Maxwell's Second Equation. The magnetic effect of current electricity was discovered by the Danish physicist Hans Christian Oersted (1777-1851) in 1819. He noticed during the course of a demonstration lecture that a compass needle, which happened to be near a wire connected to a battery, was deflected whenever the switch was closed so that the electric current flowed in the wire. He had been looking for such a deflection of the compass needle since he had noticed that sometimes a compass needle moved erratically during thunderstorms. He had tried to find the deflection of a compass needle when a current carrying wire was placed over the compass. But for a long time he had been unsuccessful, until one day in 1820 he placed the wire carrying an electric current parallel to the compass needle instead at right angles, as he had done previously.
Ampere's Current Force Law. This discovery set
off a remarkable flurry of experimentation which produced immediately
many discoveries. In fact, only a week after Oersted's discovery
was reported to the French Academy of Sciences, Andre Ampere (1775-1836)
presented the first of a series of papers on the magnetic effects
of electric currents. To the science which deals with mutual
action of currents he gave the name electrodynamics. In
his early papers Ampere showed that the currents in two parallel
wires would attract one another if the current were in the same
direction but would repel each other if the direction of the current
in one of them was in the opposite direction to the current in
the other wire. Jean-Baptiste Biot (1774-1862) and Felix Savart
(1791-1841) made quantitative measurements of the forces between
parallel currents and found that the force on the current in either
wire was directly proportional to the common length of the wire and
to the product of the currents in the wires and inversely proportional
to the distance between the wires. That is,
F ∝
(I1I2L)/d,
or, expressed as a mathematical formula,
F =
K(I1I2L)/d, (1)
where I1 and I2 are the electric currents,
measured in amperes, flowing in the two parallel wires having a common
length L between them, measured in meters, and are separated by the
distance d, also measured in meters. The quantity K
is the constant of proportionality and has a value of
2 × 10-7 newtons/amp2.
This constant is usually written as two times the
coulomb constant of magnetism, km; that is,
K = 2km,
where km = 10-7 nt/amp2.
Thus Ampere's current force law is usually written
F =
km(2I1I2L)/d. (2)
This relation is used to define the unit of current, the ampere.
Suppose that the wires are one meter apart (d = 1.0 m, exactly)
and that the two currents in them are equal
(I1 = I2 = I).
If this common current is adjusted so that, by measurement, the force
of attraction per unit length between the two wires is exactly
2 × 10-7 newtons per meter, then the current is defined to be
one ampere. By international agreement the ampere is the fourth
fundamental unit (the other three are the kilogram, as the unit of mass,
the meter as the unit of length, and the second as the unit of time).
Since all electrical and magnetic units are expressed in terms
of the coulomb as the unit of electric charge, the coulomb is
defined as the quantity of charge that crosses an cross sectional
area of an conductor when a current of one ampere passes through
this area in one second.
That is, 1 coulomb = 1 ampere × 1 second or a coulomb is an amp-sec.
Biot-Savart Law. At a meeting of the Academy of
Sciences on October 30, 1820, Biot and Savart announced that the
action experienced by the north or south pole of magnet, when
placed at any distance from a straight wire carrying an electric
current, may be thus expressed: "Draw from the pole a perpendicular
to the wire; the force on the pole is at right angles to this
line and to the wire, and its intensity is proportional to the
reciprocal of the distance." This result was soon further
analyzed and the law known as the Biot-Savart Law was found
that may be stated as follows: the magnetic force per unit pole
due to an element ds of a circuit, in which
a current I is flowing, at a point whose vector distance
from ds is r, is in rationalized
MKS (RMKS) system of units
dH =
(1/4π)[I(ds ×
r1)/r2], (3)
where r1 = r/r is the unit vector in
the direction from the current element Ids to the point where
the magnetic force per unit pole is to be determined. Now H is
the magnetic force per unit pole and is called
magnetic field intensity; in the RMKS system of units it is
measured in newtons per weber. The direction
of H is the direction of the force exerted on an
isolated point test north pole and is the direction of the vector
or cross product of the current element and the unit vector as
given by the right-hand rule.
