cphil_sr2
SPECIAL THEORY OF RELATIVITY
CONTINUED
by Ray Shelton
Minkowski Interpretation
In September, 1908, at the 80th Assembly of German Natural Scientists and Physicians meeting at Cologne, Herman Minkowski, Professor of Mathematics at Gottengen, presented the new picture of world that had been “discovered by Lorentz and further revealed by Einstein,” and called for a return to ” the idea of a pre-established harmony between pure mathematics and physics.” Minkowski attempted to formulate an effective mathematical method of blending together space and time into a single coordinate system. Einstein had regarded events as the basic data of physics. He had showed the full significance of inertial frames and inertial observers. He showed that every inertial observer has his own privately valid time and his own “instantaneous three-space” consisting of all events (x, y, z, t) with fixed time coordinate. Minkowski presented the view that the totality of events in the world are the “points” of an absolute four-dimensional manifold which is now called “space-time.” Different inertial observers draw different planes through space-time as their “instants.” Each inertial observer, with his standard x, y, z, and t, coordinatizes all of space-time, just as a choice of x and y axes coordinatizes the Euclidean plane. The world of events was not the familiar three-dimensional world of length, width, and height, proceeding along a smoothly flowing river of time. In space-time the three spatial dimensions were profoundly linked with the dimension of time by their relation to the constant velocity of light. Minkowski began his presentation,
“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independence reality.”
This union of space and time is today called “space-time”. Space-time is the arena in which stars, atoms, and people live and move and have their being. Space is different for different observers. Time is different for different observers. Space-time is the same for everyone. Space, empty and absolute, a kind of box without walls in which to fit the universe, is simply an idea, a human concept. The notion of space-time that Minkowski envisioned was not simply the empty box within which the drama of the universe was played. Space-time acts upon matter and is in turn acted upon by matter. The action of space-time on matter was not simply an illusion designed to give us variable readings on the yardsticks and clocks with which we locate events, but are real and physical. In the new world in which we find ourselves, “space by itself and time by itself are doomed to fade away into mere shadows.” In this world there are no absolute points in space nor absolute events in time but only point-events in space-time. Distances or times in this world are only relative. And the world in which we live is the continuum of all possible point-events.
Invariants
Minkowski asked himself the following question:
“Since length in Special Relativity is not an invariant quantity, nor is time an invariant quantity as between different inertial systems, is there some invariant that can replace these as demanded by relativity theory?”
In classical physics it was assumed that any two points located in space have an invariant distance between them, regardless of time. And this invariant distance was specified by the Pythagorean theorem:
Δs2 = Δx2 + Δy2 + Δz2, [1]
where Δx, Δy, Δz are the components of the distance Δs along the coordinate axes in the x, y, and z directions. This equation for the distance Δs between two points is replaced by Minkowski with an invariant “separation” between two events in space-time:
Δs2 = c2Δt2 – (Δx2 + Δy2 + Δz2). [2]
In contrast to the former equation [1], in which the quantity Δs2 varies from one inertial system to another, since according to Special Relativity the length of objects are “contracted” in the direction of their motion when they are in motion. But, according to this latter equation [2], the quantity Δs2 remains the same in all inertial systems. Any quantity which is unchanged by a general coordinate transformation is called an invariant of the transformation. Invariants play an important role in physics. They are the only entities suitable for the construction of physical laws, since the principle of relativity requires that the results of physical theories be independent of the choice of coordinate system (assuming that the system is inertial).
There are two classes of invariants: scalars and vectors.
1. Scalars are single numbers and are unaffected by the choice of coordinates.
2. Vectors are invariant under rotations of the coordinates; they are constructed to be so and by design the transformation rules assure this.
Any mathematical entity which is invariant under rotation is called a tensor. A scalar is a tensor of zeroth rank, and a vector is a tensor of the first rank. Tensors of higher rank also exists; the moment of inertia is a tensor of the second rank.
