" a distance of 34 feet along the vertical string. Gray decided to use a horizontal string tied at one end to the glass tube and an ivory ball at the other; the string was suspended near the far end by a piece of vertical cord looped over a nail in a ceiling beam. But the experiment failed completely. Gray concluded that the "electric virtue," when it reached the end of string, went up the cord to the beam. A friend suggested supporting the string along which the virtue moved by a thin silk cord and the plan worked. Gray thought that the smallness (thinness) of the supporting silk cords had prevented the "electric virtue" from being diverted from the transmission line. In order to increase the transmission distance the experimenters moved outdoors and succeeded transmitting the "electric virtue" a distance of 293 feet. When still longer distances were attempted, the silk supporting cords broke under the weight of the transmitting string. Gray replaced the silk cords with brass wires and the experiment failed completely. Gray realized he had been in error about the reason of the earlier failure; it was not the thinness of the supports but the kind of material of which they were made. So when the experimenters tried the silk supports again, but using many more of them fastened to a series of poles driven into the ground, they succeeded in transmitting the "electric virtue" a distance of 650 feet and even longer. Gray concluded that the silk was a non-conductor of the electric virtue, whereas the brass was a conductor of it.

Electrostatic discharges had been observed before Gray did; but it was not understood. It had been observed that when one attempted to build up a very large charge on an object, the charge tended to "leak off," especially at sharp points. Sailors since ancient times observed during storms at the points of masts and end of spars a glow that was called "St. Elmo's fire." But this phenomena was not understood as electrostatic discharge until Benjamin Franklin (1706-1790) performed his famous kite experiment. He recognized that lightning and other atmospheric electricity had properties identical to those of electricity produced by rubbing. And this recognition was the basis for his famous invention of the lightning rod. Having noticed in his experiments that electricity tends to discharge at points, Franklin concluded that a pointed rod would "probably draw the electrical fire silently out of a cloud before it came nigh enough to strike, and thereby secure us from the most sudden and terrible mischief."

Two technical developments during the eighteenth century helped to understand electrostatic discharge. One was the electrostatic generator, a device which produces an electric charge by rubbing an insulating material like glass mechanically instead of by hand. This made it possible to generate static electricity continuously. Sixty years after the publication of Gilbert's work the burgomaster of the German city of Magdeburg, Otto von Guericke (1602-1686), constructed a powerful electrostatic generator. He poured molten sulfur into a glass globe, let is solidify, and then broke the globe. The ball of sulfur, mounted on an axle, could be rotated by means of a crank. The rotating ball was electrified by rubbing it with a cloth. In later versions of the machine, the ball was rotated faster by a system pulleys. Von Guericke's machine was widely copied and improved. The sulfur sphere was replaced by the glass globe fitted with attachments to rub effectively the globe and to conduct the charge from the globe to other objects. In the eighteenth century electrostatic generators, like von Guericke's, became popular drawing room entertainment. Audiences were astounded by the spectacular maelstrom of feathers, paper, and whatever swirling about the electrified sphere. The air was filled with sparks and the smell of ozone, and those approaching the spinning globe felt their skin tingle. Electrostatic generators were also used for serious scientific investigation. Von Guericke, himself, noted that objects were attracted to the electrified globe until they came in contact with it, and then they were immediately repelled.

The other development was the "Leyden jar," that resulted from the discovery that a charge can be held, or "condensed" on a conductor by the presence of another, oppositely charged conductor. Consider two parallel metal plates separated by dry air, glass, or another good insulator. When these plates are oppositely charged, a large amount of charge can be "condensed" on the plates, because the opposite charges attract each other and hold each other in place. A device such as this is called an electric condenser or capacitor. The Leyden jar, invented by the Dutch mathematician, Professor Pieter van Musschenbroek (1692-1761) of the University of Leyden in Holland, in 1746, consisted of a thin-walled glass jar covered inside and out with a conducting material, such as metal foil, the inside and outside foils not connected together. The inside conductor is connected by a small chain to a rod through the stopper in the mouth of the jar. When a charge was place on inside conductor by contact at the end of the rod with a body charged by rubbing, the outside conductor also becomes charged. Sufficient charge can be accumulated, especially with the use of a mechanical electrostatic generator, to produce a severe electric shock, or a large visible electric spark. Franklin introduced the parallel plate condenser, that was an improvement on the Leyden jar.

