- #1
nonequilibrium
- 1,439
- 2
I'm a bit confused about how the Lorentz Force (as a law) stands in relation to the laws of Maxwell (independent of each other? dependent?). There are two concrete examples I can think of where they interfere with each other:
1: Lorentz Force & Faraday's Law of Induction
Imagine a rectangular conducting loop (no current) hanging in a magnetic field (in such a way that the magnetic flux through the loop is at a maximum, aka the magnetic field lines are parallel to the surface vector of the loop). One of the 4 rods of the loop is moveable and can be dragged away in such a way that the surface area of the loop is increased. Now let this rod move at a constant speed v, then the change of the magnetic flux through the loop is = B*L*v with B the external magnetic field and L the length of the moving rod. Faraday's law of induction predicts a current induced by a voltage equal in size to the changing flux.
Now you can predict the exact same current by not using Faraday's Law of Induction and instead nothing more than the Lorentz Force by noting that because the rod is moving away at a constant speed v, the charges inside the rod experience a Lorentz Force and start moving, which constitutes a current flow. (It gaves the same numerical result, yet a different interpretation)
2: Lorentz Force & Ampère's Law
Imagine a magnet standing vertically, south pole downward, north pole upward. Now hold a circular loop a few inches above the north pole. Viewed from above, there is a clockwise current in the loop. Ampère's Law predicts that the loop acts like a little magnet with its north pole aimed downward and its south pole aimed upward. You can conclude the loop will be pushed away from the magnet, because the two north poles are directed at each other.
Again, ignore Ampère's Law and simply calculate the Lorentz Force that the magnet exercises on the current (F = B*I*L). You'll have to do this for an infinitesimal piece and then integrate it and you will see that the resulting force is aimed directly upward. Conclusion: same result as with Ampère's law.
----
This is all very confusing, because practically they seem to be very different things, but in both cases they predict the same experimental result. How do you even know they are the same result? Maybe they are different phenomenons and have to be ADDED to each other. How can you know? And is it logical that sometimes the Lorentz Force can predict what different Mawellian laws say? It's like the Lorentz Force is derived from multiple laws of Maxwell, but then only for a specific case (because the Lorentz Force seems to be equivalent in these cases, but we know we can't always use it instead of the laws of Maxwell)
Very curious and grateful to all helpers,
mr. vodka
1: Lorentz Force & Faraday's Law of Induction
Imagine a rectangular conducting loop (no current) hanging in a magnetic field (in such a way that the magnetic flux through the loop is at a maximum, aka the magnetic field lines are parallel to the surface vector of the loop). One of the 4 rods of the loop is moveable and can be dragged away in such a way that the surface area of the loop is increased. Now let this rod move at a constant speed v, then the change of the magnetic flux through the loop is = B*L*v with B the external magnetic field and L the length of the moving rod. Faraday's law of induction predicts a current induced by a voltage equal in size to the changing flux.
Now you can predict the exact same current by not using Faraday's Law of Induction and instead nothing more than the Lorentz Force by noting that because the rod is moving away at a constant speed v, the charges inside the rod experience a Lorentz Force and start moving, which constitutes a current flow. (It gaves the same numerical result, yet a different interpretation)
2: Lorentz Force & Ampère's Law
Imagine a magnet standing vertically, south pole downward, north pole upward. Now hold a circular loop a few inches above the north pole. Viewed from above, there is a clockwise current in the loop. Ampère's Law predicts that the loop acts like a little magnet with its north pole aimed downward and its south pole aimed upward. You can conclude the loop will be pushed away from the magnet, because the two north poles are directed at each other.
Again, ignore Ampère's Law and simply calculate the Lorentz Force that the magnet exercises on the current (F = B*I*L). You'll have to do this for an infinitesimal piece and then integrate it and you will see that the resulting force is aimed directly upward. Conclusion: same result as with Ampère's law.
----
This is all very confusing, because practically they seem to be very different things, but in both cases they predict the same experimental result. How do you even know they are the same result? Maybe they are different phenomenons and have to be ADDED to each other. How can you know? And is it logical that sometimes the Lorentz Force can predict what different Mawellian laws say? It's like the Lorentz Force is derived from multiple laws of Maxwell, but then only for a specific case (because the Lorentz Force seems to be equivalent in these cases, but we know we can't always use it instead of the laws of Maxwell)
Very curious and grateful to all helpers,
mr. vodka