Intervals
Let us consider the interval between two events, A and B. The distance or separation in space between these two events is given by the expression
(distance)2 = (Δx)2 + (Δy)2 + (Δz)2. [3]
Thus the expression for the interval between the two events
A at (t, x, y, z) and
B at (t + Δt, x + Δx, y + Δy, z + Δz)
has the form
(interval of proper time)2 = (Δτ)2 = c2(time)2 – (distance)2 =
c2(Δt)2 – (Δx)2 – (Δy)2 – (Δz)2, [4]
when the interval is timelike, and when it is space like,
(interval of proper distance)2 = (Δσ)2 = (distance)2 – c2(time)2 =
(Δx)2 + (Δy)2 + (Δz)2 – c2(Δt)2. [5]
In relativity it is common to speak of the frame in which the observed body is at rest as the proper frame. The length of a rod in such a frame is called the proper length. Similarly, the proper time is the time interval recorded by a clock attached to the observed body. The proper time can be thought of equivalently as the time interval between two events occurring at the same place in the moving frame or the time interval measured by a single clock at one place. A nonproper (or improper) time interval would be a time interval measured by two different clocks at two different places. Similarly, the space between between two events can be thought of as the proper distance between two events.
In this new geometry, how are we to understand the minus signs? In ordinary Euclidean geometry the expression for distance contains three plus signs, but no minus signs. When in 1908 Minkowski introduced his expression for the interval of proper distance, he introduced a new quantity w to measure time so that
(interval of proper distance)2 = (Δσ)2 = (Δx)2 + (Δy)2 + (Δz)2 + (Δw)2. [6]
Thus all the signs are positive and the geometry superficially appears to be that of Euclid, but in four dimensions, instead of three. He defined this time-coordinate as w = ict, where i is the imaginary unit which is defined as
i = √(-1) so that i2 = -1. Hence,
(interval of proper distance)2 = (Δσ)2 = (Δx)2 + (Δy)2 + (Δz)2 + (icΔt)2,
or since (interval of proper distance)2 = (Δσ)2, then
(Δσ)2 = (Δx)2 + (Δy)2 + (Δz)2 – c2(Δt)2. [7]
Since time is not identical in quality to space and time is not measured in the same units as distance, to treat time on the same basis as distance but to mark this difference of character between time and space, the imaginary unit i and the minus sign is introduced. But to convert the units of the time-coordinate to space units, the time interval is multiplied by the factor c (the constant velocity of light): that is, (meters per sec) × sec = meters.
Lorentz Invariants
In Minkowski’s formulation of relativity, an event specified by x, y, z, t is viewed as a point in a four dimensional space with the coordinates x, y, z, and ict (where i = √(-1) and i2 = -1). The fourth coordinate is imaginary because time is essentially different from space. These coordinates are the components of a four dimensional vector that is known as a four-vector. Minkowski called the four dimensional space-time manifold “world,” although it has come to be called Minkowski space. A point in Minkowski space is called a world point. As a particle moves in space and time, its successive world points trace out a world line. Consider a displacement Δs between two world points of a moving particle. The location of a world point is specified by its position four-vector
s = (x, y, z, ict). [8]
The displacement between two world points is
Δs = (Δx, Δy, Δz, icΔt), [9]
or, in differential form,
ds = (dx, dy, dz, icdt). [10]
Since Δs and ds are a four-vectors, their norms are Lorentz invariants. The norm of a vector is the dot product of the vector with itself. The dot product of two vectors form a scalar. Since scalars are independent of the coordinate system, the dot product of two vectors is called a scalar invariant and the dot product of a four-vector with itself is called a Lorentz invariant.
The norm of Δs is
Δs · Δs = Δs2 = Δx2 + Δy2 + Δz2 – c2Δt2, [11]
or, in differential form, is
ds · ds = ds2 = dx2 + dy2 + dz2 – c2dt2. [12]
In the rest frame of the particle, the space coordinates are constant, and therefore,
Δx = Δy = Δz = 0.
