Electric and Magnetic Fields

In Chapter 6 we studied propagating electromagnetic fields, of which light is a special case. Electric and magnetic fields are inextricably linked to each other, and, as mentioned earlier, a changing electric field automatically creates a changing magnetic field and visa versa. Recall that an electric charge undergoing acceleration radiates off electromagnetic fields. If the electric charge is stationary in some frame, then it only generates an electric field; an electric charge moving at some constant velocity generates both electric and magnetic fields. Changing (time varying) ${\bf E}$ and ${\bf B}$ fields generate a ${\bf B}$ and ${\bf E}$ respectively. And lastly, if the electric or magnetic field is static (constant in time), it exists by itself. In this chapter, we focus on the static and stationary cases, and study in some detail the individual properties of the ${\bf E}$ and ${\bf B}$ fields. The electric field by itself is of enormous importance. Our contemporary civilization cannot even be imagined without the presence of electric power. Not only does electric power provide energy to run an almost endless list of modern devices, electricity powers modern industries and cities as well. Electric fields permeate natural phenomena. A study of atoms and molecules shows that it is primarily due to electrical forces that atoms and molecules are bound together into stable objects. Hence, in a real sense, all the varied materials around us exist due to the workings of the electric field. More close to home, the workings of the (biological) nervous system is based on the movement of charges, and the brain can sense, feel, think and so on largely due to electrical currents. Stationary magnetic fields are interesting in their own right, and we will briefly study a few examples of these. An understanding of electric and magnetic fields is the key to unlocking the inner workings of nature. Of the four fundamental forces in nature listed in Figure 2.1, except for gravity, all the other forces of nature are direct generalizations of the electromagnetic fields. Electric and magnetic fields are a prototype of the Yang-Mills fields that form the substratum of nature's laws. We now examine the individual properties of electric and magnetic fields, always keeping in mind that these two fields are two facets of a single electromagnetic field.

Electric Charges and Force

The concept of electric charge is the key to understanding the phenomena of electricity and magnetism. Electric charge, denoted by $q$, is a property of matter, similar to mass. Recall from Section 3.11 that the SI unit of electric charge is Coulomb; that is $[q]=C$. Unlike mass, charge is more difficult to directly experience since most of matter is neutral. Lightning is a case of electric charge moving from the clouds to the earth, with light being emitted due to its motion. An effect similar to lightning is the mild shock one sometimes gets in a dry climate when touching a metal object - the shock being due to the movement of electric charge from the human body to the metallic object. Recall in our discussion on the gravitational and electrostatic potential energy in Section 3.11, we had listed the similarities and differences in physical effects of mass and charge. There are two properties of charge which make it very different from mass; firstly charge comes in two varieties, namely negative and positive; and secondly, as derived in eq.(3.100), the strength of electrical attraction is more than $10^{39}$ times greater than the strength of gravitational attraction.

Figure 7.1: Force between Like and Opposite Charges

We state a few remarkable and far-reaching experimental facts about electric charge.

1. Like charges repel each other and opposite charges attract as shown in Figure 7.1.
2. Electric charge is absolutely conserved. Electric charge cannot be created or destroyed. The total electric charge in the universe is a constant. However, since electric charge comes with opposite signs, one can create equal number of positive and negative charges without violating charge conservation. And this is precisely what happens in the phenomenon of pair creation where an electron with charge $-e$ and an anti-electron with charge $+e$ are simultaneously created.
3. Electric charge is discrete, comes in units of the irreducible charge of the electron, which is denoted by $-e$. All observed charges $q$ in nature is of the form
   

   $$q=\pm N e \mbox{\rm { where }} N:\mathrm{integer}$$ (7.1)

   
   
   This ``quantization'' of electric charge is an experimental truth, and was known before the advent of quantum mechanics.

If the charges $q_1,q_2$ are separated by a distance $r$, the (Coulomb) force between them is given by

$${\bf F} = k\frac{q_1q_2}{r^2}{\bf e_r}$$ (7.2)

$$\mathrm{where}$$

$${\bf e_r} \equiv \mbox{\rm {unit vector pointing along the radial direction}}$$ (7.3)

and from eq.(3.97) we have

$$k=\frac{1}{4\pi \epsilon_0}= 9.00\times 10^9 Nm^2/C^2$$ (7.4)

Note that, like all other forces, the Coulomb force is a vector, and points along the line joining two charges, which is the radial vector. The Coulomb force has to be radial due to symmetry considerations, since the only vector in the problem is the radial vector $r{\bf e_r}$.

Figure 7.2: Electric Field of a Positive Point Charge

Electric Fields

Recall from our earlier discussion, that the force exerted on a charged particle, say $q_1$, is due to the electric field ${\bf E}$ created by the charge $q_2$. That is, from eq.(6.36) we have

$${\bf F}=q_1{\bf E}$$ (7.5)

The presence of an electrical force is the manifestation of an electric field. Since the force due to a point charge $q_2$ is radial, we conclude that the electric field ${\bf E}$ is radial, and is given by

$${\bf E}=k\frac{q_2}{r^2}{\bf e_r}$$ (7.6)

As shown in Figure 7.2, we represent the electric field by lines emanating from positive charge $q$, with the arrows on the lines indicating that the direction of the electric field ${\bf E}$ points radially outwards. Conversely, the electric field due to a negative charge $-q$ points inwards as shown in Figure 7.3, indicating the inward force on a positive test charge brought close to $-q$. The analogy for electric charge is the idea of a source and sink for fluid flow. The tap serves as a source for water, and when we open a tap, water flows out with a certain velocity. The electric charge $+q$ is the source of the electric field, with the electric field flowing out of the charge like the velocity of the water flowing out of the tap. A negative charge $-q$ is similarly analogous to a water sink, with electric field ``flowing into'' the negative charge as water flows into a sink.

