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But what we never seem to see is why the electron and positron move the way that they do. Saying "they move like they do because of the force on them" doesn't explain anything at all. It's a non-answer. The equation of motion for charge particle (electron,positron) in magnetic field is $$m\frac{d}{dt}\left(\frac{\mathbf ... 3 There are two factors at play here. The Lorentz force which causes the paths to bend with a radius proportional to the particles velocity and with a sense that dependent on both the particles charge and the direction of the particles velocity. In high energy (compared to m_e events) such as the one pictured, the particles are nearly co-linear at the ... 3 The Gauge Theory of Gravity (GTG) by Lasenby, Doran and Gull has a background spacetime with fields on it. It is basically derived from the same physical principles but as a background theory. It ends up not being the same theory, for instance it doesn't have the same isotropic solutions, and I think it does not allow time travel and such (unlike General ... 2 First, we will compute \text{div}\,F. The partial derivatives are given by$${{\partial F}\over{\partial x_i}} = q {\partial\over{\partial x_i}}\left({{x_i}\over{r^3}}\right) = q\left({1\over{r^3}} - {{3r^2 {{x_i}\over{r}} x_i}\over{r^6}}\right) = 0.$$Thus, \text{div}\,F = 0 away from the origin. Consider now a ball B of radius r centered at the ... 2 There is a grand tradition in electromagnetism to talk about the electric fields using the same terminology as we use for velocity fields. For instance we talk about the flux which rightly is a flow per area (and sometimes we multiply by the area and still call it a flux, which is even more confusing to call two things a flux) but it isn't a flow because it ... 2 What you have here is basically the B-field seen in the frame of a charge with velocity \mathbf{v}, moving in an electric field that is \mathbf{E} \perp \mathbf{v}. You can derive this expression by considering the relativistic field transformations of \mathbf{E} and \mathbf{B} in a moving frame, I'll only show you the most important steps, for a ... 2 You can find some information about that on John D. Norton's website. Einstein thought of this at the age of sixteen. Here's another article: "If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such ... 2 In terms of a dielectric, it means there is a linear constitutive relation between the vectors.$${\boldsymbol D} = \epsilon_0 {\boldsymbol E} + {\boldsymbol P}$$Or$${\boldsymbol D} = \epsilon_r \epsilon_0 {\boldsymbol E}$$where \epsilon_r \epsilon_0 is a scalar relative and vacuum permitivitty. This way, there is now a linear relation between the ... 2 I don't think there is any universal "intuition" to tap into, aside from that which comes from practice. You perhaps need to explore different physics texts in the electromagnetic department. I for example loathed Jackson as a learning text: it is comprehensive and useful as a reference for refreshing knowledge, but not good at conveying it. Volume 2 of the ... 1 I've only really done antennas in undergrad, and the most we really looked at was phased arrays of half wavelength antennas and looking at the resulting field distribution far from the source, so take what I say with a grain of salt. One interesting thing to look at might be genetic optimization processes for antenna design. I believe that there has been ... 1 Here is the misunderstanding: Now, if the magnetic force is greater than mg, the wire moves up. Now magnetic force is up and displacement is up too which means that work done by magnetic force should be positive. The statement that a magnetic field does no work is of a static magnetic field. You are positing a changing magnetic field by the word ... 1 It works just like every other kind of thermal energy. If a resistor can give out energy to the environment, it can also receive it. For example, if it gives it out by radiating, it can also absorb radiation; if it gives it out by having its fast-moving atoms smash into air molecules, then fast-moving air molecules can also smash into it. When it's in ... 1 Consider the Nordström spacetime:$$ds^{2} = - f dt^{2} + \frac{1}{f}dr^{2} + r^{2}d\theta^{2} + r^{2}\sin^{2}\theta d\phi^{2} where $f = 1 - 2M/r + Q^{2}/r^{2}$, in units where $G = c =1$ and Gaussian units are chosen for the electric charge. While it's unclear what you mean by "curvature" since, generically, the curvature of this metric is a tensor ...

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The answer to your question is the obverse of it: we assign a color to an object based on the wavelengths which are reflected to our eyes (or in the case of filters, transmited to our eyes). That means other wavelengths are absorbed. The absorption of wavelengths is based, primarily, on the chemistry of the object. Red dye applied to cotton cloth is a ...

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So you have that $\vec{B} = \frac{a}{r^2}\begin{bmatrix}0 & z & -y\end{bmatrix}$ thus the magnitude is $B = \frac{a}{r^2}\sqrt{z^2+(-y)^2},$ where $a$ is unknown. Can you write that as a function of $r?$ Can you investigate what happens as $r$ goes to zero? Are magnetic fields continuous in empty space (a vacuum)? If so, try the next five: What ...

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The magnitude of the B-field is $a/r$ and circulates around the axis. By symmetry, you understand that the magnitude is zero on-axis. But if $a$ is anything but zero, your expression gives an infinite B-field magnitude. Therefore $a$ must be zero and therefore the B-field is also zero everywhere else inside the pipe. The result also follows from Ampere's ...

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In principle it's a nice idea but there are some major flaws. First of all, even in 2D, your scheme is a super-unstable configuration (tiny fluctuation would destroy it) and the use of the beam repulsion (space charge) in the inner region just does not work. If it would be strong enough to bend the beams, they would be themselves destroyed expelling plenty ...

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GR can be recast into an equivalent but conceptually quite different form, using teleparallel gravity. This approach introduces the Weitzenboeck connection, which has no curvature, but has torsion. The presence of torsion indicates that gravity is not geometrized. Recall that in GR, we can always choose a locally inertial coordinate system such that the ...

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