Thus Δt = Δτ in the rest frame; the world points are separated only in time. Thus Δτ is the time interval measured in the rest frame, and for this reason is known as proper time. Hence,
Δs2 = –c2Δτ2, [13]
or, in differential form,
ds2 = –c2dτ2 or
–dτ2 = ds2 / c2. [14]
Hence dτ2 is a Lorentz invariant and it has the same value for the same world points in all frames.
Proper
In relativity it is common to speak of the frame in which the observed body is at rest as the proper frame. The length of a rod in such a frame is called the proper length. Similarly, the proper time is the time interval recorded by a clock attached to the observed body. The proper time can be thought of equivalently as the time interval between two events occurring at the same place in the moving frame or the time interval measured by a single clock at one place. A nonproper (or improper) time interval would be a time interval measured by two different clocks at two different places.
Time Dilation
Let us now derive the Einstein time dilation formula using four-vectors. Consider an observer at rest in the x′, y′, z′, t′ system. In this system the proper time interval between two world points is
dτ = dt′.
In the x, y, z, t system moving with velocity v relative to the first frame, the interval between the same points is given by
ds2/c2 = (1/c2) (dx2 + dy2 + dz2) – dt2, [15]
since by equation [12],
ds2 = dx2 + dy2 + dz2 – c2dt2.
Since dτ2 is a Lorentz invariant, its value for the same world points is the same in all frames. Hence, by equation [14] we can equate its value in the rest frame to its values in the second frame.
–dτ2 = ds2/c2 = (1/c2)(dx2 + dy2 + dz2) – dt2, or
(dt′)2 = dt2 – (1/c2)(dx2 + dy2 + dz2)
or dividing by dt2, we get
(dt′/dt)2 = 1 – (1/c2) [(dx/dt)2 + (dy/dt)2 + (dz/dt)2],
or since (dx/dt)2 + (dy/dt)2 + (dz/dt)2 = v2, then
(dt′/dt)2 = 1 – v2/c2, [16]
By taking the square root of both sides and solving for dt, we get
dt = dt′/√(1 – v2/c2), or
dt = γdτ, [17]
where γ = 1/√(1 – v2/c2).
Equation [17] is the Einstein’s time dilation formula.
Four-Velocity
Let us now derive the formula for velocity in four dimensions. In ordinary three dimensional space, dividing a vector by a scalar (a rotational invariant) yields another vector. Similarly, dividing a four-vector by a Lorentz invariant yields another four-vector. Consider the displacement four-vector
ds = (dx, dy, dz, icdt).
Dividing it by the Lorentz invariant dτ, we get a new four-vector:
ds/dτ = (dx/dτ, dy/dτ, dz/dτ, icdt/dτ). [18]
Comparing this with the three dimensional velocity
u = ds/dt = (dx/dt, dy/dt, dz/dt), [19]
we shall call ds/dτ the four-velocity u.
Hence we get the four-vector for four-velocity,
u = (dx/dτ, dy/dτ, dz/dτ, icdt/dτ). [20]
In the rest frame of the particle,
dx = dy = dz = 0, and dτ = dt. [21]
Thus for a particle at rest
u = (0, 0, 0, ic),
and the norm of u is
u · u = (u)2 = –c2 [22]
and it has the same value in all frames. Obviously the four-velocity u is very different from u, the familiar three dimensional velocity.
Now let us find an expression for the four-velocity u of a moving particle. Let the x, y, z, t system move with velocity –u relative to the rest frame of the particle. Then using the time dilation formula above, equation 17, that is,
dt = γdτ or dt/dτ = γ and 1/dτ = γ/dt.
where γ = 1/√(1 – v2/c2) and dt is the time interval in the moving frame,
and using this in the definition of four-velocity above, equation [20],
u = (dx/dτ, dy/dτ, dz/dτ, icdt/dτ) = (γdx/dt, γdy/dt, γdz/dt, icγ),
we get the four-velocity vector
u = γ(dx/dt, dy/dt, dz/dt, ic), or
u = γ(u, ic), [23]
where u is the three dimensional velocity ( equation 19).
