Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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12 Electrodynamics and Relativity
12.1 The Special Theory of Relativity
12.1.1 Einstein's Postulates
12.1.2 The Geometry of Relativity
12. 1.3 The Lorentz Transformations.
12. 1.4 The Structure of Spacetime
12.2 Relativistic Mechanics
12.2.1 Proper Time and Proper Velocity
12.2.2 Relativistic Energy and Momentum
12.2.3 Relativistic Kinematics
12.2.4 Relativistic Dynamics
12.3 Relativistic Electrodynamics
12.3. I Magnetism as a Relativistic Phenomenon
12.3.2 How the Fields Transform
12.3.3 The Field Tensor
12.3.4 Electrodynamics in Tensor Notation
12.3.5 Relativistic Potentials
Ether
12.1 The Special Theory of Relativity
12.1.1 Einstein's Postulates
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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Classical mechanics obeys the principle of relativity: the same laws apply in any inertial reference frame.
The laws of mechanics are not the same in accelerating reference frames.
An inertial reference is at rest or moving with constant velocity.
Galileo stated the principle of relativity is applicable to classical mechanics.
It was thought that principle of relativity is not applicable to the laws of electrodynamics.
A charge in motion produces a magnetic field, whereas a charge at rest does not.
It appears that electromagnetic theory presupposes the existence of a unique stationary reference frame, with respect to which all velocities are to be measured. This reference (medium) was called ether.
All other waves (water waves, sound waves, waves on a string) travel at a prescribed speed relative to the propagating medium.
If this medium is in motion with respect to the observer, the net speed is always greater "downstream" than "upstream."
Hints for relativity
12.1 The Special Theory of Relativity
12.1.1 Einstein's Postulates
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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Hints for relativity in electrodynamics
Applying the laws of electrodynamics produces the same effect on different inertial references although with different interpretations.
The speed of light is measured the same in all directions (Michelson-Morley experiment).
For an observer on the ground, the emf is due to the magnetic force on charges in the wire loop.
For an observer on the train the emf is induced by the chaining magnetic field in the wire loop.
Both observers obtain the same current in the loop.
Special relativity was developed out of Einstein's contemplation of electrodynamics.
Postulates for special theory of relativity
12.1 The Special Theory of Relativity
12.1.1 Einstein's Postulates
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Einstein proposed two postulates:
1. The principle of relativity. The laws of physics apply in all inertial reference systems.
2. The universal speed of light. The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source.
Einstein's velocity addition rule:
Galileo's velocity addition rule
Special relativity compels us to alter our notions of space and time.
Three consequences
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
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In this section, gedanken (thought) experiments is used to introduce the three most striking geometrical consequences of Einstein's postulates:
In the next section, the same results will be derived using Lorentz transformations.
The relativity of simultaneity
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
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A freight car travels at constant speed along a straight track.
A light bulb at the center of the car is switched on
As observed from the car
As observed from the ground
The two events in question
Event (b) happens before event (a).
As the light travels out from the bulb, the train moves forward, so the beam going to the back end has a shorter distance to travel than the one going forward.
The two events occur simultaneously.
Two events that are simultaneous in one inertial system are not, in general, simultaneous in another.
In any reference, an observer must correct for the time the signal takes to reach him from any event.
For example, he may station assistants at strategic locations, each equipped with a watch synchronized to a master clock, so that time measurements can be made right at the scene.
Time dilation
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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A freight car travels at constant speed along a straight track.
Consider a light ray that leaves the bulb and strikes the floor of the car directly below.
How long does it take the light to make this trip?
As observed from the train
An overbar denotes measurements made on the train.
As observed from the ground
Moving clocks run slow.
Because their internal clocks are running slow.
This is called time dilation.
Example
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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All moving clocks run slow by the same factor
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
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He finds that the train clock runs slow.
She concludes that the ground clock runs slow.
Who's right?
They're both right since measured different things.
Clocks that are properly synchronized in one system will not be synchronized when observed from another system.
Because moving clocks are not synchronized, when checking time dilation to focus attention on a single moving clock.
But you can use as many stationary clocks.
Example
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
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The time elapsed on earthbound clocks
The twin paradox arises when you try to tell this story from the point of view of the traveling twin. From her point of view, it would seem, she's at rest, whereas her brother is in motion, and hence it is he who should be younger at the reunion.
The two twins are not equivalent. The traveling twin experiences acceleration when she turns around to head home.
The resolution of the "paradox" is concerned, the traveling twin cannot claim to be a stationary observer.
How old is each twin at their reunion?
Lorentz contraction
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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A freight car travels at constant speed along a straight track.
