This is a diversion from the [thread=110763]shape of a relativistic wheel[/thread] thread. The idea is to consider the seemingly paradoxical outcome of having the bulk of a relativistic rolling wheel above the level of the axle. Abstract: A charged hoop of rolls at relativistic velocity past a test charge level with the axis. What is the magnitude and direction of the force on the test charge? In the hoop rest frame, the hoop charge is evenly distributed, so it seems that the force on the test charge should be horizontal. In the test charge rest frame, most of the hoop's charge is above the axis, so it seems that the force on the test charge should have a vertical component. ???? PROFIT!!! Details... (This is stilted and kind of haphazard. I'm not that familiar with the formalism, and I'm supposed to be studying for an exam tomorrow. Please ask for clarification and point out absurdities as necessary.) Note - The diagrams are kind of pretty, but I really didn't need to make them 3D, and the the z-axis separation between the hoop and the charge isn't useful. So ignore it. d=0 Choose an inertial reference frame S(x,y,t) (the hoop frame) such that: The surface is in the plane y = -R The centre of the hoop is fixed at (x,y)=(0,0) The surface and test charge are moving at velocity v(x,y) = (v,0). At t=0, the test charge is at (x,y) = (0,0) i.e. the test charge passes the centre of the hoop at t=0 Diagram: A = hoop axis, worldline (x,y,t) = (0,0,t) B = test charge, worldine (x,y,t) = (vt,0,t) Please Register or Log in to view the hidden image! Define an inertial reference frame S'(x',y',t') (the test charge frame) with velocity v relative to frame S, same axes and coincident origin. Diagram: A = hoop axis, worldline (x',y',t') = (-vt',0,t') B = test charge, worldine (x',y',t') = (0,0,t') Please Register or Log in to view the hidden image! In frame S, the hoop is circular with radius R. In frame S, the hoop is uniformly charged with lineal charge density \(\rho\). The hoop rolls without spinning - in frame S, the angular velocity of the hoop is \(\omega = v/R\) Both the hoop and the charge have negligible mass and negligible bulk. The test charge has charge q . In frame S, what is the magnitude and direction of the force on the test charge at t=0? In frame S', what is the magnitude and direction of the force on the test charge at t'=0? While I think this should be easier than the GR problem of the other thread, I still don't have the ability to solve it. But it's clear that it's going to involve but electric and magnetic forces - Moving charges make a magnetic field Magnetic fields apply forces to moving charges And maybe the resolution of this potential paradox can be similarly applied to the potential paradox in the other thread, using the analgous concept of Gravitomagnetism.
The charge may be evenly distributed, but there's also a current (since the wheel is spinning) that will create a non-zero magnetic field at the centre, oriented axially (along the z axis in your diagrams). Since the charge is moving in the hoop rest frame, it actually will get pulled up or down if it is moving horizontaly in that frame. Good luck with your exam by the way.
This problem can be somewhat ugly if you allow the hoop to be a 2-dimensional surface, so let's just consider a 1-dimensional hoop for now, with charge density \(\lambda\) as seen in frame \(S\). The current associated with the spinning hoop would then be \(2\pi R^2\omega\lambda\). In frame \(S\) you can then apply the Biot-Savart Law to find the resulting magnetic field at all points inside the hoop (and beyond), and Coulomb's law to calculate the electric field, because both fields are static in this frame. Then once you have these fields, you can apply the force law \(\vec{F}=q\left(\vec{E}+\vec{v}\times\vec{B}\right)\) and you're done for frame \(S\). Then you can just apply Relativity to calculate the force seen in \(S'\). In frame \(S\) you've got a test charge \(q\) moving with velocity \(\vec{v}\) along a plane containing a loop of current, so you expect it to undergo some magnetic deflection as well as possible electric deflection from the charges in the loop. Without applying Relativity, the problem would be a freakin' mess and you'd have to go through the whole process of directly solving Maxwell's equations in full, assuming you were given the shape and motion of the hoop and the associated charge and current densities therein as seen in frame \(S'\) as your starting point. It's enough of a pain in the ass to calculate the electric and magnetic fields of moving point charges without Relativity; doing it for a 1 or 2-dimensional charged, spinning, moving hoop would be a large homework assignment in and of itself.
Thanks przyk. Is there a similar situation with gravity in that other thread? Does the gravity of a spinning wheel exert a lateral force on a passing particle? Thanks! Three, actually... one in an hour, two more same time tomorrow and Wednesday. Interestingly, my Sciforums behaviour actually meets a number of criteria for substance dependence (a likely topic on today's exam).
