# Two movable wires in Magnetic Field

• Vibhor
In summary, the conversation discusses the concept of EMF induced in a moving wire and the forces experienced by two parallel wires due to their relative motion. The conversation also delves into the idea of momentum conservation and the implications for the movement of the wires. The main focus is on understanding the behavior of the wires and the factors that affect their motion.
Vibhor

## 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 .

Thanks

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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.

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

TSny said:
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 ?

Vibhor said:
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!

Vibhor said:
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 ?

Vibhor
cnh1995 said:
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
TSny said:

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TSny said:
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.

TSny said:

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 ??

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

TSny said:
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 .

Vibhor said:
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?

cnh1995 said:
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
TSny said:
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 ??

TSny said:
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

Vibhor said:
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?

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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 ?

TSny said:
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 .

Vibhor said:
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?

Vibhor
Vibhor said:
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
TSny said:
First investigate how the instantaneous current in the circuit depends on the instantaneous velocities of the sliders.
##i=\frac{Blv}{2R}## ??

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

Vibhor said:
##i=\frac{Blv}{2R}## ??
No, the current in the circuit is determined by the total emf induced in the circuit.

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

Vibhor
TSny said:
No, the current in the circuit is determined by the total emf induced in the circuit.
##i = \frac{Bl}{2R}(v_2-v_1)##

Vibhor said:
##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 .

Vibhor said:
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 vo/2.

Vibhor and cnh1995
TSny said:
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 vo/2.
Right!

Thanks a lot TSny .

TSny said:
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 .

Vibhor said:
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 .

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## 1. How do two movable wires behave in a magnetic field?

In a magnetic field, two movable wires will experience a force known as the Lorentz force. This force will cause the wires to move in opposite directions, with one wire moving towards the magnetic field and the other moving away from it.

## 2. What factors affect the movement of two movable wires in a magnetic field?

The movement of two movable wires in a magnetic field is affected by the strength of the magnetic field, the current flowing through the wires, and the length and orientation of the wires relative to the magnetic field.

## 3. Can the direction of the movement of two movable wires be changed?

Yes, the direction of the movement of two movable wires can be changed by altering the direction of the magnetic field or the direction of the current flowing through the wires.

## 4. What is the significance of two movable wires in a magnetic field?

Two movable wires in a magnetic field are commonly used in experiments to demonstrate the principles of electromagnetism and the behavior of charged particles in a magnetic field. They are also used in devices such as electric motors and generators.

## 5. How can the movement of two movable wires in a magnetic field be calculated?

The movement of two movable wires in a magnetic field can be calculated using the equations for the Lorentz force and the motion of charged particles in a magnetic field. These equations take into account the relevant factors such as the strength of the magnetic field, the current in the wires, and the length and orientation of the wires.

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