A lamp at one end of a boxcar and a mirror at the other, so that a light signal can be sent down and back.
How long does the signal take to complete the round trip?
As observed from the train
An overbar denotes measurements made on the train.
As observed from the ground.
The round-trip time is
Moving objects are shortened.
From the ground point of view the car is shorter.
This is called Lorentz contraction.
The observer on the train doesn't think her car is shortened-her meter sticks are contracted by that same factor, so all her measurements come out the same as when the train was standing in the station.
Lorentz contraction
12.1 The Special Theory of Relativity
12.1.2 The Geometry of Relativity
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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He finds that the stick in the car shorter.
She finds that the stick on the ground shorter.
Who's right?
They're both right since measured different things.
Assume the observer lays a ruler next to the stick.
For a moving stick, the readings next to the two ends of the stick must be read at the same instant of time.
Because of the relativity of simultaneity the two observers disagree on what constitutes "the same instant of time."
When the person on the ground measures the length of the stick in the car, he reads the position of the two ends at the same instant in his system.
But the person on the train, watching him do it, complains that he read the front end first, then waited a moment before reading the back end.
The two sticks are identical
The two sticks are identical
Galilean transformations
12.1 The Special Theory of Relativity
12.1.3 The Lorentz Transformations
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Galilean transformations
Before special relativity
Lorentz transformations
12.1 The Special Theory of Relativity
12.1.3 The Lorentz Transformations
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Lorentz transformations
12.1 The Special Theory of Relativity
12.1.3 The Lorentz Transformations
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Lorentz transformations
Lorentz transformations
Example
12.1 The Special Theory of Relativity
12.1.3 The Lorentz Transformations
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How much time elapses on the moving clock?
Example
12.1 The Special Theory of Relativity
12.1.3 The Lorentz Transformations
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The subscripts denote the right and left ends of the stick.
This is the Lorentz contraction formula
Example
12.1 The Special Theory of Relativity
12.1.3 The Lorentz Transformations
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in a time
This is Einstein's velocity addition rule.
Old notations
Four-vectors
12.1 The Special Theory of Relativity
12.1.4 The Structure of Spacetime
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The Lorentz transformations take on a simpler appearance when expressed in terms of the quantities
the Lorentz transformations read
In matrix form
Lorentz transformation matrix
Four-vectors
12.1 The Special Theory of Relativity
12.1.4 The Structure of Spacetime
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The four-dimensional scalar product is invariant under Lorentz transformations.
Upper indices are for contravariant vectors.
Lower indices are for covariant vectors.
Raising or lowering the temporal index costs a minus sign
Raising or lowering a spatial index changes nothing
The scalar product is
Summation is implied whenever a Greek index is repeated in a product-
once as a covariant index and once as contravariant.
The scalar product can be written as
This is the Einstein summation convention.
The invariant interval
12.1 The Special Theory of Relativity
12.1.4 The Structure of Spacetime
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The displacement 4-vector is
If the interval between the two events is timelike, there exists an inertial system in which they occur at the same point.
If the interval is spacelike, then there exists a system in which the two events occur at the same time
Space-time diagrams
12.1 The Special Theory of Relativity
12.1.4 The Structure of Spacetime
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The trajectory of a particle on a Minkowski diagram is called a world line.
Because no object can travel faster than light, your world line can never have a slope less than 1.
Your motion is restricted to the wedge-shaped region bounded by the two 45° lines.
Your "future" is the locus of all points accessible to you. At any moment, your “future” is the forward "wedge" constructed at whatever point you find yourself.
Your "past,“ is the backward wedge. It is the locus of all points from which you might have come.
Your “present” is the rest. It is the region outside the forward and backward wedges.
You can't get there, and you didn't come from there. There's no way can you influence any event in the present (the message would have to travel faster than light);
If you include a y axis coming out of the page, the "wedges" become cones.
Their boundaries are the trajectories of light rays. They are called the forward light cone and the backward light cone.
With respect to your present location,
all points in the past and future are timelike
all points in the present are spacelike.
Space-time diagrams
12.1 The Special Theory of Relativity
12.1.4 The Structure of Spacetime
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Time is different from the special coordinates.
The mark of its distinction is the minus sign in the invariant interval.
That minus sign imparts to spacetime a hyperbolic geometry that is much richer than the circular geometry of 3-space.
Timelike
Spacelike
When you perform a Lorentz transformation the new and old coordinates lie on the same hyperbola.
No transformation will move an event from the upper sheet of the timelike hyperboloid to the lower sheet, or to a spacelike hyperboloid.