The visible electric spark that passed between the two charged plates of the condenser so that they are no longer charged is called an electrostatic discharge. A condenser may also be discharged without generating a spark, by simply connecting the two plates with a metallic wire. During the process of discharge, the charge moves, or "flows," from one plate to other through the wire. Any movement of electric charge constitutes what is called an electric current. But this electric current during the discharge of the condenser is very brief and occurs during in an almost infinitesimally short time. This hindered the study of electric currents until it became possible to generate a steady and continuous flow of electric current.

Dynamic Electricity. The principle of the electric battery was discovered by an Italian anatomy professor at the University of Bologna, Luigi Galvani (1737-1798), who noticed by chance in 1786 that in certain circumstances convulsions were observed in frog's legs placed near an electric spark. On further investigation, he found that the electric spark was unnecessary; he found that the violent convulsions occurred when the frog was suspended from a brass hook into the spinal marrow and a rod made of a different metal was brought in simultaneous contact with the hook and frog's leg muscle. Galvani thought that the spasms was a manifestation of what was later called "animal electricity" or "galvanism." Galvani himself considered it to be the same as ordinary static electricity; others believed that galvanism to be a fluid different from ordinary electricity. It was Galvani's countryman, Alessandro Volta (1745-1827), Professor of Natural Philosophy in the University of Pavia, who in 1792 showed that animal nerve tissue served to detect, not to initiate, the electric effect that Galvani had observed. Later in 1793, Volta put forth the view that a steady electric current could be produced by connecting two dissimilar metals, such as copper and zinc, separated by any moist body, not necessarily organic. In 1800, Volta found that if he made a pile of metal disks, alternatively copper and zinc, separated by layers of a moist substance such as a moistened cloth or blotting paper soaked with a water solution of acid, lye or salt (so that the order was copper, zinc, paper, copper, zinc, paper, etc.), the effect of the pile was greater than any galvanic device. Touching the two ends of the voltaic pile, as it later was called, with moistened fingers would produce a strong continuous sensation. Furthermore, sparks could be drawn if the two ends of the pile were touched by a metal wire. When the two ends of the pile are connected by a wire, electric charge started to flow along the wire, and this flow of charge did not cease as long as chemical reaction continued between the moist substance and the two dissimilar metals. Later Volta made a device, called a battery, that consisted of a series of cups, each filled with brine and containing two separated strips of dissimilar metals, zinc and copper. The zinc strip in one cup is connected by a wire to the copper strip in another cup. When the connection was made between the copper strip in the first cup and the zinc strip in the last cup of the series, the effects of an electric current sparks and shocks, for example were observed. Thirty or more cups, arranged in the way, were required to produce the same kind of electric shock as that given by a Leyden jar or electrostatic generator. The flow of charge is continuous in Volta's battery in contrast to the instantaneous and intermittent discharges of these electrostatic devices. Each cup of the device was later called a cell of the battery.

Electric Circuit. In a single cell of Voltaic battery, the copper strip or terminal is positive with respect to the zinc terminal and the zinc terminal is negative respect to the copper terminal, and the current in the external wire can be considered to consist of a flow of positive charge from positive to negative, from the copper to the zinc terminal. The cell has the ability by the chemical reaction between the metals and the solution to impel a continuous flow of charge in the wire without altering the quantity of charge on either terminal. For this to happen, the charge must be considered to flow through the salt solution, and the cell and the wire connecting the terminals constitutes a cyclic path for electric current. Any such cyclic conducting path is called an electric circuit. The electric charge flows as long as the circuit is closed, that is, as long as the circuit is completed.

When a continuous and steady current exists in a electric circuit, the amount of electric charge flowing pass is the same at all points of its circuit; that is, electric charge is not lost and does not accumulate anywhere in the circuit. In other words, electric charge is conserved. Quantitatively this can be expressed by defining a physical quantity called electric current. Current may be defined as the amount of electric charge flowing pass a given point in a conductor per unit of time. That is,
current = charge/elapsed time or
I = Δq/Δt, (1)
where Δq is the quantity of electric charge measured in coulombs, Δt is the elapsed time measured in seconds that the quantity of charge Δq flows pass a given point in the conductor, and I is the symbol for the time rate of flow of the electric charge pass a given point in the conductor and is called electric current. According to this defining formula (1), the unit of electric current is coulombs per second, which is named the ampere. Several kinds of instruments have been invented to measure electric current, called ammeters (ampere measurers).