We shall use u to derive the momentum-energy four vector.
Four-Momentum
Let us now construct the momentum-energy four-vector. We shall also obtain the relativistic expression for force. Let us start with the observation that the classical momentum m0u is not relativistically invariant since the classical velocity u is not a four-vector. Now above we found the form of the four velocity u and, since the rest mass m0 is a Lorentz invariant, the product m0u is a four-vector. Now it is natural to identify this as the relativistic momentum, and we therefore define the four-momentum vector as
p = m0u
Using equation [23], u = γ(u, ic),
p = m0 γ(u, ic) = γm0 (u, ic).
p = (γm0u, iγm0c).
But for the classical definition of momentum as mass times velocity to hold, then the definition of mass must be modified as follows:
m = γm0,
where m0 is the mass of the body at rest with respect to the observer and m is the mass of the body moving at the speed v relative to the observer. It is obvious that this definition of mass is negligibly different from the classical concept except at very high velocities, since the ratio v2/c2 is ordinarily such a small fraction; the mass of moving body is only significantly larger when the speed of the body v approaches the speed of light c. As v approaches c,
v2/c2 becomes almost one, and
√(1 – v2/c2) becomes close to zero.
Thus,
p = (mu, imc), or,
since mc = mc2/c, we get
p = (p, imc2/c). [24]
Minkowski Force
Does this four-momentum obey a conservation law? Classically, the rate of change of momentum is equal to the applied force, so that the momentum of an isolated system is conserved. But it is not obvious whether the four-momentum is similarly conserved, since we have not developed a relativistic expression for force. Above we obtained the four-velocity u by dividing ds by the invariant dτ. Let us apply the same technique to obtain the “time derivative” of p, and then define this equal to the four-force F. That is,
F = dp/dτ = {dp/dτ, (i/c)[d(mc2)/dτ]}, [25]
where F is known as the Minkowski force.
If dt is the time interval in the observer’s frame corresponding to the interval of proper time dτ, then dt = γdτ and we get
F = γ{dp/dt, (i/c)[d(mc2)/dt]}. [26]
In classic physics, dp/dt = F. In order to conserve the momentum of an isolated system, we must retain the identification of force with time rate of change of momentum in all inertial systems. Thus the Minkowski force becomes
F = γ{F, (i/c)[d(mc2)/dt]}. [27]
The Minkowski force F is constructed in such a way that four-momentum is conserved when the four-force is zero. Like all four-vectors, F is relativistically invariant; that is, if it is zero in one frame, it is zero in every frame. This assures us that if four-momentum is conserved in one inertial frame, it must be conserved in all inertial frames.
Momentum-Energy Four-Vector
To interpret the fourth or timelike component of p = (p, imc), let us use the classical work-energy theorem, which says that F · u represents the rate at which work is done on a particle moving with velocity u, that is,
F · u = dW/dt = dE/dt,
where E is the total energy of the particle.
Since F = dp/dt, then
F · u = u · dp/dt. [28]
Let us apply this theorem to the four-dimensional Minkowski force and velocity u = γ(u, ic). We get
F · u = γ{F, (i/c)[d(mc2)/dt]} · γ(u, ic) = γ2 [F · u, –d(mc2)/dt]. [29]
Since the scalar product of two four-vectors is a Lorentz invariant, it can be evaluate in any frame. Let us evaluate F · u in the rest frame of the particle.
In this frame,
F · u = u · dp/dt = 0,
since u = 0.
Also
d(mc2)/dt = 0.