If the invariant interval between two events is timelike, their time ordering is absolute.
If the interval is spacelike, their ordering depends on the inertial system from which they are observed.
The invariant interval between causally related events is always timelike, and their temporal ordering is the same for all inertial observers.
Definitions
12.2 The Special Theory of Relativity
12.2.1 Proper Time and Proper Velocity
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Proper time is invariant, whereas "ordinary" time depends on the particular reference frame you have in mind.
The velocity relative to ground:
Proper velocity has an advantage over ordinary velocity:
it transforms simply, when you go from one inertial system to another.
The zeroth component is
The proper velocity 4-vector, or simply the 4-velocity
Transformations
12.2 The Special Theory of Relativity
12.2.1 Proper Time and Proper Velocity
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Proper velocity has an advantage over ordinary velocity:
it transforms simply, when you go from one inertial system to another.
Proper velocity
Definitions
12.2 The Special Theory of Relativity
12.2.2 Relativistic Energy and Momentum
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The relativistic momentum
The relativistic energy
Old terminology
The relativistic mass is
is the relativistic energy when the object is stationary.
The rest energy
The kinetic energy
The leading term reproduces the classical formula.
Momentum 4-vector
12.2 The Special Theory of Relativity
12.2.2 Relativistic Energy and Momentum
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The relativistic momentum
The energy-momentum 4-vector, or simply the momentum 4-vector
The relativistic energy
In every c1osed system, the total relativistic energy and momentum are conserved.
Relativistic mass is conserved
Rest mass is not conserved
An invariant quantity has the same value in all inertial systems.
A conserved quantity has the same value before and after some process.
Mass is invariant, but not conserved.
Energy is conserved but not invariant.
Electric charge is both conserved and invariant.
Velocity is neither conserved nor invariant.
Example
12.2 The Special Theory of Relativity
12.2.3 Relativistic Kinematics
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Conservation of momentum
The energy of each lump prior to the collision is
The energy of the composite lump after the collision is
since it's at rest.
Conservation of energy
The final mass is greater than the sum of the initial masses.
Mass was not conserved in this collision.
Kinetic energy was converted into rest energy,
A compressed spring is heavier than the same relaxed spring.
A hot object is heavier than the same cold object.
Massless particles
12.2 The Special Theory of Relativity
12.2.3 Relativistic Kinematics
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Indeterminate
A massless particle could carry energy and momentum, provided it always travels at the speed of light.
Indeterminate
Relativity offers no answer to this question.
Example
12.2 The Special Theory of Relativity
12.2.3 Relativistic Kinematics
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Conservation of momentum requires
Conservation of energy
In the next example, we will do an elastic scattering.
We call the process elastic if kinetic energy is conserved. In such a case the rest energy and the mass are conserved.
This means that the same particles come out as went in.
Example
12.2 The Special Theory of Relativity
12.2.3 Relativistic Kinematics
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Conservation of momentum in the "vertical" direction gives
Conservation of momentum in the "horizontal" direction gives
Conservation of energy gives
The Compton wavelength of the electron
Newton’s laws
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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Newton’s first law is built into the principle of relativity.
Newton's third law does not, in general, extend to the relativistic domain.
Only in the case of contact interactions, where the two forces are applied at the same physical point and where the forces are constant, can the third law be retained.
Example
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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The numerator is the classical answer.
Work-energy theorem
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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The work-energy theorem : “the net work done on a particle equals the increase in its kinetic energy”.
The work-energy theorem holds relativistically.
Work is the line integral of the force:
Since the rest energy is constant, it doesn't matter whether we use the total energy, here, or the kinetic energy.
Force transformations
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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Both the numerator and the denominator must be transformed.
A "proper" force, analogous to proper velocity, is the derivative of momentum with respect to proper time:
Example
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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The typical trajectory of a charged particle in a uniform magnetic field is cyclotron motion.
The magnetic force pointing toward the center provides the centripetal acceleration necessary to sustain circular motion.
The centripetal force
Example
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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Find the total momentum of all the charges in the loop.
The momenta of the left and right segments cancel.
Classically,
Relativistically,
Example
12.2 The Special Theory of Relativity
12.2.4 Relativistic Dynamics
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Hidden momentum.
A magnetic dipole in an electric field carries linear momentum,
even though it is not moving!
It is called hidden momentum and is relativistic, and purely mechanical;
it cancels the electromagnetic momentum stored in the fields.
Electrodynamics and special relativity
12.3 Relativistic Electrodynamics
12.3.1 Magnetism as a Relativistic Phenomenon
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Unlike Newtonian mechanics, classical electrodynamics is already consistent with special relativity.
