- #1

- 8

- 1

Thread moved from the technical forums to the schoolwork forums

My question arises from the following problem:

We have a uniform magnetic field into the page, decreasing at a constant rate dB/dt< 0, causing the bar to move to the right. Find the velocity of the bar as a function of the time, and in terms of the known parameters: the resistor R and the separation distance l, and the mass m. See figure below.

My solution:

This is apparently a trivial question, however I have noticed that it's not that trivial as I thought.

What will happen in this situation is that, since the magnetic filed B changes with time and therefore the flux of B through the circuit, there will be and induced current in the circuit, such direction will oppose to the decrease of magnetic flux. It is known that the flux is decreasing so the induced current will be clockwise. This does not stop here: the induced current in presence of the magnetic field will give rise to a magnetic force in the conductors, but there is only one that will move: (the sliding rod). The magnetic force on the sliding bar is to the right (Apply the right hand rule). Notice that as the rod moves the area of the the circuit increases, which is in concordance with the Lenz's Law, but that would also affect the current, the flux, and hence the velocity.

My approach to determine the velocity as a function of the time is by applying the general rule of flux:

In this case both the area and the magnetic field change with the time, so we have to apply derivative of the product rule:

Being dA/dt = (dA/dx)(dx/dt) if we define the Area as A = lxo + lx, then dA/dx = l and dx/dt = v (which is the velocity of the rod), then the emf will be:

where xo is the initial position of the rod

This will lead to an induced current i = emf/R, a magnetic Force Fm= ilB on the rod, so we can write a differential equation for the motion of the rod in virtue of Newton's second law:

We assumed the increasing direction of the x-axis to the right. Then substituting the induced current into the previous equation we have:

Rearranging the terms we get the differential equation:

Subjected to the initial conditions:

I am posting this solution, to be considered, first, to see if there is either a concept or a math mistake. I would like to know if my analysis is correct.

I am also interested in getting the solution of the differential equation, if anyone has solved this before.

Any help will be greatly appreciated

Thank you in advance.

We have a uniform magnetic field into the page, decreasing at a constant rate dB/dt< 0, causing the bar to move to the right. Find the velocity of the bar as a function of the time, and in terms of the known parameters: the resistor R and the separation distance l, and the mass m. See figure below.

My solution:

This is apparently a trivial question, however I have noticed that it's not that trivial as I thought.

What will happen in this situation is that, since the magnetic filed B changes with time and therefore the flux of B through the circuit, there will be and induced current in the circuit, such direction will oppose to the decrease of magnetic flux. It is known that the flux is decreasing so the induced current will be clockwise. This does not stop here: the induced current in presence of the magnetic field will give rise to a magnetic force in the conductors, but there is only one that will move: (the sliding rod). The magnetic force on the sliding bar is to the right (Apply the right hand rule). Notice that as the rod moves the area of the the circuit increases, which is in concordance with the Lenz's Law, but that would also affect the current, the flux, and hence the velocity.

My approach to determine the velocity as a function of the time is by applying the general rule of flux:

In this case both the area and the magnetic field change with the time, so we have to apply derivative of the product rule:

Being dA/dt = (dA/dx)(dx/dt) if we define the Area as A = lxo + lx, then dA/dx = l and dx/dt = v (which is the velocity of the rod), then the emf will be:

where xo is the initial position of the rod

This will lead to an induced current i = emf/R, a magnetic Force Fm= ilB on the rod, so we can write a differential equation for the motion of the rod in virtue of Newton's second law:

We assumed the increasing direction of the x-axis to the right. Then substituting the induced current into the previous equation we have:

Rearranging the terms we get the differential equation:

Subjected to the initial conditions:

**x (t=0) = xo , v(t=0)=0**I am posting this solution, to be considered, first, to see if there is either a concept or a math mistake. I would like to know if my analysis is correct.

I am also interested in getting the solution of the differential equation, if anyone has solved this before.

Any help will be greatly appreciated

Thank you in advance.