Electromotive Force. The flow of the electric charge must be caused by something in the circuit. The charge will not flow around the circuit of itself. Now if the battery is not present in the circuit there can be no current. What is it that the battery supplies? To arrive at the answer to this problem let us use an analogy. Just as a body, when free to move, will fall from a higher point to a lower point under the force of gravity, so the electric charge will move between two points, from the positive to the negative terminal of the battery under the force of an electric field. A body falls between the two points in a gravitational field because of .a difference of potential, meaning that there is difference of potential energy between the two points in the gravitational field. Similarly, a different of potential, that is, a difference of potential energy, between the two terminals of the battery. This difference of potential between the two terminals of the battery is called electromotive force (EMF) of the battery. That is, an electric battery is a source of electric potential energy. Since potential energy is the ability to do work by a field force, electric potential energy is the ability to do work by the forces of the electric field. In general, the electric potential difference between two points is defined as the quantity of work done by electric forces in moving a unit of charge from one point to the other, that is,
potential difference = work/charge
or, in symbols
V_ba = W_b→a/q. (2)
If the work is measured in joules and the charge in coulombs, then the unit of potential difference is one joule per coulomb, which is named one volt. This difference of potential is often called simply voltage. A device has been invented to measure the potential difference between two points in an electric circuit and it is called a voltmeter.

Each cell in Volta's battery has a certain small voltage across its terminals and as more and more identical cells are placed in the series, the battery becomes capable of producing greater currents in an external circuit. The essentials of an electric circuit are a source of potential difference, a connection between the terminals of the source and a "load" in which energy is expended, for example, the filament of a flashlight bulb or the heating elements of toaster. When the load in the circuit is by-passed by a connection of some kind between the terminals of the load, the circuit is said to be "short circuited," and is usually accompanied by sparking and intense heating. A switch in the connection between the source and load allows the circuit to be completed or broken. As long as the switch is open, no current can flow in the circuit and no work can be done in the load; when the switch is closed the potential difference produces a current in the entire circuit. If the load in the circuit is, for example, the filament of a light bulb, it has properties such that some of the work done by the charge moving through it is converted to heat but most of it is converted into light. The time rate that the electric energy is expended and converted into heat and light in the load may be calculated by multiplying the voltage (work per unit charge) by the current (the time rate of flow of charge) and is called electric power. In general, power is the time rate at which work is done. That is,
Power = work done/elapsed time or
P = ΔW/Δt, (3)
In the MKS system of units power is measured in watts and is equal to joules per second, since the work done ΔW is measured in joules and the elapsed time Δt in seconds. If both the potential difference across the load and the current flowing in the load is known, then the electric power can be calculated by the following formula, which is sometimes called Joule's Law. Since
VI = ΔW/Δq)(Δq/Δt) = ΔW/Δt = P,
then P = VI. (4)
The total electric energy expended and the work done in the load may be calculated by multiplying the electric power of the load by the length of time that the switch is closed. That is,
ΔW = PΔt, (5)
where electrical energy is measured in joules or more usually watt-sec, since electric power is measured in watts and time in seconds. Since this is a very small unit of electrical energy for commercial and practical use, a larger unit called the kilowatt-hour is commonly used. 1 kilowatt-hour = 3,600,000 watt-sec.

Electrical Resistance. The relationship between voltage and current in different conductors was first studied in the 1820's by Georg Simon Ohm (1787-1854), a German schoolmaster, who published his work in 1826. The ratio of the potential difference across the ends of a conductor to the current is called the resistance of the conductor. Ohm experimented with wires, which he made by hand, of different sizes and lengths and of different metals. He found that the resistance R of the wire varied directly with the length L and inversely as the cross sectional area A of the wire. That is,
R = ρ(L/A), (6)
where R is measured in ohms Ω (the capital Greek letter omega), the length L in meters, and the cross-sectional area A in square meters. The value of the constant of proportionality that is called resistivity ρ (the Greek letter rho) dependents upon the kind of metal as well as the system of units to measure the length and cross-sectional area of the conductor. The resistivity is measured in ohm·meters, as can be seen by solving formula (6) for ρ.
ρ = RA/L = ohms·meter²/meters = ohms·meters. (7)

The following table shows the approximate resistivities of some metals and non-metals at 0°C.