Using this law, the results that Biot and Savart announced on
October 30, 1820, can be found. Consider a long straight wire
carrying a constant current I be along the x-axis
of the rectangular coordinate system whose origin O is
at the foot of the perpendicular R to the long straight
wire from the point P where the magnetic field intensity
H is to be determined. Let θ be the angle between r,
the distance from the current element ds to the point P,
and the direction of the current element along the x-axis.
Using the scalar form the equation (3) and integrating, we get
H = ∫dH =
(1/4π)(I)∫[(sin θ
dx)/r2]. (4)
Let α be the complement of θ (α = 90° - θ),
so that x = R tan α,
the position of the current element on the x-axis, and
R is the length of perpendicular distance from the long
straight wire to the point P. Differentiating, we get
dx = R sec2 α dα.
Also let r = R sec α, so that
R/r = sin θ = cos α.
Integrating between -π/2 and π/2, we get
H = (1/4π)(I/R)[1-(-1)] =
(1/4π)(2I/R). (5)
This is called the Law of Biot and Savart for the magnetic
field intensity H about a long straight current carrying conductor.
Using this law, Biot and Savart derived Ampere's current force
law, equation (2) above, that is, the force per unit length between
two long straight parallel conductors carrying currents I1
and I2 that are separated by a distance d is
directly proportional to the common length of the wire and the
product of the currents in the wires and inversely proportional
to the distance between the wires. First, Biot and Savart restated
Ampere's current force law in differential form.
The differential force dF exerted on the current element
I2ds
in conductor two by the magnetic field of intensity H
around conductor one is
dF =
μI2(ds ×
H), (6)
where μ is called absolute permeability and is equal to
μ = μ0μr, (7)
where μ0 is called the permeability of vacuum and is
equal to 4π × 10-7 newtons per amperes2 and
μr is called the magnetic permeability or
relative permeability of the material in which the magnetic field
is present and has a value of one or more depending upon the material
(μr = 1 for vacuum and nearly one for air).
This equation (6) is sometimes called the motor principle.
Using the Law of Biot and Savart, equation (5), and taking the
x-axis of our coordinate system in the direction of current
element in conductor two, the magnitude of the differential force
on conductor two, given by equation (6), becomes
dF2 = μI2(Hdx) =
μI2(2I1/4πR)dx.
Integrating between -L/2 and L/2 and letting R
be the distance d between the conductors, we get
F2 =
(μ/4π)[(2I1I2L)/d],
which is Ampere's Current Force Law, equation (2) above, where in vacuum or air
km = μ/4π = 10-7
newtons/ampere2. (8)
Magnetic Flux Density. After Michael Faraday introduced
the concept of fields and lines of force (flux lines), the quantity
called flux density was introduced. Magnetic flux density
B at a point in a magnetic field was defined as the number
lines of force (flux lines) per unit area of a surface perpendicular
to the direction of the field at that point. That is,
B = dφ/dA, (9)
where φ is called magnetic flux or magnetic lines of force
measured in webers and A is the area of a surface perpendicular
to the direction of the field at the point where the flux density
is to be determined and is measured webers per meter2.
The magnetic flux density at a point in the magnetic field is directly
proportional to the magnetic field intensity at that point. That is,
B = μH, (10)
where μ is called absolute permeability and for vacuum
μ = μ0 = 4π × 10-7
newtons per amperes2.
Magnetic field intensity H is measured in newtons per weber
and is equal to ampere per meter, since the unit of flux is the
weber that is defined as newtonmeter per ampere. That is,
mewton/weber = newton/(newton · meter/ampere) = ampere/meter.
Both magnetic flux density and magnetic field intensity are vector
quantity; the direction of magnetic flux density is the same as
the magnetic field intensity. Hence,
B = μH, and
H = B/μ. (11)
Using this relation, the equations for magnetic field intensity
H can be converted into equations for magnetic flux
density B.
dB =
(μ/4π)[I(ds ×
r1)/r2]. (3A)
Biot-Savart Law equation.
