- #1
kmarinas86
- 979
- 1
If I have an oscillating charge inside of a sphere, will the integral of E(t), where t=proper time of the sphere, over the sphere's surface area result in a value of electric flux equal to the value of the charge?
Matterwave said:Yes, Gauss's law is fully compatible with relativity and is quite general.
Matterwave said:Certainly what you are looking for can be found, but doing it that way completely loses the point of the power of something like Gauss's law. The major power of Gauss's law is that we DON'T HAVE TO carry out that ridiculous integral, we already know the answer. If we had to carry out that ridiculous integral (where no symmetries are present), then we've completely missed the point and power of Gauss's law.
Think of it in terms of electric field lines in the same line of thought as Faraday's. No matter how that electron moves around, the number of field lines crossing a closed surface must remain constant as long as the charge itself does not move outside that surface. This is because although the field lines can get wavy or distorted, they cannot ever cross each other, they will not "multiply" (in the sense of the creation of more field lines), and they must start on the charge and end at infinity (since you have only 1 charge around). The number of field lines crossing the surface is exactly the electric flux.
A lot of people get bogged down by the specifics of what the field looks like "on the surface".
An alternative way to see this, if you are not satisfied with Faraday's picture of field lines, is to see simply that Maxwell's equations in differential form is completely in accord with special relativity, and the integral forms of Maxwell's equations can be obtained from the differential form by the generalized Stoke's theorem (which is also completely general).
One last point is that the Wigner Eckhart potentials, which are THE solution to E&M (in the sense that you can get the E&M fields from them for an arbitrary distribution of arbitrarily moving charges), are explicitly derived from Maxwell's equations. So, even if you get "time retarded potentials" that you are looking for, etc., they come from he Wigner-Eckhart potentials which come from Maxwell's equations!
Matterwave said:Yes, but the point is that the surface is closed...and so the inhomogeneities cancel.
The TOTAL flux out of a closed surface is equal to the charge.
"kmarinas86 (re-formatted)" said:Just consider the following:
If the charge is FIXED at the center, the electric field intensity should be constant at all parts of the sphere. So 50% of the total integrated value is on the left hemisphere, and the other 50% of the total integrated value is on the right hemisphere.
If the charge is FIXED very close to the azimuth of the right hemisphere, the electric field intensity should vary at all parts of the sphere. For example, IF 10% of the total integrated value is on the left hemisphere, THEN the other 90% of the total integrated value is on the right hemisphere.
The problem is, IF the charge is moving from the center, to a point very close to azimuth of the right hemisphere, THEN:
* Most of the left hemisphere will receive the field intensity from that charge as of the time it was closer to the center than the azimuth of the right hemisphere. (The 50% 50% case)
* Most of the right hemisphere will receive the field intensity from that charge as of the time it was closer to the azimuth of the right hemisphere than the center. (The 10% 90% case)
Matterwave said:The intensities may go up on that side, but the vector directions will tend to change. Remember that a moving charge's electric field tends to get stronger in the perpendicular directions and weaker in the parallel direction. In this case, the vectors, even though they are longer, rotate away from the normal of the sphere so the flux is conserved.
Just take a look at the picture you posted. No matter what circle I draw on that picture, even if I deform that circle, the same number of lines will pierce my circle.
"kmarinas86 (re-formatted)" said:Just consider the following:
If the charge is FIXED at the center, the electric field intensity should be constant at all parts of the sphere. So 50% of the total integrated value is on the left hemisphere, and the other 50% of the total integrated value is on the right hemisphere.
If the charge is FIXED very close to the azimuth of the right hemisphere, the electric field intensity should vary at all parts of the sphere. For example, IF 10% of the total integrated value is on the left hemisphere, THEN the other 90% of the total integrated value is on the right hemisphere.
The problem is, IF the charge is moving from the center, to a point very close to azimuth of the right hemisphere, THEN:
* Most of the left hemisphere will receive the field intensity from that charge as of the time it was closer to the center than the azimuth of the right hemisphere. (The 50% 50% case)
* Most of the right hemisphere will receive the field intensity from that charge as of the time it was closer to the azimuth of the right hemisphere than the center. (The 10% 90% case)
For our purposes, let's say that the E-field is recorded from all points on the sphere using sensors (fixed onto the sphere) whose clocks are synchronized. There is no "single observer" who makes the observations, but a set of observers in the same inertial frame as that of the sphere.
kmarinas86 said:Doesn't the angle between the line and the circle affect the component of the E-field that is normal to the differential barrier of that circle?
An area flux integral for the electric flux (units of V*m) is computed by the integration of the dot product of the local field vectors (units of V/m) with the corresponding vectors normal to the differential surfaces (vectors that represent those differential surfaces) (units of m^2) through which those local field vectors apply.
Matterwave said:Don't confuse electric field LINES and electric field VECTORS. These are two compatible, but distinct, models for electromagnetism.
Matterwave said:The flux out of that sphere is EITHER the integral of the VECTORS (which depends on angle, and intensity and whatnot, and is, in general, very hard to calculate directly), or simply the number of field lines that pierce the sphere (which is very easy to see).
Matterwave said:I did address your previous post directly. I told you that the angles for the VECTORS rotated away from the normal(s) to the sphere, and thereby compensate for their increased intensities. Your numbers 10%, 90%, and 50% are just example numbers, I don't get why you want me to address these 3 numbers directly?
You don't have 90% of the flux going in one sphere and 10% going the other. You may have something like 60% in one and 40% in the other (which, by the way, still adds up to 100%, so you haven't increased net flux at all) because the vectors turning has compensated for this effect.