Figure 7.3: Electric Field of a Negative Point Charge

Figures 7.4 and 7.5 show, respectively, the electric field of two like and unlike charges. From the graphical representation of the electric field, we can reach the following interesting conclusions.

Figure 7.4: Electric Fields for Like Charges

Recall, from eq.(6.26),that the energy of the electromagnetic field $u$, at the volume element around point $(x,y,z)$, is given by the strength of the electric and magnetic fields, namely

$$u=\frac{1}{2}\epsilon_0[{\bf E}^2+c^2{\bf B}^2]$$

Since we are studying the case where there in only an electric field, we have the energy per unit volume of the electric field in empty space is given by

$$u_E=\frac{1}{2}\epsilon_0{\bf E}^2$$ (7.7)

1. The conservation of charge implies that electric field lines must start from sources (positive charges) and end on sinks (negative charges), as shown in Figures 7.4 and 7.5. The electric field lines in empty space cannot be terminated, as this would imply the presence of a negative charge. The conservation of charge also implies that the electric field lines cannot cross in empty space.
2. The intensity of the electric field is proportional to the amount of charge that is generating the field. Note from eq.(7.7) that, for every volume in space, the energy of the electromagnetic field is proportional to the square of the electric field. Hence, the energy of the electric field is proportional to its intensity at some point as well as to the extent it is spread out in space.
3. If two positive charges are brought close to each other, they will repel each other. Hence the field's lines repel each other, as shown in Figure 7.4. Consequently, the field has a lower energy if the charges are moved further apart, and is an expression of the repulsion of like charges.
4. If a positive and negative charge are brought close to each other, we know they will attract. This phenomenon is realized by electric field lines emerging from the positive charge and converging on the negative charge, as shown in Figure 7.5. Since some of the electric field lines start and end on the positive and negative charges, the field's energy is reduced by bringing the charges close to each other. In other words, the force between oppositely charged particles is attractive.

Figure 7.5: Electric Fields for Opposite Charges

Electric fields obey the principle of superposition. We have already seen this in our discussion on interference in eq.(6.41). If there are a number of charges, namely, $q_1$, $q_2$, $q_3$ and so on that generate electric fields ${\bf E_1}$, ${\bf E_1}$, ${\bf E_1}$ and so on respectively, then the net electric field due to all the charges is given by

$${\bf E}={\bf E_1}+{\bf E_2}+{\bf E_3}+....$$ (7.8)

The full three dimensional structure of the electric field ${\bf E}$ of a positive charge is shown in Figure 7.6.

Figure 7.6: Electric Field due to Positive Charge

Electrostatic Potential

In general, if charges are in the presence of an electric field ${\bf E}$, they will acquire a potential energy. Recall that if the force is conservative, then, as discussed in Section 3.7, force can be replaced by potential energy, denoted by $U$. In the case of the Coulomb force, a direct demonstration can be made to show that it is a conservative force. Since force is proportional to the electric field, it follows that the (stationary) electric field can be expressed as the gradient of the ``electrostatic potential'', denoted by $V$.

The force on a charge $q$ due to an electric field ${\bf E}$ is given by

$${\bf F}=q{\bf E}$$ (7.9)

and hence

$$\Rightarrow U=q V$$ (7.10)

In other words, the potential energy $U$ is proportional to the electrostatic potential $V$. Consider the two points $d_1,d_2$ with electrostatic potential $V_1,V_2$; the electric field is then given by

$$E = -\frac{V_2-V_1}{d_2-d_1}$$ (7.11)

$$= -\frac{\Delta V}{\Delta d}$$ (7.12)

The $E$ field is the negative of the gradient of the electrostatic potential so that it points from a higher value of $V$ to a lower value, in keeping with the force pointing in the direction of decreasing potential.

Figure 7.7: Electric Field from Electrostatic Potential

Since $[U]=ML^2T^{-2}$, the dimensions of the electrostatic potential is given by $[V]=ML^2T^{-2}C^{-1}$. The units for $V$ in the SI system is called Volts (V) and is given by

$$V=\mathrm{Joule/Coulomb}\equiv JC^{-1}$$ (7.13)

In terms of volts, we have a simpler expression for the units of electric field. Recall that $[{\bf E}]=NC^{-1}$. In term of volts, we have

$$[{\bf E}]=\mbox{\rm {volts per meter}}\equiv VM^{-1}$$ (7.14)