B = (μ/4π)(2I/R). (5A)
Biot-Savart Law for long straight wire.
dF = I2(ds ×
B), (6A) The Motor Principle.
As an example of the application of the motor principle,
let us compute the torque on a rectangular current circuit in
a uniform magnetic field. Consider a rectangular coil suspended
vertically at the middle of its top so that its plane is parallel
to the magnetic flux density B. Let the length
of the coil perpendicular to the field be L and its width
W, and the current I in the coil. Using the equation
(6A), let us compute the force on each side of the coil. At the
top and bottom of the coil the force is zero, since ds
and B then are parallel and
ds × B = 0.
On the side of the coil in which
the current is flowing downward and which is perpendicular to
B, the force per turn will be ILB normal
to the plane of the coil. On the opposite side in which the current
is flowing upward, the force will have same magnitude, but in
the opposite direction. The result is a torque τ whose magnitude
is give by
τ = NILBW = NIAB,
where A = LW is the area of the coil and N
is the number of turns in the coil.
If the coil is allowed to turn, the torque due to the field will
decrease and the restoring torque of the suspension fiber will
increase unit equilibrium is reached. Letting be the angle turn
trough and k be the stiffness constant for the fiber, we get
τ = NIAB cos θ = kθ.
If N, A and k are known and if θ is measured,
we can calculate the magnetic flux density B in terms of
the current I; that is,
B = kθ/(NIA cos θ).
According to equation (3A), dB is directly proportional to
(ds × r1)/r2
and may be integrated over all the current elements of the
entire circuit of which Ids is a part. Then, since
the current I is constant, we can integrate equation (3A)
and we get
B =
(μ/4π)(I)∫S[(ds ×
r1)/r2] (12)
for the magnetic flux density due to a current circuit. The lines
of flux representing B form closed circles about
the conductor, as given by the following right-hand fist rule:
If you imagine that the wire is grabbed by your right hand with
your fingers curled about the wire and your thumb pointing in
the direction of the positive current, then the fingers will curl
in the direction of the lines of flux representing the magnetic field.
Ampere's Circuit Law. Ampere's Circuit Law is an
alternative to the Biot-Savart Law equation (12) for electric currents
and the magnetic field they generate. In some ways it is analogous
to Gauss' Law for electrostatics, which is an alternate formulation
of Coulomb's law. Ampere's Circuit Law is particularly useful
for finding the magnetic field in situations of geometric symmetry.
Consider equation (5A) that gives the magnetic flux density produced
by an infinitely long straight conductor. Rearranging the equation, we get
B(2πR) = μI.
On the left side of this equation, the magnetic flux density B,
at all points a distance R from the conductor, is multiplied by
2πR, the circumference of a circular loop at the distance R.
In general this expression can be found by the line integral of
B around a closed path λ which we designate as
∫λ(B · dλ) =
B(2πR),
where dλ is an element of the path. This line integral
implies that the components of B along the path
element dλ are to be taken around a closed path. In the
case of a circular loop about a long straight conductor, the direction
of B will always coincide with the direction dλ,
if the loop is traverse in the same direction as the magnetic
lines of force. The current I is the current flowing in
the conductor enclosed by the loop path. That is,
∫λ(B · dλ) =
μI, or,
∫λ(H · dλ) =
I, (13)
since by equation (11), H = B/μ.
This equation (13), which is a mathematical statement of Ampere's Circuit
Law, holds for any closed path about any configuration of electric
conductors where the current I is taken to be the net current
enclosed by the loop path. Let any surface S whose
periphery is the path of integration λ. Then the sum of the currents
linking the path is the total current passing through S.
In terms of the current density J (the current
per unit area), this total current through S is
I =
∫∫S(J · dλ).
Hence, we get
∫λ(B · dλ) =
μ∫∫S(J · dS), or
∫λ(H · dλ) =
∫∫S(J ·
dS). (14)
This is the usual mathematical form of Ampere's Circuit Law.