Hence, F · u = 0, and
F · u – d(mc2)/dt = 0, or F · u = d(mc2)/dt. [30]
Since by classical work-energy theorem:
F · u = dE/dt, then
dE/dt = d(mc2)/dt. [31]
Thus the relativistic equivalent of total energy is
E = mc2. [32]
Thus the four-momentum vector becomes
p = (p, imc2/c) = (p, iE/c). [33]
This four-vector is also called the momentum-energy four-vector, or 4-momentum vector, where the p is the ordinary 3-dimensional momentum and E is the total energy of the mass m.
Mass-Energy Equivalence
Let us generate a Lorentz invariant by taking the norm of p.
First, by using equation [33],
p · p = (p, iE/c) · (p, iE/c) = p2 – E2/c2,
and then by using the definition of the four-momentum vector
p = m0u and
the four-velocity vector ( equation [23]): u = γ(u, ic), we get
p · p = m0u · m0u = [m0γ(u, ic)] · [m0γ(u, ic)] = m02γ2 (u2 – c2).
Therefore,
p2 – E2/c2 = m02γ2 (u2 – c2). [34]
But since γ2 = 1/(1 – u2/c2) = 1/[(c2 – u2)/c2] = c2/(c2 – u2) = –c2/(u2 – c2), then
p2 – E2/c2 = –m02c2. [35]
Hence, solving for E2, we get
E2 = p2c2 + (m0c2)2 = p2c2 + E02. [36]
This equation is called mass-energy equivalence, where E0 = m0c2 is the rest energy of the particle.
The Minkowski approach of generating four-vectors leads in a natural way to relativistic correct expressions for momentum and energy. With this approach the conservation laws for energy and momentum appear as a single law; the conservation of four-momentum. In relativity, momentum and energy are different aspects of a single entity. This is a significant simplification over classical physics, where the concepts of momentum and energy are essentially unrelated.
Summary of Minkowski Relativity Dynamics
Minkowski beginning with the four-dimensional world developed the dynamics of special relativity. Classical physics began with the fundamental principle F = ma in which both space and time are separate and absolute. Minkowski replaced this classical fundamental principle with the four-dimensional world in which the invariance requirement of relativity is satisfied by uniting space and time in a four-dimensional world in which time is the fourth dimension and is combined with the three dimensions of space. That is, every mechanical phenomenon is given by four coordiantes x, y, z, w. The element of the world-line of any material particle will be a four dimensional vector ds, the square of whose magnitude is given by
ds2 = dx2 + dy2 + dz2 + dw2,
where dw = icdt; that is,
ds2 = dx2 + dy2 + dz2 – c2dt2.
By dividing this equation by dt2, we get
(ds/dt)2 = (dx/dt)2 + (dy/dt)2 + (dz/dt)2 – c2.
This can be be expressed in terms of the instantaneous velocity components of the particle
dx/dt, dy/dt, dz/dt so that,
v2 = (dx/dt)2 + (dy/dt)2 + (dz/dt)2
Hence, we get
(ds/dt)2 = v2 – c2.
or dividing both sides of this equation by (ic)2, we get
(ds/dt)2/(ic)2 = (v2 – c2)/(ic)2 = [-(v2 – c2)/c2] = [(c2 – v2)/c2] = (c2/c2 – v2/c2) = (1 – v2/c2).
since (ic)2 = –c2. Simplifying,
[(ds/dt)/(ic)]2 = (1 – v2/c2) = (1 – β2),
since v2/c2 = (v/c)2 = β2, and β = v/c.
Taking the square root each side, we get
(ds/dt)/(ic) = √((1 – β2).
Taking the reciprocal of both sides of this equation (dividing each side into one), we get
icdt/ds = 1/√(1 – β2), or
icdt/ds = γ,
where γ = 1/√(1 – β2).
Consider another inertial frame where the v << c, so that Then β = 0 and γ = 1.
If dt = dτ, where τ is the proper time, then
icdτ/ds = 1, or
icdτ = ds
in that inertial frame. Therefore, in the other frame moving with velocity v with respect to it,
icdt/icdτ = γ, or
dt = γdτ,
which is the Einstein Time Dilation formula.