Note that the electrostatic potential $V$ is not a vector; rather at every point in space $x$ only a single number $V(x)$ completely specifies the $V$. A field such as $V(x)$ is called a scalar field. Clearly it is much easier to analyze a scalar field such as $V$ compared to a vector field like ${\bf E}$, and this is one of the main reasons for working with potentials. The contours of constant $V$ lines are plotted in Figure 7.7. Since the electrostatic force is conservative, the energy of a particle at some point is independent of the path it took to get there. As shown in Figure 7.7, the difference in the energy of a charged particle is the same whether it takes path 1 or path 2 in going from the surface $V=5V$ to the surface $V=6V$. Figure 7.8 shows the lines of equipotential in three dimensions. The lines with a high value of $V$ are analogous to a point of high elevation for a body moving under the influence of gravity. Just as in gravity a mass $m$ can gain potential energy in going from a lower to a higher height, a positively charged particle also gains potential energy in moving from a point with a lower to a point with a higher electrostatic potential $V$; however, and this is what makes electromagnetism so different from gravity, negatively charged particle loses energy in moving from a point with a lower value of $V$ to one with a higher value.

Figure 7.8: Equipotential Surface in Three Dimensions

A charged particle will take the shortest path in going from a point at a higher to a point at a lower potential. In effect, the charge moves under the force created by the electric field derived from the potential $V$. The direction of the electric field at some point on a contour points in the direction of the shortest distance to close-by contour points, as shown in Figure 7.7. The magnitude of ${\bf E}$ is determined by the steepness of the contours. What is the rationale for separating out the electrostatic potential from the potential energy of two charges? In contrast to say gravity, for which mass is always a positive quantity, electric charge can either be negative or positive. Hence, by separating off the electrostatic potential $V$ from potential $U$, we can study the field $V$ generated by a given charge, and then analyze how it affects other charges. Suppose $V$ is generated by a positive charge; a negative charge will move towards increasing values of $V$ whereas a positive charge will move towards decreasing values of $V$, as shown in Figure 7.9. A potential $V$ that looks like a ``mountain'' to a positively charged particle $+q$ looks like a ``crater'' to a negatively charged particle $-q$.

Figure 7.9: Positive and Negative Charge Moving in the Same V-field

Recall that the fundamental irreducible charge in nature is the electron charge; by convention, the charge of the electron is taken to be negative and is denoted by $-e$. It is a natural constant, with the numerical value of $e$ in the SI units given by

$$e=1.60\times 10^{-19} C$$ (7.15)

The value of the electron charge is extremely small. In a typical light bulb, every second over $10^{19}$ electron charges enter and leave the bulb's filament. Protons carry charge equal to $+e$, and the proton seldom enters the processes that we will be interested in. The electron charge is the basis of another unit for energy. The energy gained by an electron, in moving from say a point with $V=5$ volts in Figure 7.7 to a point with $V=6$ volts, such that the electrostatic potential difference between the two points is 1 volt is called an electron-volt, and is denoted by $eV$. We consequently have

$$1 eV = e \times 1 \mathrm{volt}$$ (7.16)

$$= 1.60 C\times 10^{-19} \times JC^{-1}$$ (7.17)

$$\Rightarrow 1 eV = 1.60\times 10^{-19}J$$ (7.18)

The electron-volt (eV) is the appropriate unit of energy to measure energies involved in atomic, chemical and (molecular) biological processes. The energy of a single molecule of air at room temperature is about $$\frac{1}{40}$$ eV.

Potential due to Point Charges

The potential energy of charges $q_1$ and $q_2$, separated by a distance $r$, is given by Coulomb potential

$$U_{\mathrm{Coulomb}}(r)=k\frac{q_1q_2}{r} \mbox{\rm { : Electrostatic Potential Energy}}$$ (7.19)

From eq.(7.17) the electrostatic potential of a point charge $q_2$ is given by

$$V(r) = \frac{U_{\mathrm{Coulomb}}(r)}{q_1}$$ (7.20)

$$\Rightarrow V(r) = k\frac{q_2}{r}$$ (7.21)

Figure 7.10 shows the Coulomb potential. Note the striking difference in the potentials due to a positive and negative charge, with the positive charge giving a ``mountain'' and the negative charge yielding a ``valley''.

Figure 7.10: Coulomb Potential for Positive and Negative Charge

Electric Current; Resistance

We have seen that a positive charge will move from a higher to a lower electrostatic potential. Subjecting charges to appropriate potential configurations gives us the means to control their movement. The movement of electric charge is called an electric current, and is denoted by $I$. The electrostatic potential is the cause, and the electric current is the effect. Why are we interested in electrical currents? One of the over-riding considerations in our study of nature is to find new forms of energy, such as the electrostatic potential energy. The next question that naturally arises is whether we can transform electrical energy into other forms of energy? In particular, can we move objects and power our homes and industries with this (new) form of energy? Recall work adds to energy, and is one way of transforming energy from one form to another. To do work, we must apply a force $F$ on an object over some distance $d$, with the resulting work done given by $Fd$. Movement in space under the action of the electrical force is hence essential in converting electrical energy into mechanical energy. Only electrical charges can be moved by electrical forces, or what is the same, by electric fields. An electrical current is an instance of such a movement of electric charges, and is therefore a necessary step in doing electrical work. Consider a stream of charges flowing past a fixed point. Consider an imaginary surface $S$ placed orthogonal to the flow of charges. The flow of charge is measured by the current $I$, which is the amount of charge that flows across the surface $S$ per unit area and in unit time interval . Current is given by the charge flux $\Delta q =q_2-q_1$ in time $\Delta t=t_2-t_1$. Hence

$$I=\frac{q_2-q_1}{t_2-t_1}\equiv \frac{\Delta q}{\Delta t}$$ (7.22)