Just as Gauss' Law enabled us to get quickly the expression for
the electric field intensity in problems involving a high degree
of symmetry, so Ampere's Circuit Law offers us, in some cases,
the easiest method of finding the expression for B.
The Magnetic Field about a Linear Current Conductor. Ampere's
Circuit Law gives a simpler method for obtaining equation (5A).
Consider the magnetic field about a very long straight conductor.
Let us find the B at some point P external
to the conductor, that is a distance R from the conductor.
In equation (13) let us take our path λ a circle of radius R
from the conductor passing through P and with its center
at the conductor. From symmetry consideration B
must have the same magnitude at all points on this path. From
Ampere's Circuit Law we know that B cannot be radial,
but must be circular about the conductor, so that B and
dλ are parallel. Hence,
∫λ(B · dλ) =
B∫λ(dλ) =
2πRB.
Thus equation (13) becomes 2πRB = μI, or solving for
B, we get
B = (μ/2π)(I/R),
which is equation (5A), Biot-Savart Law for long straight wire.
The Magnetic Field of a Toroidal Solenoid. Consider a
doughnut-shaped coil of uniformly wound with wire carrying a current.
Let there be N total turns, or N/L turns
per unit length, where L is the mean or average circumference
around the interior of the coil. In applying equation (13), let
us take for the path λ to be this circular path of length L.
From symmetrical considerations B must be constant and
in the direction of the path. Hence
∫λ(B · dλ) =
B∫λ(dλ) =
BL.
Since the path λ threads the current N times,
equation (13) becomes
BL = μNI,
or solving for B we get
B = μ(NI/L). (15)
As L increases, a toroidal solenoid becomes more and more
similar to a very long straight solenoid. Hence equation (15)
also gives B for a long straight solenoid with
μNI/L ampere-turns per meter.
The Curl of B. From equation (14) we may easily
obtain the expression for ∇ × B
(and ∇ × H).
Using Stokes' Theorem, which says that for any vector function
A:
∫λ(A · dλ) =
∫∫S(∇ × A) ·
dS,
where S is a surface whose periphery is λ.
Thus equations (14) becomes
∫∫S(∇ × B) ·
dS =
∫∫S(μJ · dS), or,
∫∫S(∇ × H) ·
dS =
∫∫S(J · dS).
Since both surface integrals refer to the same surface, we may
equate the integrands, and get
∇ × B = μJ, or,
∇ × H = J, (16)
which is the differential or point form of Ampere's Circuit Law.
Thus the curl of B is zero only where there is no
current density, that is, if J = 0,
then ∇ × B = 0.
If the current density J is due to the motion of
a charge density ρ with a mean velocity v (all quantities
refer to the immediate neighborhood of the point where
∇ × B
or ∇ × H is being evaluated), then
J = ρv
and
∇ × B = μρv, or,
∇ × H =
ρv, (17)
for a steady flow of charge.
The Divergence of B. Having adopted the Amperian
approach to electromagnetism, it is assumed that all magnetic
fields can be attributed to the motion of electric charges.
The magnetic flux density B may be considered to be
the sum or integral of fields as those given by equation by equation
(3A); that is,
dB =
(μI/4π)[(ds ×
r1)/r2]. (3A)
This equation tells us that the magnetic lines representing the
direction of the field dB are circles about
the axis of the current element Ids. These lines
do not start at (diverge from) any point, nor do the stop at (converge
toward) any point. Thus the field dB is source-free
or solenoidal, that is its divergence is zero. Furthermore,
the sum of any number of such fields will be solenoidal. That is,
∇ · B = 0. (18)
This equation is one of the fundamental equations of field theory
and is one of Maxwell's Electromagnetic Equations. We shall now
prove mathematically this equation.
Using the identity from Vector Analysis that says that for any
two vectors such as a and b,
∇ · (a × b) =
b · (∇ × a) -
a · (∇ ×
b), (19)
let a = (μ/4π)Ids and
b = r1/r2, so that
dB = a × b = .
