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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) and fields generate a and
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 and 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
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
Electric and Magnetic Fields
The concept of electric charge is the key to understanding the
phenomena of electricity and magnetism. Electric charge, denoted by , is a
of matter, similar to mass. Recall from Section 3.11 that
the SI unit of electric charge is Coulomb; that is .
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 times
greater than the strength of gravitational attraction.
Electric Charges and Force
We state a few remarkable and far-reaching experimental facts about electric charge.
Force between Like and Opposite Charges
If the charges are separated by
a distance , the (Coulomb) force between them is given by
- Like charges repel each other and opposite charges attract
as shown in Figure 7.1.
- 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 and an anti-electron with charge are simultaneously created.
- Electric charge is discrete, comes in units of the
irreducible charge of the electron, which is denoted by . All observed
charges in nature is of the form
This ``quantization'' of electric charge is an experimental truth, and was
known before the advent of quantum mechanics.
and from eq.(3.97) we have
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 .
Electric Field of a Positive Point Charge
Recall from our earlier discussion, that the force exerted on a
charged particle, say , is due to the
electric field created by the charge . That is, from eq.(6.36)
The presence of an electrical force is the manifestation of an electric field.
Since the force due to a
point charge is radial, we conclude that the electric
field is radial, and is given by
As shown in Figure 7.2, we
represent the electric field by lines
emanating from positive charge , with the arrows on the
indicating that the direction of the electric field points radially
Conversely, the electric field due to a negative charge
points inwards as shown in Figure 7.3, indicating
the inward force on a positive test charge brought close to .
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 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 is similarly analogous to a water sink,
with electric field ``flowing into'' the negative charge as water flows into a
Figures 7.4 and 7.5 show, respectively, the electric field
of two like and unlike charges. From the graphical representation of
electric field, we can reach the following interesting conclusions.
Electric Field of a Negative Point Charge
Recall, from eq.(6.26),that the energy of the
electromagnetic field , at the volume element around
point , is given by the strength of the electric and
magnetic fields, namely
Electric Fields for Like Charges
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
- 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.
- 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.
- 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
- 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.
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, , , and so on that
generate electric fields , , and so on
respectively, then the net electric field due to all the charges
is given by
Electric Fields for Opposite Charges
The full three dimensional structure of the electric field of a positive charge
is shown in Figure 7.6.
Electric Field due to Positive Charge
In general, if charges are in the presence of an electric field , 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 . 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 .
The force on a charge due
to an electric field is given by
In other words, the potential energy is proportional to the electrostatic
Consider the two points with electrostatic potential
; the electric field is then given by
The field is the negative
of the gradient of the electrostatic potential so that it points from
a higher value of to a lower value, in keeping with the force
pointing in the direction of decreasing potential.
, the dimensions
of the electrostatic potential is given by
. The units for in the SI system is called
Volts (V) and is given by
Electric Field from Electrostatic Potential
In terms of volts, we have
a simpler expression for the units of
electric field. Recall that
term of volts, we
Note that the electrostatic potential is not a vector; rather
at every point in space only a single number completely
specifies the . A field such as is called a scalar
field. Clearly it is much easier to analyze a scalar field such
as compared to a vector field like , and this is one
of the main reasons for working with potentials.
The contours of constant 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
to the surface . Figure 7.8 shows the lines of
equipotential in three dimensions.
lines with a high value of are analogous to a point of high
elevation for a body moving under the influence of gravity. Just
as in gravity a mass 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
electrostatic potential ; 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 to one with a higher value.
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 . 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 is determined by the steepness of the
What is the rationale for separating out the electrostatic
potential from the potential energy of two charges? In contrast to say
for which mass is always a positive quantity, electric charge can either be negative or
positive. Hence, by separating off the electrostatic
potential from potential , we can study the field generated by
a given charge, and then analyze how it affects other charges.
Suppose is generated by a positive charge; a
negative charge will move towards increasing values of whereas
a positive charge will move towards decreasing values of
, as shown in Figure 7.9. A
potential that looks like a ``mountain'' to a positively charged
looks like a ``crater'' to a negatively charged particle .
Equipotential Surface in Three Dimensions
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 . It is
a natural constant, with the numerical value of in the SI units given by
Positive and Negative Charge Moving in the Same V-field
The value of the electron charge is extremely small. In a typical light bulb, every
second over electron charges enter and leave the bulb's filament.