The dimensions of current is $[I]=CT^{-1}$, and the SI unit of current is Amperes, denoted by $A$. That is

$$1 \mathrm{ampere}= 1 \mbox{\rm {coulomb per second}}$$ (7.23)

Electric current $I$ in general depends on the surface through which the flow of charge is taking place. In one-dimension $I$ has a sign, with say $+I$ signifying current flowing in one direction and $-I$ for flow in the opposite one. Conservation of charge holds good for moving charges, and is expressed by the laws of current conservation. If a current $I$ flowing in a wire bifurcates into two wires with currents, say $I_1$ and $I_2$, then charge conservation requires

$$I=I_1+I_2$$ (7.24)

So how does one go about creating the flow of an electrical current? Since we first need to get our hands on some electrons, we start with a conductor, say a copper wire, since it has a supply of free electrons. We next need to create an electric field that will exert a force on electric charges and make them move. To create a constant electric field, all we need to do is to create an electrostatic potential difference between two points in the wire; this will in turn exert force on the charges so as to create a current. If the potential difference between two points in a wire separated by distance $d$ is $V$, the electric field along the wire is then given by

$$E=\frac{V}{d}$$ (7.25)

An electrical battery is a device that creates such an electrostatic potential difference, called the voltage $V$, between its two terminals. The battery converts chemical energy into electrical energy. Figure 7.11 shows a copper wire connected in a closed circuit with a resistance $R$ and a battery with voltage $V$. The electrostatic potential, say $V_+$ at the positive terminal(+) is taken to be higher than the electrostatic potential, say $V_-$ at the negative terminals (-); the voltage of the battery is the potential difference given by $V=V_+-V_-$. Due to the potential difference $V$, free charges in the conductor (copper wire) are set in motion and yield a current $I$ in the circuit. The convention in labeling the terminals of a battery means that the positive charges flows from the higher potential at the positive terminal to the lower potential at the negative terminal. Consider piped water that is pumped to a height $h$, and flows back to the ground. The electrical current $I$ can be compared to the kinetic energy of the water as it hits the ground. The battery is the analogue of the pump, the potential difference $V$ created by the battery is the analogue of the height $h$ to which the water is raised. The resistance $R$ is analogous to the radius of the pipe, and whether it is clean, or whether there is accumulation of gravel and sand in the pipe that hinders, or even blocks, the flow of water.

Figure 7.11: A Circuit with Resistance and Voltage

What is the relation between the current $I$ in the circuit, the resistance $R$ and the voltage $V$? From the analogy of water flow, we expect that the larger the voltage $V$ the higher the current, and the higher the value of the resistance $R$ the lower the current. We hence expect that

$$I=\frac{V}{R} \mbox{\rm { : Ohm's Law}}$$ (7.26)

Ohm's law states that if we create a potential difference of $V$ in a conductor having resistance $R$, then a current $I$ given by eq.(7.33) will flow in the conductor. The dimension of resistance is $[R]=ML^2T^{-1}C^{-2}$; the SI unit of resistance is ohms and

$$1\mathrm{ohm} = 1\mbox{\rm { volt per ampere}}$$ (7.27)

$$= 1V/A$$ (7.28)

Electrical currents can flow in vacuum as well as in a medium. The resistance $R$ measures the relative ease with which a currents flows in a medium. Let a material of length $L$ and area $A$ have resistance of $R$; the resistivity $\rho$ of the material is defined by

$$R=\rho\frac{L}{A}$$ (7.29)

Note $\rho$ has units of ohm-m. Table 7.1 shows the resistivity of different kinds of material. Resistance also depends on temperature, and increases with increasing temperature.

Table 7.1: Resistivity at Temperature $20^0$C

Material	$\rho$
	ohm-meter
Conductors
Silver	$1.59\times 10^{-8}$
Copper	$1.68\times 10^{-8}$
Mercury	$98\times 10^{-8}$
Semiconductors
Graphite (carbon)	$(3-60)\times 10^{-5}$
Germanium	$(1-500)\times 10^{-3}$
Silicon	$0.1-60$
Insulators
Glass	$10^9-10^{12}$
Rubber	$10^{13}-10^{15}$

Worthy to note is that superconductors are materials whose properties are explicable only by quantum theory, and have zero resistance to the flow of electrical currents. Conductors and insulators allow and block the flow of currents respectively, and can be understood based on principles of classical physics. Semiconductors lie in-between conductors and insulators, and their workings are based on quantum theory. Conductors have electrons that are not bound to the nuclei, and can hence move in response to an external electric field or a potential difference. All parts of a conductor at equilibrium are at the same potential, since any potential difference would cause currents to flow. One can have many resistors, and which can be combined in series and in parallel, with the value of the resultant resistance being derivable from Ohm's law. Energy is constantly lost due to the heating of the resistance $R$, called Ohmic heating. How much power is expended by the battery in keeping the current flowing in the circuit? In $\Delta t$ time, charge of amount $\Delta q$ flows from the high electrostatic potential to the lower potential, hence losing potential energy $\Delta U$. The difference in the electrostatic potential is given by the voltage of the battery, namely $V$, and hence

$$\Delta U=\Delta q V$$ (7.30)