(μ/4π)Ids ×
r1/r2 =
(μI/4π)[(ds ×
r1)/r2],
which is the Biot-Savart Law, equation (3A).
In equation (19) the ∇ is an operator involving space differentiation
with respect to the coordinates of the point P where the
field is being computed. As P moves, the distance r
and the direction of unit vector r1 may change but μ,
I, and ds remain constant. Hence,
∇ × a = 0.
The vector function b is in the radial direction
with a magnitude inversely proportional to r2. This is
the same type of vector function as electric field intensity E,
the electric field due to a point charge. In a similar way as
∇ × E was zero,
∇ × b = 0.
Thus the right side of equation (19) is zero, so that,
∇ · (a × b) = 0.
Since a × b = dB, then
∇ · dB = 0.
And since
∇ · B =
∇ · ∫S(dB) =
∫S(∇ · dB),
and ∇ · dB = 0, it follows that
∇ · B = 0, which is equation (18),
Maxwell's Second Equation of Electromagnetics.
Comparing this with the electrostatic equation
∇ · D = ρ, we see that the attempt
to find the source of magnetic fields in magnetic "poles,"
analogous to isolated electric charges, the volume density of such poles
must be everywhere zero. This older approach interpreted this as meaning
that north and south poles always appear in pairs. The Amperian approach
interprets this as meaning that the concept of poles are unnecessary and
need not be introduced as the source of magnetic fields.
The Vector Potential. In Vector Analysis it was
shown that a vector function whose divergence is always zero can
be expressed as the curl of another vector function. That is,
if ∇ · B = 0, then
B = ∇ × A, (20)
where A is called the vector potential.
We shall now derive an expression for A at the point
P, given the current I in the circuit produces the
magnetic field B. First, let us show that
if ds has the components dsx,
dsy, and dsz, and
r = xi + yj + zk and
r =
(x2 + y2 +
z2)½, then
∇ × (ds/r) =
(ds ×
r1)/r2. (21)
Using the definition of the curl,
∇ × A =
i[∂Az/∂y -
Ay/∂z] +
j[∂Ax/∂z -
∂Az/∂x] +
k[∂Ay/∂x -
∂Ax/∂y],
substituting (ds/r) for A, we get
∇ × (ds/r) =
i[∂(dsz/r)/∂y -
∂(dsy/r)/∂z] +
j[∂(dsx/r)/∂z -
∂(dsz/r)/∂x] +
k[∂(dsy/r)/∂x -
∂(dsx/r)/∂y]
∇ × (ds/r) =
i[-(y/r3)dsz +
(z/r3)dsy] +
j[-(z/r3)dsx +
(x/r3)dsz] +
k[-(x/r3)dsy +
(y/r3)dsx]
∇ × (ds/r) =
(1/r3)[i(ydsz -
zdsy) +
j(zdsx - xdsz) +
k(xdsy - ydsx)]
∇ × (ds/r) =
(1/r3)(r × ds) =
(1/r3)(ds × r) =
ds × (1/r3)(r) =
ds × (1/r2)(r/r) =
(ds × r1)/r2, or,
∇ × (ds/r) =
(ds × r1)/r2,
where r1 = r/r is the unit vector
in the direction of r.
As in a previous section, ∇ operates differentially upon
r, but not on ds. Now substituting this
into equation (12), we get
B =
(μ/4π)∫S[∇ ×
(ds/r)].
Since the differentiation involved in ∇ and the integration around
the circuit are independent of each other, then the curl can
be taken outside of the integral sign and we get
B = ∇ × A, where
A =
(μ/4π)(I)∫S(ds/r). (22)
is the expression for the vector potential that we were seeking.
The vector potential A given by equation (22) is
not the only one whose curl is equal to B. If another
vector function A′ is added to A, such as
A′ = ∇Φ, where Φ is a scalar function,
then we find that
∇ × (A + A′) =
∇ × A +
∇ × A′ =
∇ × A +
∇ × ∇Φ =
∇ × A = B,
since the curl of the gradient is always zero. Therefore, the
potential function A must not be considered to be completely
determined by equation (22).
