EOE-002 B from E

Electostatic-field (right) causing the magnetic-force (left) of a neutral-wire's current on a moving-charge: This figure shows two views of the frame-invariant proper-force [1] (dark green-arrow) on a positive moving-charge (blue-dot), due to a current of negative electrons (red-dots) in a neutral wire (shown in the left panel). The electrons in the wire are moving in the same direction as the moving-charge, and (for simplicity) at the same speed.

Notice that the magnetic force on the moving-charge in the frame of the neutral-wire (left) results from an electric field in the frame of the moving-charge itself (right). This is true in general.

In standard symbols the frame-invariant proper-force Fo ≡ mα = qE = γ(qv×B) = γdp/dt = γF, where α is the proper-acceleration (were the force unbalanced) felt by the moving-charge, relativistic momentum p ≡ mw in terms of synchrony-free proper-velocity w ≡ dx/dτ = γv where τ is frame-invariant proper-time on the clocks of the moving-charge, Lorentz-factor γ ≡ dt/dτ, coordinate-velocity v ≡ dx/dt, and frame-variant force F ≡ dp/dt (if unbalanced) is the force observed (lighter green-arrow above) in the neutral-wire frame. Of course x and t are 3-vector position and time (respectively) in the local map-frame, m and q are our moving-ion's rest-mass and charge, while B and E are local magnetic and electric fields.

Note also that:

(i) we are not allowing the force on the moving-charge to change its constant-velocity trajectory e.g. as though it is being counteracted by thrust T = mα downward from an "ion-engine" (pun intended),

(ii) we are not showing the magnetic field from the current in the panel at right because it has no effect when the moving charge is stationary, plus

(iii) we are ignoring E-field distortion-effects due to high-speed motion in the wire, if indeed they are an issue for the case of a current-carrying wire. 

Questions that students might try to answer based on their own real-time observations and measurements:

Neutral-wire panel questions

1a. Which way is the current flowing in the neutral-wire frame?

1b. How many electrons pass a given point in the neutral-wire, per second?

1c. What is the current (in Amperes) through that neutral-wire?

1d. How long (in seconds) does it take any single electron to traverse the image field?

Moving-charge panel questions

2a. From the moving-charge's perspective, which way is the current flowing?

2b. From that perspective, what type of charges are carrying that current?

2c. How many charges per second does the moving-charge see passing the point on the wire beneath it?

2d. As a result, what is the magnitude of the current (in Amperes) from the moving-charge's perspective?

Time-is-local questions

These use relations involving the traveler-kinematic Lorentz-factor or "speed-of-map-time" equation γ ≡ dt/dτ, where t is map-frame time and τ is proper-time on the clocks of a traveler. Since speeds are the same in both panels, this translates to the simple relations: γ = Δte/Δtleft = Δtleft/Δtp = Δxe/Δxleft = Δxleft/Δxp, where subscript left refers to (temporal & spatial) separations in the left panel, while subscripts p and e refer to (blue) proton and (red) electron separations in the right panel. Here "length-contraction" arises most simply from measurable time-ratios between comparable events, in this case for objects traveling at the same speed.

Note that γ in turn can be converted into absolute values for proper-velocity w ≡ δx/δτ = c Sqrt[γ2-1] and coordinate-velocity v ≡ δx/δt = c Sqrt[1-1/γ2] = w/γ. These relations follow directly from the flat space-time version of Pythagoras' theorem, namely (c δτ)2 = (c δt)2 - (δx)2 and the lightspeed space-time constant value of c ≡ 299,792,458[m/s] ≈ 3×108[m/s].

3a. How does the time Δtleft for electrons to traverse the distance between protons in the left panel compare to the times in the moving-charge frame: Δte for a proton to pass two electrons and Δtp for an electron to encounter two protons? What speed-of-map-time does this suggest for charge-motion in our wires?

3b. Can you similarly estimate the absolute speed of charge-motion (for both the moving-charge and the current-carrying electrons) from the spacings Δxleft, Δxe and Δxp between charges in the left and right panels?

Absolute-distance questions

4a. If you know the speed of the charges, and the time it takes to traverse the image field, can you also determine the width of the field-of-view in each panel?

4b. If you know the width of the field-of-view, then, how far away is the moving-charge from the wire?

Field-strength questions

These might use standard textbook-equations e.g. for magnetic-fields like B = μoI/2πr a distance r away from a wire with current I, for electric-fields like E = λ/2πεor a distance r away from a line-charge density λ, and for associated frame-invariant proper-forces like Fo = qE + qw×B:

5a. Given panel dimensions, how strong do you expect is the magnetic field in the left (neutral-wire frame) panel?

5b. Similarly, what is the net charge per unit length on the wire in the right (moving-charge frame) panel?

5c. Given that, how strong is the electric field in the right (moving-charge frame) panel?

5d. If the moving-charge (like any singly-ionized molecule) has +1.6×10-19[C] of charge on it, what is the magnitude of the green force Fo on it at any point in time (using data from either panel)?

What else? 

Note that the answers to the above questions for the most part depend on your limited-precision measurement of time & distance intervals. As a result everyone's answers will differ in the measurement-events, and if sufficiently-precise they will differ in value as well.

To make responses to questions even more robust, therefore, we recommend that students describe their measurement-process in detail (not just the results). An analysis of measurement-uncertainties would be helpful, as well as consideration of their choice of, and assumptions in, underlying models.

In fact it is so easy to generate new figures where the parameters have changed that, in places where you want to encourage "solution-independence" it would be very straightforward to e-mail each student their own figure (generated with different parameters) for their analysis. Regardless, of course, sharing of strategies (with appropriate citation) can perhaps be encouraged in EOE challenges of this sort. 

Sample attempts by others to address some of these questions may be found on the talk page here.

Note that in FireFox and Chrome, right-clicking on the image and then clicking "copy" when ready may capture the desired animation frame e.g. for pasting into Word or Paint or ImageJ for measurement on-screen or on-paper.

References

• 1. P. Fraundorf (2016) Frame-invariant proper-force (preprint) URL:http://www.umsl.edu/~fraundorfp/anyspeed/properForce.pdf. Accessed: 2016-04-16. (Archived by WebCite® at http://www.webcitation.org/6goiKW1ND).