If ds = dσ, where σ is the proper distance, then
icdt/dσ = 1, or
icdt = dσ, or
dt = dσ/ic,
in that inertial frame.
Therefore, in the other frame moving with velocity v with respect to it,
(ic/ds)(dσ/ic) = γ, or
dσ = γds, or
ds = dσ/γ = αdσ,
where 1/γ = α = √(1 – β2),
which is the Einstein Length Contraction formula.
In Newtonian dynamics in the absence of forces, the momentum mv of a particle was conserved. Thus the mass m of the particle is invariable, that is, it does not increase or decrease when its velocity v changes. But in special relativity, which regards ict as a fourth coordinate, the differentiation of the vector displacement of the particle with respect to this fourth coordinate cannot produce anything like a new vector. By multiplying both sides of the equation
icdt/ds = γ
by the rest mass of the particle m0, we get
m = m0icdt/ds = γm0,
which is Einstein Mass Increase formula.
Classical electromagnetism is consistent with special relativity. Maxwell’s equations are invariant under a Lorentz transformation and do not need to be modified. Indeed, Lorentz originally arrived at his transformation equations by requiring the invariance of Maxwell’s equations. In a statement sent to a meeting in 1952 honoring the centenary of Michelson’s birth, Einstein wrote:
“The influence of the crucial Michelson-Morley experiment upon my own efforts has been rather indirect. I learned of it through H.A. Lorentz’s decisive investigation of the electrodynamics of moving bodies (1895) with which I was acquainted before developing the special theory of relativity. … What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body in motion in a magnetic field was nothing else but an electric field.”
Current electricity was understood and treated as a self-contained subject before Einstein’s work on relativity. And this is still done in many text-books. The connection of electromagnetism to the special theory of relativity is usually confined to the proof of the invariance of Maxwell’s equations under a Lorentz transformation; we will not do that here. We shall instead investigate the concept of magnetism as a relativistic correction to electrostatics and the nature of magnetism as relativistic effect.
Measurement of time, length and mass are modified by relative velocity. These have been treated above, and we will focus our attention on the fact that observers in uniform relative motion agree about the magnitude of a momentum at right angles to their relative motion. Suppose that some outside force causes a moving particle to receive a small amount of momentum δp at right angles to its velocity. From Newton’s definition of force as the time rate of change of momentum, an observer moving with the particle who observes momentum δp acquired in a time δt0, will report that the force was
F0 = δp/δt0. [37]
An observer in the laboratory will attribute the same magnitude δp to the momentum but will think it was acquired over a longer time; that is,
δt = γδt0, [38]
where γ = 1/√[1 – (v2/c2)] = 1/√[1 – β2] and β = v/c.
He will therefore say that the force acting on the particle was:
F = δp/δt = δp/γδt0 = (1/γ)F0, or
F0 = γF. [41]
Thus we see that an observer moving with the particle (that is, in the “rest-frame” of the particle) is the one who attributes the smallest value to a time-interval, and the largest value to a sideways force experienced by the particle.
Forces Between Electric Charges in Relative Motion
When two charges q and Q are at rest, the force on q is given by the equation
F = qE,
where E is the electric field intensity at the position of q due to Q, and is given by the equation:
E = (1/4πε0) (Q/r2)(r/r). [42]
Now suppose that q remains at rest while Q and the observer are moving with velocity v at right angles to the line joining Q and q. If measurements are made by the observer, the electric field intensity at the position of q produced by Q is still E and the force on q is still qE, independent of v. Electric charge is assumed to be relativistic invariant, independent of its motion and appearing the same to all observers. But to an observer at rest in the laboratory, which is the rest-frame of q, the force on q will appear to be larger by a factor γ; this follows from equation [41].