Protons carry charge equal to , 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 volts
in Figure 7.7 to a point with volts, such that
the electrostatic potential difference between the two
points is 1 volt is called an electron-volt, and is denoted by .
We consequently have
The electron-volt (eV) is the appropriate unit of energy to measure
involved in atomic, chemical and (molecular) biological
processes. The energy of a single molecule of air at room
temperature is about
The potential energy of charges and
, separated by a distance , is given by
From eq.(7.17) the electrostatic
potential of a point charge is given by
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''.
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 . 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 on an object over some distance , with the resulting work
done given by . 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
Consider a stream of charges flowing past a fixed point. Consider
an imaginary surface placed orthogonal to the flow of charges.
The flow of charge is measured by the current , which is
the amount of charge that flows across the surface per unit area and
in unit time interval . Current is
given by the charge flux
Coulomb Potential for Positive and Negative Charge
The dimensions of current is , and the SI
unit of current is Amperes, denoted by .
Electric current in general depends on the surface through which
the flow of charge is taking place. In one-dimension has a
sign, with say
signifying current flowing in one direction and 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
flowing in a wire bifurcates into two wires with currents, say and , then charge
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 is , the electric field along the wire is then
An electrical battery is a device that
creates such an electrostatic potential difference, called the voltage , 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 and a battery with voltage . The electrostatic potential, say
at the positive terminal(+) is taken to be higher than the
electrostatic potential, say at the negative terminals (-); the voltage
of the battery is the potential difference given by . Due to the
potential difference , free charges in the conductor (copper
wire) are set in motion and yield a current 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 , and flows back to the ground.
The electrical current 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
created by the battery is the analogue of the height to which
the water is raised. The resistance 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
What is the relation between the current in the circuit, the resistance and
the voltage ? From the analogy of water flow, we expect that
the larger the voltage the higher
the current, and the higher the value of the resistance the
lower the current. We hence expect that
A Circuit with Resistance and Voltage
Ohm's law states that if we create a potential difference of in a conductor having
resistance , then a current given by eq.(7.33) will
flow in the conductor.
The dimension of resistance is
; the SI unit of
resistance is ohms and
Electrical currents can flow in vacuum as well as in a medium.
measures the relative ease with which a currents flows in a medium.
Let a material of length and area have resistance of ; the
resistivity of the material is defined by
Note 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.
Resistivity at Temperature C
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
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 , called Ohmic heating. How much power is expended by the battery in
keeping the current flowing in the circuit? In time, charge of amount
flows from the high electrostatic potential to the lower potential, hence
losing potential energy . The difference in the
electrostatic potential is given by the voltage of the battery, namely
, and hence
Power is rate of loss of energy, in this case of potential energy
Note that the power loss in a circuit is proportional to . In
other words, if we reverse the flow of current, the sign of
will change to , 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
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.
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
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
To obtain the total energy of the field, one has to integrate (sum
over) the contributions of the over all points of
If the electric field has a constant value, say , in a finite cube
with volume , then, from eq.(7.43), the total energy in
the electric field is given by
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 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
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.
A parallel plate capacitor is composed out of two conducting plates placed
parallel to each
other and separated by a distance 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.
the conducting plates with opposite charges of amount and , a potential
difference of amount is created. A measure of how much charge
is stored in a capacitor is the change in the
potential difference of the capacitor if charge 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 ,
and is a measure of the volume of the storage tank, which is the
analog of capacitance .
Clearly, the simplest case for a capacitor is when the voltage difference
is proportional to the charge ; the proportionality
constant is the capacitance of the capacitor, and is denoted
by (not to be confused with the SI unit of charge, namely the
coulomb ). We hence have
The SI unit of capacitance is the farad, denoted by ; we
A capacitor is charged by
connecting it in a circuit with a battery with voltage , 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 by the flow of electrons
to the battery, leaving a net charge on the conducting plate. And similarly
for the conducting plate
connected to the (-) terminal; it will have the potential and charge . The
charged capacitor will therefore
have a voltage difference equal to the battery, namely equal to
, with charges and 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.
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 across a
distance , 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
Charging a Capacitor
The energy stored in the capacitor is, from eq.(7.44), proportional to
the volume of the capacitor.