Matterwave said:The lagged behind vectors don't rotate as much, but they don't grow, while the vectors in front of the electron grow but rotate away from the normal.
Matterwave said:1) Electric field lines and electric field vectors are 2 fundamentally different (although compatible) models for the electric field.
Electric field vectors are always tangent to field lines, but the method by which they represent the STRENGTH of the electric field differ.
The field vectors represent STRENGTH by the vector's MAGNITUDE (longer=stronger), whereas the electric field lines represent STRENGTH by the DENSITY of lines (denser=stronger).
top row v/c
left column phi in degrees from the +z direction
all other cells z position particle according sensor at phi a time t
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
10.0 1.00 0.99 0.98 0.97 0.97 0.96 0.95 0.94 0.93 0.92 0.90 0.89 0.88 0.86 0.84 0.82 0.79 0.76 0.70 0.59
20.0 1.00 0.98 0.97 0.95 0.93 0.91 0.90 0.88 0.86 0.84 0.82 0.80 0.77 0.74 0.71 0.68 0.63 0.58 0.50 0.36
30.0 1.00 0.97 0.95 0.92 0.90 0.87 0.85 0.82 0.80 0.77 0.74 0.71 0.68 0.64 0.61 0.56 0.51 0.44 0.36 0.24
40.0 1.00 0.97 0.93 0.90 0.87 0.84 0.81 0.78 0.74 0.71 0.68 0.64 0.60 0.56 0.52 0.47 0.41 0.35 0.27 0.16
50.0 1.00 0.96 0.92 0.88 0.84 0.80 0.77 0.73 0.69 0.66 0.62 0.58 0.54 0.49 0.45 0.40 0.34 0.28 0.21 0.12
60.0 1.00 0.95 0.90 0.86 0.82 0.77 0.73 0.69 0.65 0.61 0.57 0.52 0.48 0.44 0.39 0.34 0.29 0.23 0.16 0.09
70.0 1.00 0.94 0.89 0.84 0.79 0.74 0.70 0.65 0.61 0.57 0.52 0.48 0.43 0.39 0.34 0.29 0.24 0.19 0.13 0.07
80.0 1.00 0.94 0.88 0.82 0.77 0.72 0.67 0.62 0.57 0.53 0.48 0.44 0.39 0.35 0.30 0.26 0.21 0.16 0.11 0.06
90.0 1.00 0.93 0.87 0.81 0.75 0.70 0.64 0.59 0.54 0.50 0.45 0.41 0.36 0.32 0.27 0.23 0.19 0.14 0.10 0.05
100.0 1.00 0.93 0.86 0.79 0.73 0.67 0.62 0.57 0.52 0.47 0.42 0.38 0.33 0.29 0.25 0.21 0.17 0.13 0.08 0.04
110.0 1.00 0.92 0.85 0.78 0.72 0.66 0.60 0.55 0.50 0.45 0.40 0.36 0.31 0.27 0.23 0.19 0.15 0.11 0.07 0.04
120.0 1.00 0.92 0.84 0.77 0.70 0.64 0.58 0.53 0.48 0.43 0.38 0.34 0.29 0.25 0.21 0.18 0.14 0.10 0.07 0.03
130.0 1.00 0.91 0.83 0.76 0.69 0.63 0.57 0.51 0.46 0.41 0.37 0.32 0.28 0.24 0.20 0.17 0.13 0.10 0.06 0.03
140.0 1.00 0.91 0.83 0.75 0.68 0.62 0.56 0.50 0.45 0.40 0.35 0.31 0.27 0.23 0.19 0.16 0.12 0.09 0.06 0.03
150.0 1.00 0.91 0.82 0.75 0.68 0.61 0.55 0.49 0.44 0.39 0.34 0.30 0.26 0.22 0.19 0.15 0.12 0.09 0.06 0.03
160.0 1.00 0.91 0.82 0.74 0.67 0.60 0.54 0.49 0.43 0.38 0.34 0.30 0.25 0.22 0.18 0.15 0.11 0.08 0.05 0.03
170.0 1.00 0.91 0.82 0.74 0.67 0.60 0.54 0.48 0.43 0.38 0.33 0.29 0.25 0.21 0.18 0.14 0.11 0.08 0.05 0.03
180.0 1.00 0.90 0.82 0.74 0.67 0.60 0.54 0.48 0.43 0.38 0.33 0.29 0.25 0.21 0.18 0.14 0.11 0.08 0.05 0.03
The time-retarded electric field is an electromagnetic field that is calculated at a specific time based on the position and movement of charged particles at an earlier time. It takes into account the finite speed of light, so the field is "retarded" or delayed in time compared to the actual position of the charged particles.
A Gaussian surface integral is a mathematical method used to calculate the electric field at a point due to a distribution of charges. It involves choosing a closed surface, known as a Gaussian surface, and integrating the electric field over the surface to determine the total electric flux passing through it.
The time-retarded E-field and Gaussian surface integral are related because the Gaussian surface is used to determine the electric field at a specific point in time. By integrating the electric field over the Gaussian surface, we can take into account the time delay of the field and calculate its value at a specific time.
The time-retarded E-field and Gaussian surface integral are important concepts in electromagnetism and are used to understand and predict the behavior of electric fields in various situations. They are essential for calculating the forces and interactions between charged particles and are crucial in many areas of physics, including optics, electronics, and particle physics.
Yes, there are many real-world applications of the time-retarded E-field and Gaussian surface integral. They are used in the design and analysis of electrical circuits, antennas, and other devices that use electric fields. They are also essential in understanding the behavior of electromagnetic waves, which are used in various technologies such as radio, television, and wireless communication.