If the force on q, as measured by the stationary observer, is
then we see that this observer will describe the situation by saying that there is, in the neighborhood of q, an electric field
Thus different observers, each defining electric field intensity as force per unit charge on a test charge at rest with respect to himself, will attribute to the electric field intensity at a point, values which differ in the same way as do the values attributed to a force.
Force Between Two Moving Charges
1. Total Force
If both charges q and Q are moving with velocity v in the same direction, at right angles to the line joining them, the force on q (measured by an observer moving with the charges) will be qE. But this observer is now in the rest frame of q; so according the equation [41], an observer at rest in the laboratory will attribute to the force on q a smaller value, that is,
2. Comparison to the force on a stationary charge.
The force F0 on a similar stationary charge at the same place and time is given by equation [43], so if we want to describe the force F ′ on the moving q as a force F0, which is independent of its movement, plus an extra force Fm associated with the movement; that is,
F ′ = F0 + Fm.
In order to obtain Fm, F0 must be subtracted from F ′ that is,
Fm = F ′ – F0,
and using equation [45] and equation [43], we get
Fm = (1/γ)qE – γqE = qE[(1/γ) – γ] = qE[(1 – β2)1/2 – (1 – β2)-1/2].
For sufficiently small values of β = (v/c), we may expand the terms on the right-hand side by the binomial theorem, ignoring all terms higher that the first power of β2 = (v2/c2). Thus we get
Fm = qE[(1 – ½β2) – (1 + ½β2)] = qE[-β2] = –qE(v2/c2). [46]
We thus see that the total force on the moving charge q may be expressed as the resultant of two forces: F0 which is independent of its velocity, and Fm which gives the effect of the velocity; the minus sign indicates that the extra force Fm is in the opposite sense to the electrostatic force F0. Since material objects cannot move with velocities larger than that of light, the factor β2 = (v2/c2) is always less than unity. This means that when Q and q are of the same sign, making F0 repulsion, Fm is an attraction which can reduce the magnitude of the net repulsive force, but can never lead to a net attraction, however great the velocity.
The Magnetic Force
The subscript m has been given to the extra force, Fm, because the latter is usually attributed to magnetic effect. The moving source charge Q is said to produce a magnetic field with flux density B. The characteristic of this field is that a test charge q moving through it with velocity v experiences a force
Equating the right side of equation [47] to the right side of equation [46],
qvB = –qEv2 / c2,
and solving for B, we see that the moving charge Q must produce a magnetic field with flux density
Although equations [47] and [48] relate the magnitude of vectors, they are not written in vector notation because they do not relate directions of the vectors; we shall see later that B must be considered as a vector perpendicular to E and to the velocity v.
Definition of Magnetic Field
1. The case of unequal velocities.
In the calculation of the magnetic force between two charges with the same velocity in the previous section, one factor v was included in the expression [48] for B and another in equation [47] which gave the force on a charge moving through a region of given magnetic flux density B. This device not only describes the special case of equal velocities, but also gives the correct result (no magnetic force) when either velocity is zero. A little algebra shows that it also correctly covers the general case of unequal velocities, if v in equation [47] is taken to be the velocity of q, but u is put as the velocity of Q in equation [48], which becomes
B = (1/c2) Eu. [49]
2. The case of unequal velocities in different directions.
The still more general case, in which the charges move in different directions, requires careful treatment by relativistic vector methods. The calculations involved are rather laborious when the velocities have components parallel to E, but a calculation using three-dimensional vectors with relativistic rules for adding them, is given by Rosser. An alternative calculation, using four-dimensional vectors, is given in Appendix C of Basic Electricity by W.M. Gibson. These calculations both leads to the normal solution, which is to consider B as a vector perpendicular to E and u; it is obtained by writing the right-hand side of equation [49] as a vector product:
B = (1/c2) (u × E). [50]
Now the electric flux density D is defined as equal to ε0E in free space, that is,
D = ε0E.