Substance (metals)	Resistivity (Ω·m)	Substance (non-metals)	Resistivity (Ω·m)
silver	1.47 × 10-8	carbon	4 × 10-5
copper	1.59 × 10-8	germanium	2 × 100
gold	2.27 × 10-8	silicon	3 × 104
aluminum	2.60 × 10-8	boron	1 × 106
tungsten	5.00 × 10-8	wood (maple)	3 × 108
iron	11.0 × 10-8	celluloid	4 × 1012
platinum	11.0 × 10-8	glass	1011 × 1013
constantan (60 Cu, 40 Ni)	49 × 10-8	amber	5 × 1014
mercury	94 × 10-8	sulfur	1 × 1015
Nichrome (60 Ni, 24 Fe, 16 Cr)	100 × 10-8	mica (colorless)	2 × 1015
		fused quartz	5 × 1017

Ohm's Law. Ohm's Law states that for given conductor the current in the conductor is directly proportional to the voltage across the ends of the conductor and inversely proportional to its resistance. In symbols,
I = V/R, (8)
where the voltage V is measured in volts, the current I is measured in amperes, and the resistance R is measured in ohms. The unit of electrical resistance, the ohm, is defined as that resistance which will allow a current of one ampere to flow through a conductor when the potential difference across its ends is one volt.

Joule's Law. The English physicist, James Prescott Joule (1818-1889), in his determination of the mechanical equivalence of heat, investigated the relationship between the heating effect and the current flowing in the conductor. He found that the rate of production of heat energy in a metallic conductor is directly proportional to the square of the current. Now the rate of production of energy is power, P, and is measured in the MKS system of units in joules per second, that is called the watt. Thus Joule's Law states that the power expended in a conductor is directly proportion to the square of the current flowing in it.
P = I²R. (9)
The constant proportionality is the resistance R of the conduction and is measured in the MKS units in ohms when the current I in the conductor is measured in amperes.
Since by formula (4) P = VI and, solving formula (8) for voltage V, V = IR, then
P = VI = IV = I(IR) = I²R.
Therefore, formula (4) is equivalent to formula (9), where the voltage V is the potential difference across the load in the circuit.

Series Resistance. If the load in a circuit consists of a series of resistance elements, like for example a string of series connected Christmas tree lights, then the potential difference across the entire string is equal to the sum of potential differences across each resistance element. That is,
V = V₁ + V₂ + V₃ + ..., (10)
where V is the voltage across the whole series, and V₁, V₂, V₃ and so forth, are the voltages across each resistance elements. Since by Ohm's Law,
V = IR, then V₁ = I₁R₁, V₂ = I₂R₂, V₃ = I₃R₃, and so forth.
Substituting into the equation (10), we get
IR = I₁R₁ + I₂R₂ + I₃R₃ + ....
But since the current in a series circuit is the same in each of the series element, that is,
I = I₁ = I₂ = I₃ = ..., then IR = IR₁ + IR₂ + IR₃ + ....
Therefore, dividing by the common current, we get
R = R₁ + R₂ + R₃ + .... (11)
That is, the total resistance of series of resistance elements is equal to the sum of the resistances of each element.

Parallel Resistance. If the load in a circuit consists of resistance elements connected in parallel, then the sum of the current in the branches is equal to the total current entering the parallel elements. That is,
I = I₁ + I₂ + I₃ + ..., (12)
where I is the total current entering the parallel arrangement, and I₁, I₂, I₃, and so forth, are the currents flowing in each of the branches of the parallel arrangement. Since by Ohm's Law,
I = V/R, then I₁ = V₁/R₁, I₂ = V₂/R₂, I₃ = V₃/R₃, and so forth.
Substituting into equation (12), we get
V/R = V₁/R₁ + V₂/R₂ + V₃/R₃ + ....
But since the voltage across the parallel arrangement of resistance elements is equal to the voltage across each of the branch elements, that is, V = V₁ = V₂ = V₃ = ..., then
V/R = V/R₁ + V/R₂ + V/R₃ + ....
Therefore, dividing by the common voltage V, we get
1/R = 1/R₁ + 1/R₂ + 1/R₃ + .... (13)
That is, the reciprocal of the total resistance of a parallel arrangement of resistances is equal to the sum of the reciprocals of the resistance of each element that are in parallel.