The vector potential may also be expressed in terms of the current
density J, or the product of the volume density ρ
of charge and the velocity v of the moving free
charges producing the magnetic field at point P. Suppose
that the current element Ids has a cross
section area A and volume dV = Ads. Then
Ids = JAds = JAds
= JdV,
since ds is in the direction of J.
Hence, equation (22) can be written as
A =
(μ/4π)∫∫∫(JdV/r), (23)
where the volume integral is taken over all regions containing
a current density ρ. Since J = ρv,
then the equation (23) may be written as
A =
(μ/4π)∫∫∫(ρv/r)dV. (24)
While ρ and v may vary throughout space, they must
be subject to the restriction that they are independent of time,
so that the magnetic field will be steady. In the case
of a single charge, the magnetic field that it produces at a point
P will vary as the charge moves along; such a field is
not steady. For a charge q moving with the constant velocity
v, we may put qv for Ids
or ρvdV, provided that the charge is moving with
a speed considerably less than the speed of light c, so
that relativity effects may be neglected. Let us find the magnetic
flux density B from the vector potential A.
Using equation (24), we get
A = (μ/4π)(qv/r). (25)
Let us take the curl of this vector potential A,
remembering the ∇ operates differentially only on the 1/r
factor, since q and v are independent of
the coordinates. Then we get
B =
∇ × A =
(μq/4π)[∇ × (v/r) =
(μq/4π)[(v ×
r1)/r2], (26)
since by the same procedure that obtained equation (21), we get
∇ × (dv/r) =
(dv × r1)/r2.
Equation (26) gives us the magnetic flux density due to a slowly
moving charge. This is the same result that we get when we substitute
qv for Ids in equation (12).
Force Exerted on a Moving Charge. At the end of
the nineteenth century the Dutch physicist Hendrik Antoon Lorentz
(1853-1928) in 1892 published his first memoir in which proposed
a theory that all electrodynamic phenomena were ascribed to the
agency of moving electric charges. This theory of Lorentz, like
those of Weber, Riemann and Clausius, was a theory of electrons,
which were supposed in a magnetic field to experience forces that
proportional to their velocities, and to communicate these forces
to the ponderable matter with which they might be associated.
His theory is based on the four equations of Maxwell to which
he added the equation which determines the ponderomotive force
on a charged particle. This equation is now known as the Lorentz
Force Law. This equation was first proposed by Oliver Heaviside
(1850-1925) in his 1889 paper on moving charges in the Philosophical
Magazine. Heaviside gave for the first time the equation
for a mechanical force exerted on an electric charge which is
moving in a magnetic field. That is, the force is equal to the
charge multiplied by the vector product of the velocity v
of the charge and magnetic flux density B.
F =
q(v × B). (27)
This equation can be derived from the Motor Principle, equation
(6A), where the current element Ids is replaced
with vdq. Assume that the current is constant and
in the current element there is a relative motion of a certain
amount of charge, q. Also assume that charge has a constant
forward velocity v and moves a displacement ds
in the time interval dt. Then,
Ids = (dq/dt)ds =
(ds/dt)dq = vdq, or
Ids = vdq.
Substituting vdq for Ids in the Motor
Principle, equation (6A), we get
dF = I(ds × B) =
Ids × B =
vdq × B
dF = dq(v × B).
Integrating both sides of this differential equation between the
limits from 0 to F, and from 0 to q.
F = q(v × B).
Lorentz added to this force exerted by the magnetic field the
force exerted by an electric field.
F =
qE + q(v × B) =
q(E + v ×
B). (28)
To illustrate Lorentz Force Law consider the magnetic deflection
of a beam of charged particles. Suppose that we have a beam of
particles, each with a mass m, a charge e, and a
velocity v. Let the beam move through a region
where the magnetic flux density has a uniform value B.
Using the equation (27) the force on each particle is
F = ev × B. (29)
Let B= and B⊥ be the components of
B parallel and perpendicular to v, respectively.
