Two movable wires in Magnetic Field

[Vibhor]

Homework Statement

Homework Equations

EMF induced in a moving wire =Bvl

The Attempt at a Solution

When right wire is given a velocity v , an initial emf Bvl is induced in right wire due to which current starts flowing in the loop .

Initial current i = Bvl/(2R)

The left wire experiences a repulsive force due to right wire towards left .It also experiences a force of magnitude ilB towards right due to the magnetic field .

Similarly , the right wire experiences a repulsive force due to left wire towards right .The speed of left wire changes which means the emf induced varies .As a result the current flowing is not constant .

I am unable to proceed .

Please help me with the problem .

Thanks

 

[TSny]

I suspect that you are meant to neglect the force of one wire on the other wire compared to the force that each wire experiences from the external B field.

 

[cnh1995]

I believe forces on wires due to each other should also be neglected since distance between them is not mentioned here.

 

[Vibhor]

I suspect that you are meant to neglect the force of one wire on the other wire compared to the force that each wire experiences from the external B field.

Ok .

In that case force on left wire would be towards right and that on right wire would be towards left . The left will accelerate towards right and right would decelerate i.e v decreases. Right ?

 

[cnh1995]

Ok .

In that case force on left wire would be towards right and that on right wire would be towards left . The left will accelerate towards right and right would decelerate i.e v decreases. Right ?

Yes. Right wire will stop eventually(momentarily) while left wire will have some velocity towards right. This problem is interesting!

 

[TSny]

Ok .

In that case force on left wire would be towards right and that on right wire would be towards left . The left will accelerate towards right and right would decelerate i.e v decreases. Right ?

Yes. If you think about this problem physically, I believe you can get the answer without much calculation.

 

[TSny]

Yes. Right wire will stop eventually(momentarily) while left wire will have some velocity towards right. This problem is interesting!

Will the right wire ever stop?

 

[Vibhor]

Yes. If you think about this problem physically, I believe you can get the answer without much calculation.

 

[cnh1995]

Will the right wire ever stop?

I believe there will be a moment when emf induced in the right wire will be 0 and left wire will have a velocity(maximum) towards left. From that moment onwards, left wire will act as a generator and push the right wire again towards right.
Edit: No, I think it won't stop at all. When velocity of the left wire will become greater than that of the right wire, it will push the right wire again towards right.

 

[Vibhor]

Yes. If you think about this problem physically, I believe you can get the answer without much calculation.

Momentum conservation ?

mv = 2mv' i.e final speed =v/2 .

Both the sliders would move together towards right with speed v/2 after long time .

Is that what you are hinting at ??

 

[TSny]

It's your problem . Give me a reason why the momentum should be conserved.

 

[Vibhor]

It's your problem . Give me a reason why the momentum should be conserved.

Because no net horizontal force on the system of two rails . The force due to magnetic field is always equal in magnitude and opposite in direction .

So , mv = 2mv' i.e final speed =v/2 .

Both the sliders would move together towards right with speed v/2 after long time .

 

[TSny]

Because no net horizontal force on the system of two rails .

Are you saying that the net force on the system consisting of the two wires is zero? Why?

So , mv = 2mv' i.e final speed =v/2 .

Why are the final speeds of the two wires equal to each other?

 

[TSny]

Edit: No, I think it won't stop at all. When velocity of the left wire will become greater than that of the right wire, it will push the right wire again towards right.

Will the speed of the left wire ever become greater than the speed of the right wire?

 

[Vibhor]

Are you saying that the net force on the system consisting of the two wires is zero? Why?

Force ilB acts leftwards on right wire and force ilB acts rightwards on left wire . On the whole ,they cancel each other ??

Why are the final speeds of the two wires equal to each other?

The left wire accelerates towards right and right accelerates towards left till they meet . Afterwards they move together (presuming momentum conservation )

I am bereft of ideas

 

[TSny]

Force ilB acts leftwards on right wire and force ilB acts rightwards on left wire . On the whole ,they cancel each other ??