Power is rate of loss of energy, in this case of potential energy $U$; hence

$$P = \frac{\Delta U}{\Delta t}$$ (7.31)

$$= \frac{\Delta q}{\Delta t}V$$ (7.32)

$$\Rightarrow P = IV$$ (7.33)

$$= I^2R=\frac{V^2}{R}$$ (7.34)

Note that the power loss in a circuit is proportional to $I^2$. In other words, if we reverse the flow of current, the sign of $I$ will change to $-I$, but the loss due to Ohmic heating will be the same, no matter in which direction the current flows. The loss of power of the battery due to heating is not a net loss, but rather an example of the transformation of energy. Light bulbs that use a filament emanate light when the filament is heated by the flow of current, and over 95% of the power $I^2R$ is converted into light, and similarly for electric ovens and so on. Hence, electrical currents serve as a vehicle for converting the energy expended by the battery into other forms of energy. The current we have considered so far is called a D.C. (Direct Current) since its direction does not change. An A.C.(Alternating Current) is one in which the current changes direction, and hence its sign, with some fixed frequency.

Capacitor : Storing Electrical Energy

A device that stores electrical energy based on opposite charges is called a capacitor. Storing the various forms of energy is an essential link in transforming energy from one form into another. The capacitor is a device for storing electrical energy. The storage of energy has another, even more vital function, and that is in the amplification of energy. If one stores energy in small amounts and accumulates a large amount over (a long) time, then one can arrange to release the stored energy in a short burst, and in effect generate an enormous amount of power. The capacitor also plays an indispensable role in integrated circuits, and together with semi-conductors, is one of the main devices for controlling and manipulating electrical energy and charge. Recall the energy per unit volume of the electric field in empty space is given by

$$u_E=\frac{1}{2}\epsilon_0{\bf E}^2 \nonumber$$

To obtain the total energy $U_E$ of the field, one has to integrate (sum over) the contributions of the ${\bf E}$ over all points of space.

If the electric field has a constant value, say $E_0$, in a finite cube with volume $vol$, then, from eq.(7.43), the total energy in the electric field is given by

$$U_E=\frac{1}{2}\epsilon_0E_0^2 \times vol$$ (7.35)

In other words, the electric field contributes energy for every volume element of space where it is non-zero, and the total energy of the field is found by adding together the contribution to the total energy $U$ from all the volume elements of space. As can be seen from the above derivation, a capacitor is a device that performs an integration of the electric field, and in effect integrates the energy of the electrical field. The term integrated circuit partly derives its terminology from this feature of a capacitor. Electric charge generates electric field, and hence to store electrical energy, we need to collect charges into a storage device. Since most objects found in nature are electrically neutral, to obtain electrical energy, it is logical to try and store positive and negative electric charge separately . However, if we bring together a large collection of positively (or negatively) charged particles, the force of electrical repulsion is very large, the charges would all tend to fly apart, and an enormous amount of energy would be expended to merely keep them in place. Hence, a more efficient arrangement for generating electric field is to bring together equal amounts of positive and negative charges, but separate them so that we are not back to an electrically neutral object. And this in essence is the principle behind the design of all capacitors.

Figure 7.12: Capacitor

A parallel plate capacitor is composed out of two conducting plates placed parallel to each other and separated by a distance $d$ filled with an insulator. One can let air fill up the space between the conducting plates, but, as shown in Figure 7.12, a dielectric material is usually placed instead of air in order to increase the capacitance of the capacitor. By charging the conducting plates with opposite charges of amount $Q$ and $-Q$, a potential difference of amount $V$ is created. A measure of how much charge is stored in a capacitor is the change in the potential difference of the capacitor $V$ if charge $Q$ is placed on the conducting plates. The analog of a capacitor is a water storage tank; if a certain volume of water is poured into the tank - the analog of electric charge - the increase in the height of the water is the analog of the increase in potential $V$, and is a measure of the volume of the storage tank, which is the analog of capacitance $C$. Clearly, the simplest case for a capacitor is when the voltage difference $V$ is proportional to the charge $Q$; the proportionality constant is the capacitance of the capacitor, and is denoted by $C$ (not to be confused with the SI unit of charge, namely the coulomb $C$). We hence have

$$Q=CV$$ (7.36)

The SI unit of capacitance is the farad, denoted by $F$; we have

$$1\mathrm{farad} = \mbox{\rm { coulomb per volt}}$$ (7.37)

$$\Rightarrow 1F = 1 C/V$$ (7.38)

A capacitor is charged by connecting it in a circuit with a battery with voltage $V$, as shown in Figure 7.13. Note the capacitor is shown by two parallel lines with a gap to indicate that there is an insulator between the two conducting plates. A insulator in a D.C. circuit breaks the closed circuit, with no current flowing in the circuit. The conducting plate connected to the (+) terminal of the battery will attain the potential $V_+$ by the flow of electrons to the battery, leaving a net charge $+Q$ on the conducting plate. And similarly for the conducting plate connected to the (-) terminal; it will have the potential $V_-$ and charge $-Q$. The charged capacitor will therefore have a voltage difference equal to the battery, namely equal to $V$, with charges $+Q$ and $-Q$ on the two respective conducting plates. After the (transient) process of charging is over, there is no more flow of charges and the current in the circuit is zero.

