Hi. Sorry, I am not following your question. It seems that if the velocity of the crossbars is the same, no emf appears in the middle crossbar. Moreover, I cannot understand how the contradiction occurs. Would you please explain more?

Mohammad Javanshiry

Dear Professor Javanshiry,

Thank you very much for your question.

It's to do with the strictures of Relative Motion.

Because you can say (for the left hand circuit alone,) that either :

The crossbar is moving,

or

The U-shaped wire is moving,

then it follows that the same magnitude emf has to be able to appear across each opposite crossbar,

Cf.

https://www.researchgate.net/post/Faradays_Law_and_Relative_Motion

The same can be said for the right-hand circuit, and so this is where the contradiction arises, because we would then have two emfs in the same piece of wire --(the middle crossbar.)

Because this is impossible, -- is a contradiction, then the light (electromagnetic wave) organises itself to cease the effect altogether : no induction takes places in either circuit, if both crossbars are in motion. The voltage across the middle crossbar drops to zero.

You can replace the moving crossbars with two batteries of different voltages at rest relative to the middle crossbar. Does the contradiction appear again according to your configuration?!

Mohammad Javanshiry

Dear Professor Javanshiry,

Good question. I'd say the two setups were not equivalent, because

i) you are removing the "relative motion" aspect of it.

ii) and so you are also, in effect, removing the "Faraday induction" aspect from it too.

and

iii) thirdly, you are removing the magnetic field from from the arrangement.

As far as I understand, when you have two electro-chemical batteries in a circuit, then the resultant is just the net voltage ie. the sum of each of them. And this would then determine the current around the circuit.

Each moving crossbar drives currents in the centre bar and in the other crossbar. If both bars are moving outwards the currents in the centre bar partially or completely cancel, and the currents in each end bar add.

The voltage drop does not only appear across the centre bar but continuously along each wire, so when there is no current in the centre wire, the top and bottom wires have all the voltage drop across them, that adds up to the induced voltage in each moving bar, and there is no voltage across the centre bar because it joins places on the top and bottom wires where the voltage is the same.

Malcolm White

Dear Professor White,

Thanks for your comment. What you say sounds reasonable, but what sounds reasonable or even likely, is not always the case in circuits.

For example, it sounds reasonable that two voltmeters connected to the same pair of terminals will show same potential difference between these two points, but this has been shown (experimentally) not to be the case.

Each voltmeter is held at a different value, even a different sign, notwithstanding them being connected to the same two terminals.

Cf. https://www.researchgate.net/post/Electrodynamic_Paradox--How_can_the_voltmeters_in_the_above_circuit_register_different_values_both_being_attached_to_the_same_pair_of_terminals

If the middle voltmeter reads zero when both crossbar "runners" are in motion--- it could be viewed as the converse of the above effect. ie. the middle crossbar would be "held" at zero volts.

You have changing magnetic fields around these wires. The fields will also be changing around the wires leading to the voltmeters so that you can measure different voltages depending on where the wires to the voltmeters go. The voltmeters in the article you reference are in wire loops that form coils that can have different amounts of flux in them so can have different induced voltages.

Malcolm White

Dear Professor White,

Thanks for your comment.

---"You have changing magnetic fields around these wires. The fields will also be changing around the wires leading to the voltmeters so that you can measure different voltages depending on where the wires to the voltmeters go."---

In the above experiment, the magnetic field is static-- held at a constant value by a D.C supply fed into a large coil a couple of hand-spans in diameter. So the voltmeter leads connected across the middle crossbar shouldn't be affected by this static magnetic field.

---"The voltmeters in the article you reference are in wire loops that form coils that can have different amounts of flux in them so can have different induced voltages."---

That's right, in that article-- it is possible to split the circuit into two Faraday induction circuits. It's then obvious-- or more obvious-- why the connected voltmeters are "held" at different values, notwithstanding them being connected to the same pair of terminals.

The applied magnetic field may not change, but there will be a changing magnetic field from the currents in the wires that are caused by moving the sliding crossbars. These magnetic fields will induce voltages in the voltmeter leads if any of the changing fields go through the loop formed by the voltmeter leads.

Malcolm White

Dear Professor White,

Thanks for your comment.

Only if induction takes place in this circuit. And it's not certain that it does --- it doesn't appear to do so in the case of the middle region of the two-paradox circuits stuck together. I say when one crossbar is in motion, and you begin to move the other crossbar, the voltage across the middle crossbar immediately drops to zero, and remains at zero so long as both crossbars are in motion.

But I am taking to steps to approximate the apparatus, so we will soon find out.