Basics of Electromagnetism and special relativity

Charge moving perpendicular to the wire experiences a force parallel to the wire, perpendicular to its direction of motion. Positive ions cannot cause a horizontal field at the test charge position. The x' component of the field from an ion on the left is exactly cancelled by the x' component of the field of a symmetrically positioned ion on the right. The effect we can see is caused by the electrons. Electrons are all moving diagonally in the test charge frame, downward and toward the right. Consider the two symmetrically located electrons e₁ and e₂. Their electric fields, relativistically compressed in the direction of the electrons' motion, are represented by a field lines. You can see that, although e₁ and e₂ are equally far away from the test charge, the field of electron e₂ is stronger than the field of electron e₁ at that location. That is because the line from e₂ to the test charge is more nearly perpendicular to the direction of motion of e₂. In other words, the angle θ' that appears in the denominator of equation for the magnitude of the field of electron in the frame moving relative to the electron is here different for e₁ and e₂, so that sin²θ'₂>sin²θ'₁. That is true for any symmetrically located pair of electrons on the line, as you can verify with the aid of picture. The electron on the right always acts stronger. Therefore, summing over all the electrons must give a resultant field E' in the x direction. The y' component of the electrons' field is exactly cancelled by the field of the ions. That E'_y is zero is guaranteed by Gauss's law, for the number of charges per unit length of wire is the same as it was in the lab frame. The wire is uncharged in both frames. The force on the test charge, qE'_x, transformed back into the lab frame is a force proportional to v in the x direction, which is the direction of v×B if B is a vector in the z direction, pointing out of the page.

Suppose a charge q has been moving at constant speed v₀ in the x direction for a long time. Suddenly it stops after a short period of constant deceleration. Electric field of a charge travels radially outward and moves with the velocity that the charge had while emitting that field.

The velocity versus time graph

The electric field

Assuming that the v₀ ≪ c, we can neglect the relativistic compression of the field lines. Time t=0 it was the moment when deceleration began, and position x=0 it was the position of the particle at that moment. The particle has been moved a little farther on before coming to a stop, Δx=1/2*v₀Δt₁. That distance is very small compared with the other distances in the picture.

We now examine the electric field at a time t=Δt₂≫Δt₁. The electric field reaches a distance R=cΔt₂. Thus, the field at a distance greater than R=cΔt₂, in region I must be a field of a charge which has been moving and is still moving at the constant speed v₀. That field appears to come from the point x=v₀Δt₂ on the x axis. That is where the particle would be now if it hadn't stopped. On the other hand, the field at a distance less than c(Δt₂-Δt₁), in region II must be a field of a charge at rest close to the position x=0 (exactly at x=1/2*v₀Δt₁).

What must the field be like in the transition region, the spherical shell of thickness cΔt₁ between region I and region II? A field line segment AB lies on a cone around the x axis which includes a certain amount of flux from the charge q. Due to the Gauss' law, if CD makes the same angle θ with the axis, the cone on which it lies includes that same amount of flux. (Because the v₀ ≪ c, we can neglect the relativistic compression of the field lines.) That's why AB and CD must be parts of the same field line, connected by a segment BC. The line segment BC shows us the direction of the filed E within the shell. This field E within the shell has both a radial component E_r and a transverse component E_θ. From the geometry of the figure their ratio is easily found.

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(2)

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A remarkable fact is here revealed: E_θ is proportional to 1/R, not to 1/R²! As time goes on and R increases, the transverse field E_θ will eventually become very much stronger than E_r. Accompanying this transverse (that is, perpendicular to R) electric field will be a magnetic field of equal strength perpendicular to both R and E. This is a general property of an electromagnetic wave.

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According to current theory, the electrons in the atoms occupy only the allowed energy levels and emit radiation only when they jump between energy levels, but it is only a simplification. Every accelerating charged particle emit electromagnetic radiation. Electrons continously emit and absorb electromagnetic radiation. By adding up the waves emitted by the hydrogen atom, we can see that the source is the expiring vibrations.

According to current theory, the beam of light is not a wave propagating through space, but rather a collection of discrete quantized wave packets (photons), each with energy hν, but it is only a simplification. The energy of light depends on the intensity of the electric field of the electromagnetic wave. The frequency of the electromagnetic wave depends on the velocity and acceleration of the electrons that emit it. The higher the frequency of the wave and the movement of electrons, the greater the accelerations affect the electrons, and thus the intensity of the electric field of the electromagnetic wave emitted by electrons is higher.

In addition to the time dilation related to the motion (relative velocity) there is also gravitational time dilation. Gravitational time dilation is a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of spacetime.

In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of the twins makes a space journey at a speed close to the speed of light and when returns home he find that the twin who stayed on Earth is older. This result appears puzzling because each twin sees the other twin as moving, and so, according to an incorrect naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged more slowly. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey, and so there is no symmetry between the spacetime paths of the two twins. Therefore the twin paradox is not a paradox in the sense of a logical contradiction.

Yes, two scientists took four atomic clocks and set them to be at exactly the same time. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at ground. When they got back, they found out that clocks disagree with one another, and their differences were consistent with the predictions of special and general relativity.

Clocks on the ISS space station run slightly slower than reference clocks on Earth, while clocks on GPS and Galileo satellites run slightly faster.