Figure 7.13: Charging a Capacitor

The energy stored in a capacitor can be found by determining the electric field that has been created due to charging up of the capacitor. The capacitor has a potential difference of $V$ across a distance $d$, and, for now, let air be placed between the conducting plates. Hence the electric field is constant and points from the positively to the negatively charged conductors as shown in Figure 7.12. From eq.(7.18), the constant electric field is given by

$$E_0=\frac{V}{d}$$ (7.39)

The energy stored in the capacitor is, from eq.(7.44), proportional to the volume of the capacitor. For conducting plates with area $A$, the energy stored in the capacitor is consequently given by

$$U_{\mathrm{capacitor}} = \frac{1}{2}\epsilon_0\times E_0^2\times \mbox{\rm { volume of capacitor}}$$ (7.40)

$$\Rightarrow U_{\mathrm{capacitor}} = \frac{1}{2}\epsilon_0(\frac{V}{d})^2\times (Ad)$$ (7.41)

$$= \frac{1}{2}\epsilon_0\frac{A}{d}V^2$$ (7.42)

It is known from charging up the capacitor that the total energy stored is given by

$$U_{\mathrm{capacitor}} = \frac{1}{2}CV^2$$ (7.43)

$$= \frac{Q^2}{2C}$$ (7.44)

Hence, from the above equations, we see that the capacitance of a parallel plate is given by

$$C=\epsilon_0\frac{A}{d}$$ (7.45)

Note that capacitance $C$ depends only on the geometrical shape of the capacitor, as well as the material serving as the insulator. By changing the insulator from air to a dielectric material, one simply changes in $$U_{\mathrm{capacitor}}$$ the permittivity of the vacuum $\epsilon_0$ to that of the medium, namely $\epsilon$, obtaining the general result

$$C=\epsilon\frac{A}{d}$$ (7.46)

As shown in Table 7.2, a properly chosen dielectric can increase the capacitance by a few order of magnitude.

Table 7.2: Some Dielectric Materials

Material	$$\epsilon/\epsilon_0$$
Vacuum	$1$
Air	$1.00054$
Paper	3.5
Silicon	12
Germanium	$16$
Water($25^0$)	$78.5$
Titania ceramic	130
Strontium titanate	$310$

One can replace the insulating dielectric in a capacitor with a semi-conductor, and create a more complex device that stores and discharges electrical energy depending on the state of the semi-conductor.

*RC Circuit

The charging and discharging of a capacitor takes place in a finite time, and we briefly discuss the salient aspects of these processes. Consider a $RC$ circuit as shown in Figure 7.14. When the switch $S$ is closed, the circuit is completed and current flows until the capacitor is fully charged, after which the current ceases to flow.

Figure 7.14: Circuit with a Battery, Capacitor and Resistor

Since a capacitor gets charged in a finite amount of time, the first thing one does is to look for a constant which has the dimensions of time, and, within an order of magnitude, this would be the time taken to charge the capacitor. Consider the following combination.

$$[R][C] = \frac{V}{I}\times \frac{Q}{V}$$ (7.47)

$$= \frac{Q}{I}$$ (7.48)

$$= \frac{\mathrm{coulomb}}{\mathrm{coulomb/sec}}$$ (7.49)

$$= \mathrm{sec}$$ (7.50)

Again, the power of dimensional analysis! There is an inherent time scale in the problem, namely, $RC$. Suppose we have a battery of voltage $V$, the charging of the capacitor commences at $t=0$. Let the current at time $t>0$ be $I(t)$, and the charge on the capacitor be $q(t)$. The potential drop across the resistor, from Ohm's law, is $$IR$$, and the potential drop across the capacitance is $$\frac{q}{C}$$. At time $t$ the total drop across the $R,C$ circuit elements must equal the voltage of the battery, and is given by

$$V=IR+\frac{q}{C}$$ (7.51)

Also

$$I=\frac{dq}{dt}$$ (7.52)

$$\Rightarrow \frac{dq}{dt}=\frac{V}{R}-\frac{q}{RC}$$ (7.53)

The charge on the capacitor plates is zero when the charging starts, and at final time of $t=\infty$ the charge reaches the equilibrium value of $Q$. In other words

$$q(0)=0 \mbox{\rm { ; }} q(\infty)=Q$$ (7.54)

By direct substitution, one can verify that

$$q(t) = Q[1-e^{-t/RC}]$$ (7.55)

$$\Rightarrow I(t) = \frac{dq}{dt}$$ (7.56)

$$= \frac{Q}{RC}e^{-t/RC}$$ (7.57)

$$= \frac{V}{R}e^{-t/RC}$$ (7.58)

where in obtaining last equation above we have used $Q=CV$. Eqn.(7.71) is a typical case of an exponential dependence on time. For $t»RC$, that is, for time much larger than the characteristic time $RC$ for a capacitor, to a very good approximation the current is zero. One can conclude that for all practical purposes, the charging is over $RC$ seconds after the the charging starts. To discharge the capacitor, the same circuit as the one for charging is used, except that one removes the battery. The capacitor is first charged and then, at $t=0$ the switch $S$ is closed, connecting the positive and negative plates of the capacitor through a resistor $R$. It can be shown by an analysis similar to the one for charging the capacitor that the charge and current on the capacitor, at time $t>0$, is given by

$$q(t)=Qe^{-t/RC}$$ (7.59)

$$I(t)=\frac{Q}{RC}e^{-t/RC}$$ (7.60)

Similar to the discussion on the charging of a capacitor, from the equation above we conclude that the capacitor is fully discharged in $RC$ seconds after the start of the discharging process.

