Recent content by cwilkins

Here is some typical reasoning: 1. The spatial coordinates themselves are independent of time, so performing time differentiation before or after has no effect on the result (provided the integration is definite). 2. The integrals are over the same *arbitrary* volume. If they are equal then the...

Intuitively you might assume that the light from each star shouldn't reach the other, but that isn't the case. Here's the problem with your reasoning: velocities don't add linearly. The formula you would use to find the observed velocity between the two stars is v_{obs} = \frac{v_1 + v_2}{1+v_1...

It is not true that \frac{0}{|\mathbf{r}|^5}=\delta ^3( \mathbf{r}) (See below.) Regarding the definition of the Dirac delta, you can create a delta from basic functions as long as it's clear what you are limiting. The limit of a Dirac delta as you approach the active point is always zero--not...

It comes from the Divergence Theorem and relating the expression \frac{\hat{r}}{r^2} over the volume/surface of a sphere. In particular, you find that the surface integral of the divergence of that quantity over a sphere of any radius is equal to 4\pi. Normally the effect of increasing radius...

The idea of a spacetime, which is really what I am getting at by suggesting that an observed process can be viewed from either the time or spatial perspective, is one that Einstein based his general theory on. By studying the equations you will find that time and space are equivalent in a...

You gave the expression for the force in your post. The values you gave could have been plugged into it.

Measuring the length is a time-dependent process. To be clear, I am not saying that the rest length of the object changes as a function of time. If you are moving relative to another frame it is not sufficient to place a meterstick next to the object. The way Einstein thought about it was to...

I don't agree. Length measurement is a time-dependent process in the relativistic sense, since you must have an associated moving frame for length contraction to occur (in SR). If the frames were moving at different rates, you would detect a change in length. The usual way length would be...

What is the distinction between "slowing all time-dependent processes down" and "slowing time down"?

To add to what mfb said, I'm pretty sure the only way you could use the phase velocity to transmit information would be to detect changes in the phase velocity. Those changes will propagate at a speed less than or equal to the speed of light in whichever medium you're considering.

This is not true. You can easily see that by evaluating the field at points on the sphere. The magnitude of the separation vector is no longer the same at each point on the sphere, so the field cannot be the same at all points on the sphere. You would be correct in saying that the net flux is...

Why would it be constant at a face? You would need a special configuration of charges for that to be true. Also, Gauss's Law is the general method. You just need to choose your coordinate system appropriately for the charge distribution of interest. In general, you are calculating a double...

I suspect that either nothing will happen, or there will be a layer on top that is negatively charged, followed by a layer immediately below (on atomic scales) that is positively charged. The configuration will somehow adjust itself so that there is no field inside the bulk of sphere. (Note: I...

The way I think about it is you had to put some energy into forming that configuration of charges. The energy went into separating the charges and putting them in place, which means there will be some potential energy associated with holding those charges apart from each other. The result will...

The field due to the point charge is constant on the surrounding sphere because every point on that sphere is at the same distance away from the charge. You can think of r^2 as the (squared) distance from the point where the charge is to wherever you're measuring the field. Every point on the...