For conducting plates with area , the energy stored in the capacitor is
consequently given by
It is known from charging up the capacitor that the total energy
stored is given by
Hence, from the above equations, we see that the capacitance
of a parallel plate is given by
Note that capacitance 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
the permittivity of
the vacuum to that of the medium, namely ,
obtaining the general result
As shown in Table 7.2, a properly chosen dielectric can
increase the capacitance by a few order of magnitude.
Some Dielectric Materials
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.
The charging and discharging of a capacitor takes place in a
finite time, and we briefly discuss the salient aspects of these
Consider a circuit as shown in Figure 7.14.
When the switch is closed, the circuit is completed and current flows
until the capacitor
is fully charged, after which the current ceases to flow.
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
Circuit with a Battery, Capacitor and Resistor
Again, the power of dimensional analysis! There is an inherent
time scale in the problem, namely, .
Suppose we have a battery of voltage , the charging of the capacitor
commences at . Let the current at time
be , and the charge on the capacitor be . The potential
drop across the resistor, from Ohm's law, is
the potential drop across the capacitance is
At time the total drop across the
circuit elements must equal the voltage of the battery, and is given by
The charge on the capacitor plates is zero when the charging
starts, and at final time of the charge reaches the
equilibrium value of . In other words
By direct substitution, one can verify that
where in obtaining last equation above we have used .
Eqn.(7.71) is a typical case of an exponential dependence
on time. For , that is, for time much larger than the
characteristic time for a capacitor, to a very good approximation
the current is zero. One can conclude that for all practical
purposes, the charging is over seconds after the the charging
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 the switch is closed,
the positive and negative plates of the capacitor through a resistor .
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 , is given by
Similar to the discussion on the charging of a capacitor, from the
equation above we conclude that the capacitor is fully
discharged in seconds after the start of the discharging
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
Table 7.3 gives the strengths of the magnetic fields
typically encountered in nature.
Some Approximate Magnetic Fields
|At Neutron Star's surface
|Small Bar Magnet
|At Earth's surface
|Quantum flux of magnetic field
|Detectable magnetic field
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.
A moving charge creates a magnetic field . Consider the example of
a current flowing along an infinitely long wire lying along
the -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 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.
Magnetic Field Due to a Current
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 points along the direction of the curled
What is the magnitude of ? 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 ;
Ampere's law then yields
We have obtained the -field that is created by a current.
A magnetic field external to the
, can exerts a force on a
current. Suppose the particles in the current carry charge , and are
traveling at a velocity of . Then the force exerted on the
charged particle by
, as indicated
in eq.(6.37), is given by
If two currents are brought close to each other, they will exert a
magnetic force due to the operation of the above equation.
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
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 . Suppose we have a constant
pointing in the -direction. We draw a closed circuit or contour (loop)
in say the
-plane of area . The magnetic flux going through the area is
then given by
Current due to a Magnetic Field
In other words, magnetic flux measures the amount of field
flowing through an area .
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 the flux changes by
. If the circuit has - turns of the wire, the
induced emf is then given by
We apply Faraday's law to a simple example. Consider a constant magnetic
, say due to a permanent magnet, pointing in the -direction.
Suppose for simplicity that the field in the -direction, and has
been shown in Figure 7.16 by crosses indicating that the field
points into the paper. The field extends from
towards the positive -axis as shown in Figure 7.16.
Insert a rectangular
loop of wire, having resistance a and width , and
with a distance inside the
field. We apply a force along the negative -axis to pull the
loop of wire out of the field at a constant
velocity . It is easy to show (Lenz's law) that
the current induced in the wire is such as to create a 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
The flux at an instant is given by
Ignoring a minus sign, we have from eq.(7.81)
We have a circuit in which an emf is driving a current
. One should note that, unlike the stationary case where a
potential difference 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 , we can still apply Ohm's law to obtain
The rate of heating gives the power being expended in keeping the
current flowing and is given by
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 , 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 . 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 and a consequent electrical
current. This ``catalytic'' role of magnetic fields is a rich
resource for the invention of new devices and technologies.
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
So what are the Maxwell's equations? They can be enumerated in the
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!
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.''
- 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.
- 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.
- There is no magnetic monopole. This is an experimental fact
that continues to hold.
- 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.
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