Magnetic Fields

Magnetic fields are ubiquitous in nature. Magnetic fields appear everywhere in astrophysics. Spinning black holes generate very strong magnetic fields that cause intense X-ray radiation from charged objects that fall into it. Magnetic fields determine the properties of quasars and neutron stars, as well as erupting as magnetic storms during solar flares. The earth's mild magnetic field has been known since antiquity, and magnetic fields manifest in more subtle forms as permanent magnets, in electric motors and electric power generators, in permanent magnetic information storage devices, and down to the quantized magnetic fields that are trapped in superconducting material. Magnetism is one of the less known wonders of nature. Magnetic fields are more difficult to study than electric fields. Unlike the electric field that, for the stationary case, is reducible to the much simpler (scalar) electrostatic potential, magnetic fields, even for the simplest stationary cases, can only be represented by vectors. The ultimate reason for the asymmetry between electric and magnetic fields is that, unlike electric charges, nature chooses not to have any ``magnetic charges'' - otherwise known as magnetic monopoles. Instead of being generated by magnetic charges, all magnetic fields in nature are generated by moving charges. Given the complexity of the magnetic field B, we will study only two examples as these illustrate the intimate connection of the electric and magnetic fields. Recall from eq.(6.32) that the units of the magnetic field B is given by newton per (coulomb-meter/sec), which is abbreviated to tesla(T). We have

$$1T = \frac{N}{Cms^{-1}}$$ (7.61)

$$= 1\frac{N}{Am}$$ (7.62)

Table 7.3 gives the strengths of the magnetic fields typically encountered in nature.

Table 7.3: Some Approximate Magnetic Fields

Phenomena	Magnetic Field
At Neutron Star's surface	$10^6 T$
Large Electromagnet	$1.5 T$
Small Bar Magnet	$10^{-2} T$
At Earth's surface	$10^{-4} T$
Quantum flux of magnetic field	$4 \times 10^{-17}T-m^2$
Detectable magnetic field	$10^{-14} T$

Magnetic fields are most commonly experienced in the form of permanent magnets, which have a south (S) and a north (N) pole. Lines of magnetic field emanate from the north pole and converge on the south pole. Similar to charge, like poles repel and opposite poles attract. Since in the final analysis there are no magnetic monopoles, the permanent bar magnets are the result of microscopic electric currents that generate the observed magnetic fields.

Magnetic Field Due to a Current

A moving charge creates a magnetic field ${\bf B}$. Consider the example of a current $I$ flowing along an infinitely long wire lying along the $x$-axis. From the symmetry of the problem, the magnetic field can only point in two directions, namely either radially outwards from the wire, or in concentric circles girdling the wire. Since magnetic monopoles do not exist, the ${\bf B}$ field cannot emanate from the wire. Hence, the magnetic field must form concentric circles about the wire. If one has a current in a closed loop, as shown in Figure 7.15, the generated magnetic field looks like the magnetic field of a permanent magnet, with one side the North Pole and the other the South Pole.

Figure 7.15: curloo.eps

The direction of the magnetic field can either be clockwise or counter-clockwise with respect to the direction of flow of the current. Ampere's law determines the magnitude and direction of the magnetic field due to a current. If one points one's right hand thumb along the direction of the current, then ${\bf B}$ points along the direction of the curled fingers. What is the magnitude of ${\bf B}$? Again, from the symmetry of the problem, B can depend only on the orthogonal distance from the wire, and should fall off to zero at a large distance from the wire. Let the orthogonal distance from the wire be denoted by $r$; Ampere's law then yields

$${\bf B}_{\mathrm{current}} = B_C\times \mbox{\rm {unit vector in angular direction}}$$ (7.63)

$$\mathrm{where}$$ (7.64)

$$B_C = \frac{k}{c^2}\frac{2I}{r}$$ (7.65)

We have obtained the ${\bf B}$-field that is created by a current. A magnetic field external to the current, namely ${\bf B}_{\mathrm{external}}$, can exerts a force on a current. Suppose the particles in the current carry charge $q$, and are traveling at a velocity of ${\bf v}$. Then the force exerted on the charged particle by ${\bf B}_{\mathrm{external}}$, as indicated in eq.(6.37), is given by

$${\bf F}=q{\bf v \times B}_{\mathrm{external}}$$ (7.66)

If two currents are brought close to each other, they will exert a magnetic force due to the operation of the above equation.