The Dry Cell. An ordinary dry cell consists essentially of a carbon rod and a zinc can separated by a moist paste containing ammonium chloride and manganese dioxide, which can be considered an electrolyte solution. When the dry cell is on the shelf with nothing connected to it, experiment shows that there is slight excess of positive charge on the carbon rod and a slight excess negative charge on the zinc can. To move a charge of one coulomb from the positive to the negative terminal through the air requires 1.5 joules of work; hence, the potential difference between the terminals of the dry cell is 1.5 joules per coulomb or 1.5 volts.

Capacitance. If the terminals of the dry cell are connected by wires to a pair of parallel plates, the plates become charged in a small fraction of second and then a potential difference of 1.5 volts appears between the two plates. This device consisting of parallel plates separated by air or some insulating material is a simple form a capacitor. Each of the two plates has an electric charge Q, but of opposite sign. This charge Q is directly proportional to the impressed potential difference V. That is,
Q = CV, (14)
where the constant of proportionality C is called the capacitance of the capacitor. The symbol for capacitance is C and solving formula (14) for the capacitance, we find that the capacitance of a capacitor is the ratio of its charge to the potential difference across its plates. That is,
C = Q/V. (15)
The unit of capacitance is the farad, abbreviated with letter f, and is equal to a coulomb per volt. The farad is so large that, for practical purposes, capacitance is usually measured in microfarads (μf) or micromicrofarads (μμf) that is also called a picofarads (pf), whose values are
1 μf = 10^-6, and 1 μμf = 1 pf = 10^-12 f.
In general a capacitor is a device that stores electrical potential energy in the form of an electric field. We can calculate the electric potential energy U of a charged capacitor by computing the amount of work W that must be performed to charge the capacitor to an electric potential V. Using the definition of electric potential as the instantaneous rate that work is done per unit of charge, that is,
V = dW/dq, (16)
and, solving equation (16) for dW, the work done in charging the capacitor from q = 0 to q = Q is found by integrating the differential equation
dW = Vdq = (q/C)(dq) = qdq/C.
That is,
U = W_0→Q = ∫₀^QdW = ∫₀^QVdq = ∫₀^Qqdq/C = ½Q²/C,
since at any time when the charge q is being transferred, the potential difference V between the plates is
V = q/C.
Thus, the electric potential energy U of a capacitor of capacitance C whose plates have the charge Q is
U = ½(Q²/C), (17)
or, using formula (14), in terms of capacitance and voltage,
U = ½[(CV)²/C] = ½CV², (18)
or in terms of charge and voltage,
U = ½[(CV)V] = ½QV, (19)
where the capacitor has electric potential energy U because work was done to charge the capacitor. This potential electric energy is stored in the electric field between the plates of the capacitor and the density of this potential electric energy can be calculated. Energy density u is defined as the energy per unit volume. For a parallel plate capacitor (with small plate separation and large plate area), since the capacitance
C = εA/s
and the potential difference across its plates
V = Es,
the electric potential energy is
U = ½CV² = ½(εA/s)(Es)² = ½εE²(As),
where the factor As is the volume occupied by the electric field between the plates of the capacitor. Thus the electric energy density is
u = U/As = [½εE²(As)]/(As) = ½εE². (20)
That is, the electric energy density of an electric field is directly proportionality to the square of the electric field intensity. While this relation is derived for the specific case of a parallel plate capacitor, it is a completely general result.

Parallel Capacitance. If two capacitors are connected in parallel, then their total capacitance is
C = Q/V = (Q₁ + Q₂)/V = Q₁/V + Q₂/V = C₁ + C₂. Thus,
C = C₁ + C₂. (21)
That is, the total capacitance of a parallel arrangement of capacitance elements is equal to the sum of the capacitances of each element.

Series Capacitance. If two capacitors are connected in series, then the total capacitance is
Q/C = V = V₁ + V₂ = Q₁/C₁ + Q₂/C₂.
But since Q = Q₁ = Q₂, then
1/C = 1/C₁ + 1/C₂. (22)
That is, the reciprocal of the total capacitance of a series arrangement of the capacitance elements is equal to the sum of the reciprocals of the capacitance of each element in the series.