OK. At any instant of time the two wires (sliders) have the same magnitude of current and the currents are opposite in direction. So, the net force on the system of the two sliders is zero. So, linear momentum is indeed conserved! Good.

The left wire accelerates towards right and right accelerates towards left till they meet .

Until they meet? You mean the two wires are going to collide? If so, you would need to know some information about the collision (elastic or inelastic, etc.) I think you can assume that the wires never touch each other. [EDIT: There's a good argument for why they never meet. See my question in post #14.]

I am bereft of ideas

What happens to the magnitude of the induced emf as time increases? Why?

 

[Vibhor]

I think there will also be an induced emf in the left wire as it starts to move. Polarity of emf in left wire will be opposite to that of right wire . Do I need to consider this ?

 

[Vibhor]

Will the speed of the left wire ever become greater than the speed of the right wire?

Why not ? left wire accelerates rightwards wheras right wire decelerates .What stops left wire from gaining higher speed than right one .

 

[TSny]

I think there will also be an induced emf in the left wire as it starts to move. Polarity of emf in left wire will be opposite to that of right wire . Do I need to consider this ?

Consider the total instantaneous emf induced in the circuit that consists of the two slider and the rails. What law is relevant?

 

[TSny]

Why not ? left wire accelerates rightwards wheras right wire decelerates .What stops left wire from gaining higher speed than right one .

First investigate how the instantaneous current in the circuit depends on the instantaneous velocities of the sliders.

 

[Vibhor]

First investigate how the instantaneous current in the circuit depends on the instantaneous velocities of the sliders.

$i=\frac{Blv}{2R}$ ??

 

[Vibhor]

Consider the total instantaneous emf induced in the circuit that consists of the two slider and the rails. What law is relevant?

Faraday's Law ??

 

[TSny]

$i=\frac{Blv}{2R}$ ??

No, the current in the circuit is determined by the total emf induced in the circuit.

 

[TSny]

Faraday's Law ??

Yes. Or, equivalently, you can get an expression for the total emf by combining the motional emfs of the two sliders

 

[Vibhor]

No, the current in the circuit is determined by the total emf induced in the circuit.

$i = \frac{Bl}{2R}(v_2-v_1)$

 

[TSny]

$i = \frac{Bl}{2R}(v_2-v_1)$

Yes. So, by inspection of this formula and knowing what happens to the velocities, describe what happens to the current as time increases.

 

[Vibhor]

The current varies as long as relative speed of the sliders is non zero. Just when the relative speed become 0 ,the current ceases and the force acting on the sliders also vanish .After that the two sliders move with same constant speed with non zero separation between them . Applying momentum conservation the final speed is v/2 .

 

[TSny]

The current varies as long as relative speeds of the sliders is non zero. Just when the relative speeds become 0 ,the current ceases and the force acting on the sliders also vanish .After that the two sliders move with constant speeds with non zero separation between them . Applying momentum conservation the final speed is v/2 .

Yes, that's it. Now, it may be that the speeds of the two wires do not become equal for any finite time, but only asymptotically as t increases to infinity. But, either way, you get that after a "long time" the final speed of each slider is v_o/2.

 

[cnh1995]

Yes, that's it. Now, it may be that the speeds of the two wires do not become equal for any finite time, but only asymptotically as t increases to infinity. But, either way, you get that after a "long time" the final speed of each slider is v_o/2.

Right!

 

[Vibhor]

Thanks a lot TSny .

 

[Vibhor]

I suspect that you are meant to neglect the force of one wire on the other wire compared to the force that each wire experiences from the external B field.

Hi ,

Don't you think even if we do not neglect the force of one wire on the other wire , it would not affect the final speed of the sliders ? The two forces would cancel each other as far as momentum conservation is concerned .

 

[TSny]

Hi ,

Don't you think even if we do not neglect the force of one wire on the other wire , it would not affect the final speed of the sliders ? The two forces would cancel each other as far as momentum conservation is concerned .

Yes, I think that's right. The momentum would still be conserved and the final velocities of the sliders would still be equal. Nice!

 

[Vibhor]

Ok . Thanks .