Current due to a Magnetic Field

We have seen that currents can generate magnetic fields. By a relation of reciprocity, we expect (correctly) that a changing magnetic fields in turn should also generate electric currents. Consider a closed circuit with an ammeter that measures electric current. If one moves a magnet in and out of the circuit, the ammeter registers an oscillating current, which ceases the moment the magnet stops moving. Such a current is called an induced electric current. To understand this phenomenon, we need to first understand the idea of magnetic flux, denoted by $\Phi$. Suppose we have a constant magnetic field pointing in the $z$-direction. We draw a closed circuit or contour (loop) in say the $xy$-plane of area $A$. The magnetic flux going through the area $A$ is then given by

$$\Phi=BA$$ (7.67)

In other words, magnetic flux $\Phi$ measures the amount of ${\bf B}$ field flowing through an area $A$. Faraday's Law of Induction explains the phenomenon of an induced current: The change of magnetic flux through a closed circuit induces an emf in the circuit. The emf drives a current in a circuit and is the result of a changing magnetic field inducing an electric field. The induced electric field does not arise from charges, and hence the emf is not the same as the voltage for stationary electric fields discussed earlier. Suppose that in time interval $\Delta t$ the flux changes by $\Delta \Phi$. If the circuit has $N$- turns of the wire, the induced emf $\cal E$ is then given by

$${\cal E}=-N\frac{\Delta \Phi}{\Delta t}$$ (7.68)

We apply Faraday's law to a simple example. Consider a constant magnetic field ${\bf B}$, say due to a permanent magnet, pointing in the $z$-direction. Suppose for simplicity that the ${\bf B}$ field in the $z$-direction, and has been shown in Figure 7.16 by crosses indicating that the ${\bf B}$ field points into the paper. The ${\bf B}$ field extends from $x=0$ towards the positive $x$-axis as shown in Figure 7.16.

Figure 7.16: emf.eps

Insert a rectangular loop of wire, having resistance a $R$ and width $L$, and with a distance $x$ inside the ${\bf B}$ field. We apply a force $F$ along the negative $x$-axis to pull the loop of wire out of the ${\bf B}$ field at a constant velocity $v$. It is easy to show (Lenz's law) that the current induced in the wire is such as to create a ${\bf B}$ field that opposes the force, since otherwise we could violate energy conservation. The power being expended by the force in moving the wire loop is

$$P=Fv$$ (7.69)

The flux at an instant $t$ is given by

$$\Phi = BA=BLx$$ (7.70)

$$\Rightarrow \frac{\Delta \Phi}{\Delta t} = BL\frac{\Delta x}{\Delta t}$$ (7.71)

$$= BLv$$ (7.72)

Ignoring a minus sign, we have from eq.(7.81)

$${\cal E} = \frac{\Delta \Phi}{\Delta t}$$ (7.73)

$$= Blv$$ (7.74)

We have a circuit in which an emf $\cal E$ is driving a current $I$. One should note that, unlike the stationary case where a potential difference $V$ drove a current in the circuit, in the case of induced emf we have time dependent electric and magnetic fields, and hence the electric field can no longer be reduced to an electrostatic potential. Since we have a circuit with a current $I$, we can still apply Ohm's law to obtain

$$I=\frac{\cal E}{R}=\frac{Blv}{R}$$ (7.75)

The rate of heating gives the power being expended in keeping the current flowing and is given by

$$P = I^2R$$ (7.76)

$$\Rightarrow P = \frac{B^2l^2v^2}{R}$$ (7.77)

From energy conservation, the power expended in pulling the wire loop out of the magnetic field must exactly equal the energy appearing as Ohmic heating, and this can be proven. If devices are placed in the circuit which are driven by the current $I$, energy conservation then requires that additional mechanical energy will be required to pull the wire out of the magnetic field. The point of the exercise is to illustrate the essential principles of an electric power generator that transforms mechanical energy into electrical energy. Using the properties of magnetic fields, we can transform mechanical work into electrical energy as embodied in the induced current $I$. The crucial point to note is that, without any mechanical contact, the magnetic field acts as an intermediate agency in transforming mechanical energy (involved in pulling out the wire loop) into an induced emf $\cal E$ and a consequent electrical current. This ``catalytic'' role of magnetic fields is a rich resource for the invention of new devices and technologies.

Maxwell's Equations

The discussion in Chapters 7 and 6 all revolved around a piece meal exposition of Maxwell's equations. We summarize our discussion here so as to unify our understanding of electromagnetism. So what are the Maxwell's equations? They can be enumerated in the following manner.

1. Electric fields are generated by positively and negatively charged particles. We discussed the relation of electric field to charge in Sections 7.3 and 7.2.
2. A changing magnetic field generates an induced electric field. In Section 7.9 this equation of Maxwell was called Faradays' law of induction, and we derived the emf and induced electric current from a time varying magnetic field.
3. There is no magnetic monopole. This is an experimental fact that continues to hold.
4. An electric current or a changing electric field gives rise to an induced magnetic field. In Section 7.8 we studied the induced magnetic field due to a current.

Note that the second and fourth of Maxwell's equations explain the phenomenon of electromagnetic radiation, since, as mentioned earlier, in empty space a changing electric field induces a magnetic field which in turn induces an electric field and so on, and provides the mechanism for the propagation of radiation over virtually infinitely large distances! Just to give a sense of how important are Maxwell's equations, we quote Richard Feynman: ``From a long view of the history of mankind - seen from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics.''