Therefore, E may be replaced in equation [50] by D / ε0, we get
B = (1/ε0c2) u × D. [51]
This introduces a new constant μ0 which is defined as
μ0 = 1/ε0c2. [52]
In the magnetic field, magnetic field intensity H is defined as
H = B/μ0,
and substituting into equation [51] gives
H = u × D. [53]
This is a conventional expression for the magnetic field due to a moving charge, in terms of the electric field which it causes in its own rest-frame. If we wish to express H in terms of the magnitude and position of the moving charge, we may, using equation [42], multiplied it by ε0 to give D; that is,
E = (1/4πε0) (Q/r2)(r/r), or
D = ε0E = (1/4π)(Q/r3)r.
and substituting for D into equation [53]; we get
H = (1/4π)(Q/r3)(u × r). [54]
3. The Lorentz Force.
Now that we have defined the magnetic field, in direction as well as magnitude, we must re-write equation [47] in vector form; to obtain Fm along the same line E, but in the opposite sense, vB must be replaced by a vector product v × B, giving:
This part of a well-known equation and the force which it describes is often called the Lorentz Force Law. This law states that the force on a charge q, moving with velocity v through a region where from unspecified sources there is the presence of an electric field intensity E and a magnetic flux density B; that is,
Fm = q(E + v × B). [56]
This Lorentz Force Law summarize the following experimental facts about an electric charge moving in a magnetic field:
a. The magnetic force is proportional to the charge q and the speed v of the particle.
b. The magnitude and direction of the magnetic force depends upon the velocity of the particle and the magnitude and direction of the magnetic field.
c. When a charged particle moves in a direction parallel to the magnetic field vector, the magnetic force F on the charge is zero.
d. When the velocity vector makes an angle θ with respect to the magnetic field, the magnetic force acts in a direction perpendicular to both v and B; that is, the force F is perpendicular to the plane of v and B.
e. The magnetic force on a positive charge is in the direction opposite the force on a negative charge moving in the same direction.
f. If the velocity vector makes an angle θ with respect to the magnetic field, the magnitude of the magnetic force is proportional to sin θ.
These experimental observations are summarized in the equation [55] where the direction of magnetic force is in the direction of v × B, which by definition of the vector cross product is perpendicular to both v and B. This is specified by the right-hand rule, which says, using the four fingers of the right hand with the palm facing the vector B with the four fingers rotating from the direction of the velocity vector v into the direction of the magnetic flux density vector B, the thumb then points in the direction of the magnetic force F. This assumes that the charge q is positive; if the charge is negative, then the direction of the magnetic force will be in the opposite direction to that of the positive charge. The value of the magnitude of the magnetic force F is given by the formula:
F = qvB sin θ, [57]
where θ is the angle between v and B. Note that the force is zero when v is parallel to B (θ = 0° or 180°) and the force is maximum value of F = qvB when v is perpendicular to B (θ = 90°).
Equation [55] is an operational definiton of a magnetic field at a point in space. That is, the magnetic field is defined in terms of a sideway force acting on a moving charged particle. There are several important differences between electric and magnetic forces that should be noted:
a. The electric force is always in the direction of the electric field, whereas, the magnetic force is always perpendicular to the magnetic field.
b. The electric force acts on a charge particle independent of its velocity, whereas, the magnetic force acts on a charged particle only when the particle is in motion.
c. The electric force does work in displacing a charged particle, whereas, the magnetic force associated with a steady magnetic field does no work when the charge particle is displaced.
This last statement is a consequence of the fact that when a charged particle moves in a steady magnetic field, the magnetic force is always perpendicular to the displacement. That is,
F · ds = (F · v) dt = 0, [58]
since the magnetic force is a vector perpendicular to v. From the property and the work-energy theorem, we conclude that the kinetic energy of a charged particle cannot be altered by a magnetic field alone. In other words, when a charge particle moves with a velocity v, an applied magnetic field can alter the direction of the velocity vector, but cannot change its speed.