Charging a Capacitor. Now consider a circuit consisting of a capacitor with a capacitance C in series with a resistance R connected through a switch to a battery with an EMF of V volts. Assume that the capacitor is initially uncharged (the charge on the two plates are equal) and there is no current flowing in the circuit when the switch is open. When the switch is closed at time t = 0, charges begin to flow, setting up a current in the circuit and the capacitor begins to charge. Since the plates of the capacitor are separated by an insulator of some kind so that no current flows between the plates. As the charge builds up on the plate connected to the positive terminal of the battery, the charge on the other plate of the capacitor connected to the negative terminal of the battery is repelled as the current flows toward the negative terminal of the battery. This continues until the capacitor is fully charged. The value of the maximum charge depends upon the EMF of the battery. Once the maximum charge is reached, the current in circuit is zero.

To analyze quantitatively what happens after the switch is closed, let us start with the fact that the EMF voltage of the battery is equal to the sum of potential differences across the capacitor
V_C = q/C and across the resistor
V_R = IR. That is,
V = V_R + V_C = IR + q/C. (23)
Note that the charge and current are variables whose instantaneous values change as the capacitor charges. At time t = 0, when the switch closed, the charge on the capacitor is zero, and the initial value of current I₀ is a maximum and is equal to I₀ = V/R. At this time the applied voltage is entirely across the resistor. Later, when the capacitor is fully charged to its maximum value Q, the charges cease to flow (the current is zero), and the applied voltage is entirely across the capacitor. The charge Q on the capacitor is then equal CV. To determine an analytical expression for the time dependence of the charge and current, the equation (23) must be differentiation with respect to time.
dV/dt = R(dI/dt) + (1/C)(dq/dt),
since I = dq/dt.
Since the EMF voltage is zero, then dV/dt = 0. Thus
0 = R(dI/dt) + (1/C)(dq/dt) or
R(dI/dt) = -(1/C)I,
Separating the variables and rearranging the equation, we get
dI/I = (1/RC)dt.
Since R and C are constants, this can be integrated using the initial conditions that at t = 0,
I = I₀. Integrating, we get
ln(I/I₀) = -t/RC. (24)
Solving this equation for I, we get a formula for current as function of time.
I(t) = I₀e^-t/RC = (V/R)e^-t/RC, (25)
where e is the natural logarithmic base and the I₀ = V/R is the initial current.

In order to find the charge on the capacitor as a function of time, substitute I = dq/dt into equation (25),
I(t) = dq/dt = (V/R)e^-t/RC
or, separating the variables, .
dq = (V/R)e^-t/RCdt
Integrating, using the condition q = 0 at t = 0, we get
q(t) = CV[1 - e^-t/RC] = Q[1 - e^-t/RC], (26)
since Q = CV is the maximum charge on the capacitor.

Note that the charge is zero at t = 0 and approaches exponentially the maximum value of Q at time t . Also the current has its maximum value I₀ = V/R at t = 0 and decays exponentially to zero as t → ∞. The quantity RC, which appears in the exponentials of equations (25) and (26), is called the time constant τ of the circuit. It represent the time it takes for the current to decrease to 1/e of its initial value; that is, in the time equal to the time constant τ, the current decays to
I = I₀e^-1 = 0.37I₀,
and the charge on the capacitor will increase to
q = Q[1 - e^-1] = Q[1 - 0.37] = 0.63Q.
The following dimensional analysis shows that the product RC has the unit of time
[τ] = [RC] = [(V/I) × (Q/V)] = [Q/I] = [Q/(Q/T)] = [Q].

Discharging a Capacitor. Now consider a circuit consisting of a capacitor C with an initial charge Q, a resistor R, and switch, but no source of EMF. When the switch is open, there is a potential difference of
V = Q/C across the capacitor and zero potential difference across the resistor since I = 0. When the switch is closed at t = 0, the capacitor begins to discharge through the resistor. At some time during the discharge, the current in the circuit is I and the charge on the capacitor is q. Now the voltage across the resistor, IR, must equal the potential difference across the capacitor, q/C.
V_R = V_C, or
IR = q/C.
However, the current in the circuit must equal the rate of decrease of charge on the capacitor; that is,
I = -dq/dt.
-R(dq/dt) = (q/C)
or, separating the variables and rearranging the equation, we get
ln(q/Q) = -(t/RC).
Integrating this equation using the fact that q = Q at t = 0, we get
q(t) = Qe^-t/RC. (27)
In order to get the current as a function of time, differentiate equation (27).
I = dq/dt = (Q/RC)e^-t/RC, (28)
where the initial current I₀ = Q/RC.
Both the charge on the capacitor and the current decay are exponential at a rate characterized by the time constant τ = RC.