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• SR Theory of Electrodynamics for Relative Moving Charges

By James Keele, M.S.E.E.

October 27, 2012

• Special Theory of Relativity (SRT)

Basic Postulates

1. Relativity Principle(RP): all inertial frames are totally equivalent for the performance of all physical experiments

2. light travels rectilinearly with constant speed c in vacuum in every inertial frame

• Logic applied to the 2nd Postulate: Constancy of the Speed of Light.

If Physicists accept the law, , and believe that and are constant in all inertia frames then must be constant in all inertial frames. Otherwise, if and/or are/is not constant in all inertial frames, because is not constant, then Postulated 1 is invalid because the electrodynamics laws of nature contain these constants.

o oo

1c

o

o o oc

o

o

oc

• Other Basic Considerations

1. The charge of an electron or proton is mathematically considered herein to be a point charge and does not have a finite size.

2. An uniform electric field exists about a stationary electric point charge, pervading the space equally in all directions and falling off in intensity at 1/r2.

3. Velocities are relative between interacting particles.

• Other Basic Considerations Contd 4. The electric field of a moving charge pervades all the space of its inertial frame, thus having instant reaction with a charge in contact with it. Acceleration of a charge creates a new velocity that changes the electric field that spreads out over the new inertial frame at the speed of light.

5. The charge value, q, is invariant from one inertial frame to another.

6. A positive sign on the overall magnetic force represents repulsion while a negative sign represents attraction.

7. A negative sign must be entered into the equations for negative charges such as electrons. A positive sign must be entered into the equations for positive charges such as protons. This makes the direction of the overall force appear correctly as in 6 above.

• SRT Formalism Employed

1. Lorentz Transformation

2. Three Vectors: a(a1, a2, a3) (lower case)

3. Four Vectors: V(V1.V2.V3,V4) (Upper case)

Four vector force formula:

4. The four vector force is good for transforming force between inertial frames and creating force laws.

5.

F

2c,

vffF v

22 c/1/1 vv

• Mathematical Setup for Analysis of Force Between Relative Moving Charge Particles

• Lorentz Force LawStarting Point for the Four-Force SRT Transformation

Relativists Starting Point Lorentz Force Law (Wrong)

(1)

Keeles Starting Point (Right)

(2)

f=3-force, q=charge, e=electric field, v=relative velocity, h=magnetic field, c=velocity of light

2cq

hvef

ef q

• Why Relativists Starting Point is Wrong

1. When we are just trying to derive the force between a stationary charge and a moving charge the Lorentz Force Law that contains the magnetic field is the wrong starting point. If we include the term that has the magnetic field and assume it comes from the moving charge itself, then we would be doing a transform from two different inertial frames which is a no-no in SRT. If the term involving the magnetic field arises from a separate source other than the moving charge, then the transform would work for the transform of magnetic field. But we are not doing that, so we can expect the magnetic force to fall out of the transform of just the moving charges electric field. This was found to be the case. The is the most important point of this whole presentation that shows where errors are made.

2. The regular Lorentz Force Law could be applicable in Cathode Ray Tube or accelerators where the source of the magnetic field is separate from the magnetic field created by the moving charge. You dont start out in two different inertial frames.

3. An experiment performed by Keele with his results shows that his is the correct starting point.

4. The magnetic field h in (1) is derived from the Biot-Savart Law which was derived before the Old Amperes Law from current flowing in a wire. The magnetic field from a moving isolated charge is different from the magnetic field of a current element as will be shown in a later slide.

• Transformation Results

Relativists Results

(3)

Keeles Results

(4)

evh c

1

3/222233 sinc/-1q

vr

re

• Length Contraction of e-field of Moving Electron as seen by a Stationary Particle

• Magnetic Force Between Relative Moving Isolated Charges

Total electric field and magnetic forces between the relative slow moving charges (after mathematical manipulations of (4) using the Binomial Theorem and eliminating higher orders of ) :

(5)

Subtracting the stationary electric field force from (5) we have the magnetic force:

(6)

22 c/v

22

2

3

1221

12mcos5.15.0

c

qkq

v

r

rf

2

2

2

3

122112 cos5.15.0

c1

qkq v

r

rf

• Current Element

Length contraction of the spacing between charges in the current element

increases the charge line density of the current element as seen by a stationary

observer ().

Stationary Electrons

Stationary Protons

Moving Electrons --- showing length contraction of each charges field and the length contraction of the spacing between the charges in the current element

• The Magnetic Force Law Between a Stationary Charge and a Stationary Current Element

Include one more relativistic effect to (4) for a current element (current flowing in a short piece of a conductor): This effect is length contraction of the spacing between the current carrying electrons in the current element as seen by the stationary charge. This has the effect of increasing the charge density of the current carrying electrons in the current element. The result is:

(7) Where is the line charge density.

Notice there is a small magnetic force on a stationary charge with respect to the current element.

22

2

3

1221

12mcos5.11

c

dskq

v

r

rf

• Eq. (7) is applied three times to the cross combinations of charges in the two current elements and then the resulting forces are added. The charges in (7) are replaced by resulting in the Old Amperes Law:

(8)

A study of this law reveals that successive current elements with current in the same direction repel each other. This fact has been demonstrated by experiments of Peter Graneau. This is an effect the Biot-Savart Law does not allow.

Eq. (8) obeys Newtons 3rd Law whereas most other similar laws do not. They generally have to be integrated around a closed loop to have any applicability.

The Magnetic Force Law Between Stationary Current Elements (Found to Be the Old Amperes Law)

sv'ds/ I

12212222121

1212m32

c

dsdsIkI rsdrsdsdsdrf 111

r

• Equivalent Mathematical Form of Old Amperes Law

Eq. (9) below is mathematically equivalent to Eq. (8) in the previous slide:

(9)

where = unit vector in the direction of r12 and r12 = magnitude of the vector r12 joining the two current elements. The constants are k = 1/40 (0 = permittivity of free space) and c = speed of light. The I1 and I2 are current magnitudes and ds1 and ds2 are current element lengths. The angles are: 1 = angle between ds1 and r12; 2 = angle between ds2 and r12; = angle between the plane of ds2 with r12 and the plane of ds1 with r12.

2121212

2

21211212m

coscoscossinsin2c

dsdsIkI

rrf

12r

• Experiment With the Old Amperes Law

A simple experiment was performed on various shapes of one-turn coils. The inductance L of these coils was computer calculated using Amperes Law. Then the Inductance was measured by determining resonance of the inductor with a calibrated capacitor. The measured value of L is then compared with the computed value.

• Method employed to measure inductance (Lm) of a coil

Calibrated Capacitor

Formulas: , Resonance

Sine Wave Generator

One-turn coil

Frequency Meter

Scope

2

LIE

2

Cf2

1L

2m

• One-turn solenoid set up for calculating its inductance using the old Amperes Law

• Math showing how inductance may be calculated using a computer

The energy stored between two current elements

and is:

(3)

If both differential lengths and are imagined to move

outward with r to keep angle variables constant, then all the

variables in the () function are constant and only r varies.

The () function can then be moved outside the integral sign,

and the result of the integration is then:

(4)

• Math for one-turn solenoid inductance contd

Inserting Eq. (1) into (4), we get:

(5)

From Fig. 1 we know , ,

. To set up (4) for computer integration,

we further set 1 degree, so

(6)

For the present case, . Substituting (6) into (5), we

get:

(7)

where K is a constant.

• Math for one-turn solenoid c

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