Maxwell's equations and the Lorentz force law can be applied in any inertial system.
What one observer interprets as an electrical process another may regard as magnetic,
but the actual particle motions they predict will be identical.
Calculate the magnetic force between a current-carrying wire and a moving charge without invoking the laws of magnetism.
A current-carrying wire and a moving charge
12.3 Relativistic Electrodynamics
12.3.1 Magnetism as a Relativistic Phenomenon
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Because the two line charges cancel,
there is no electrical force on q.
q is at rest
The velocities of the positive and negative lines are
The wire carries a net negative charge!
A current-carrying wire and a moving charge
12.3 Relativistic Electrodynamics
12.3.1 Magnetism as a Relativistic Phenomenon
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Because the two line charges cancel,
there is no electrical force on q.
The magnetic force on q
A current-carrying wire and a moving charge
12.3 Relativistic Electrodynamics
12.3.1 Magnetism as a Relativistic Phenomenon
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As a result of unequal Lorentz contraction of the positive and negative lines, a current-carrying wire that is electrically neutral in one inertial system will be charged in another.
A current-carrying wire and a moving charge
12.3 Relativistic Electrodynamics
12.3.1 Magnetism as a Relativistic Phenomenon
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The magnetic force on q
The electrical force on q,
Using the transformation rules for forces
Transformations
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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We will show
Assume charge is invariant.
Electric field of a moving capacitor
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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Consider a uniform electric field in the region between the plates of a large parallel-plate capacitor
The plates are moving to the left.
Magnetic field of a moving capacitor
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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The surface currents
The magnetic field
Ampere’s law
K is the current per unit width-perpendicular-to-flow.
Surface current density
Magnetic field due to a flat sheet
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
Aljalal-phys306-162-ch12: Electrodynamics and Relativity
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The capacitor is at rest.
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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Change components
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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Consider a uniform electric field in the region between the plates of a large parallel-plate capacitor
The plates are moving to the left.
So the charge per unit area does not change
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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A long solenoid at rest.
The magnetic field within the coil is
Special cases
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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Example
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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From the Lorentz transformations
The same result as in Chapter 10 using the retarded potentials.
The field points away from the instantaneous (as opposed to the retarded) position of the charge.
Example
12.3 Relativistic Electrodynamics
12.3.2 How the Fields Transform
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From previous example
The Biot-Savart law applied to a point charge.
Definition
12.3 Relativistic Electrodynamics
12.3.3 The Field Tensor
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We will use antisymmetric, second-rank tensor
to write these equations in a compact form
A 4-vector transforms by the rule
symmetric tensor
Antisymmetric tensor
Transformation
12.3 Relativistic Electrodynamics
12.3.3 The Field Tensor
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The field tensor
12.3 Relativistic Electrodynamics
12.3.3 The Field Tensor
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The dual tensor
12.3 Relativistic Electrodynamics
12.3.3 The Field Tensor
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The current density 4-vector
12.3 Relativistic Electrodynamics
12.3.4 Electrodynamics in Tensor Notation
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The charge density is
The current density is
Because one dimension (the one along the direction of motion) is Lorentz-contracted,
The charge density and current density make the current density 4-vector:
The proper velocity 4-vector
The continuity equation
12.3 Relativistic Electrodynamics
12.3.4 Electrodynamics in Tensor Notation
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The continuity equation
The continuity equation
The continuity equation states that the current density 4-vector is divergenceless.
Maxwell's equations
12.3 Relativistic Electrodynamics
12.3.4 Electrodynamics in Tensor Notation
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Maxwell's equations
Maxwell's equations
12.3 Relativistic Electrodynamics
12.3.4 Electrodynamics in Tensor Notation
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Maxwell's equations
The Lorentz force law
12.3 Relativistic Electrodynamics
12.3.4 Electrodynamics in Tensor Notation
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the Minkowski force
The Lorentz force law
4-vector potential
12.3 Relativistic Electrodynamics
12.3.5 Relativistic Potentials
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The field tensor
4-vector potential
12.3 Relativistic Electrodynamics
12.3.5 Relativistic Potentials
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Maxwell's equations
The potential formulation automatically takes care of the homogeneous Maxwell equation
4-vector potential
12.3 Relativistic Electrodynamics
12.3.5 Relativistic Potentials
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Maxwell's equations
The inhomogeneous equation
becomes
This is an intractable equation
The Lorentz gauge condition
becomes, in relativistic notation,
In the Lorentz gauge
becomes
The d' Alembertian
The most simplest formulation of Maxwell's equations.
using