Thanks Captain. What about a simpler scenario (also posted in the other thread, but with gravity)... Consider two infinite straight parallel rods with uniform proper linear charge \(\rho\), in constant relative inertial motion such that they remain parallel and separated by proper distance 2L. Choose an inertial (x,y,t) reference frame S, such that: The upper rod is positioned at y=L, and has velocity v parallel to the x-axis. The lower rod is positioned at y=-L, and has velocity -v parallel to the x-axis. S' (the upper rod rest frame) has velocity v parallel to the x-axis relative to S. S'' (the lower rod rest frame) has velocity -v parallel to the x-axis relative to S. In S, the charge density is \(\gamma\rho\) for each rod, the upper rod carries current \(\gamma\rho v\) the lower rod carries current \(-\gamma\rho v\) In S', the upper rod has charge density \(\rho\) and zero current the lower rod has charge density \(\gamma'\rho\) and carries current \(\gamma'\rho v'\). In S'', the lower rod has charge density \(\rho\) and zero current, the upper rod has charge density \(\gamma'\rho\) and current \(-\gamma'\rho v'\) Where \(v' = 2v/(1 + v^2/c^2)\) And \(\gamma' = (1 + v^2/c^2)/\gamma^2\) So, three questions: What is the acceleration of a test charge A at rest in S at (x,y,t)=(0,0,0)? What is the acceleration of a test charge B with velocity v along the x-axis? What is the acceleration of a test charge C with velocity -v along the x-axis?
You've gotta be careful here- the charge densities are not the same as seen in all reference frames. Assuming there is no charge flow across the boundaries of a given volume, the net charge contained in that volume is the same for all inertial frames (I will demonstrate this at some point in my Maxwell thread, already worked it out on paper for the most general possible case), but the volume itself looks different in different frames, and the times associated with the points in a volume will also be different. I.e. what you need to consider is this: At a given time in frame \(S\), i.e. \(t=0\), you've got a collection of charges along a line with some given density. In \(S'\) and \(S''\), the points associated with these charges at \(t=0\) will correspond to different times as seen in these frames as well as different distances, and what you really need to ask is what the line charges look like in these frames at \(t'=0\) and \(t''=0\) (or some other arbitrary fixed time); times \(t'=t''=0\) do not coincide with \(t=0\) unless \(x=x'=x''=0\) as well. Not too difficult to handle, especially in the case of infinite line charges, but it is a caveat you need to keep in mind, and can help clear any confusion when considering which lengths get contracted with respect to what frames and when. When you take all this into account and apply the correct transformation rules for charge and current densities, I believe you'll find that the currents in each frame are still proportional to charge density times velocity, but the charge densities look different in each frame, as do the velocities (I note you already accounted for the differences in velocity, along with the proper Relativistic addition formula). Then once you've got the charge and current densities for each frame, it should be fairly straight forward to apply the Biot-Savart and Coulomb laws as before (Maxwell's equations always reduce to the static case, provided there's a symmetry guaranteeing that the electric and magnetic fields remain constant), and the problem should be fairly straightforward to solve in any of the three chosen reference frames, although it's probably easiest to work in frame \(S\).
Thanks - that was all supposed to be included, but I left out the gamma factors on the charge density in S. Fixed now. Will do the EM homework in a bit Please Register or Log in to view the hidden image!
The test charge is stationary in the ground frame S: \(u_x=u_y=u_z=0\) Let \((F_x,F_y,F_z)\) be the (unknown) three-force acting on the charge in frame S. The three-force acting on the charge in the frame S' comoving with the axle of the hoop is: \(F'_x=F_x+\frac{u_y V}{c^2+u_x V}F_y+\frac{u_z V}{c^2+u_x V}F_z\) \(F'_y=\frac{c^2 \sqrt{1-(V/c)^2}}{c^2+u_x V}F_y\) \(F'_z=\frac{c^2 \sqrt{1-(V/c)^2}}{c^2+u_x V}F_z\) Therefore: \(F'_x=F_x\) \(F'_y=F_y/\gamma\) \(F'_z=F_z/\gamma\) In the frame S', \(F'_y=F'_z=0\) due to the radial symmetry. It follows immediately that : \(F_y=F_z=0\) \(F_x=F'_x\) In other words, there is no transverse component to the force, hence, no "paradox".