The B=, being parallel to the direction of the velocity,
will not contribute to the force and may be ignored. But since
B⊥ is perpendicular to the direction of the velocity,
a force of evB will act at right angles to the direction of motion and to
that of B. As a result, the motion in the direction of
B will remain unchanged, but the vector v will
rotate in a cone about the direction of B. In other words,
the path of the beam will be a helix (spiral) with its axis parallel
to B. Charged particles traveling in the earth's
magnetic field move in just such paths.
When the velocity v is perpendicular to B,
the force F = evB. Since the force is at right angles
to v, the motion is circular. From Newton's Second
Law of Motion (F = ma) and the equation for centripetal
acceleration (a = mv2/R), we get
evB = (mv2)/R,
or
(e/m) = (v/BR). (30)
This is a very useful equation when the mass or speed of an atomic
or subatomic particle needs to be determined.
Magnetic Field of a Moving Charge. According to
Lorentz Force Law, the electric charge must be in motion in order
that a force exerted on it by a magnetic field; if the charge
is at rest (v = 0), then there will be no
force exerted on the charge by the magnetic field. But it is
also true that the source of magnetic field must also be charges
in motion. Experiments have shown that electric currents, or
moving charges are needed to produce a magnetic field. This fact
is expressed by the following equation.
B =
(1/c2)(v ×
E), (31)
where B = magnetic flux density at a point r
meters from q,
E = electric field intensity produced by
q at a point r meters from q,
v = velocity of q with respect
to some observer, and c = the speed of light.
We will now derive this equation.
Substituting vdq for Ids in Biot-Savart Law,
equation (3A), we get
dB = (μ/4π)
[(v ×
r1)/r2]dq.
Integrating both sides of this differential equation between the
limits from 0 to B, and from 0 to q.
B = (μq/4π)
[(v ×
r1)/r2], (32)
where B = magnetic flux density at a point r
meters from q,
q = moving source charge,
v = velocity of q1
with respect to some observer.
Let μ = 1/(εc2) =
4π × 10-7 (nt/amp2),
where c = the speed of light, then
B =
(1/εc2)(q/4π)
[(v ×
r1)/r2] =
(1/c2)(v ×
[q/(4πεr2)]r1. (33)
Now the electric field intensity E at any point
about an isolated point source q is given by the equation.
E =
[q/(4πεr2)]r1.
Hence, substituting this into equation (33) we get the equation
B =
(1/c2)(v × E),
which is equation (31) that we wanted to derive.
Magnetic Flux. As electric fields are pictured
by drawing lines of force to show the direction and to represent
the strength of the field by the concentration of these lines,
the same can be done for magnetic fields. This is done by defining
the total magnetic flux φ crossing a surface S as
φ =
∫∫S(B ·
dS), (34)
where B is the magnetic flux density that is defined
as the number of lines of force or flux per unit area through
a surface S whose normal is in the direction of
B and dS is an element of the
surface S.
Gauss' Law for Magnetism. When the surface S
is a closed surface, by convention the vector dS
is taken as positive when it points toward outside of the closed
surface. Then, the magnetic flux is positive when the magnetic
line pass outward through the surface and is negative when the
lines pass inward. If the total magnetic flux through a close
surface is zero, then this implies that an equal number of magnetic
lines enter and leave the closed surface. This is precisely what
is found experimentally and is expressed in the following equation
that is called Gauss' Law for Magnetism.
φ = ∫∫S(B · dS) =
0. (35)
This implies that magnetic lines of force do not terminate on
any thing but its own "tail" and that magnetic lines
of force are always closed loops. This also implies that there
are no isolated magnetic poles in the same way that electric lines
of force terminate on electric charges. On a permanent magnet
the magnetic lines of force do not originate on the north pole
and terminate on the south pole; the lines of force passes through
the interior of the magnet and exits at the north pole and enters
the magnet at the south pole, forming closed loops. Neither do
monopoles exist in the physical world, as the most subtle experiments
confirm.