Physics: Calculate Emf, Electric Field in Magnetic Field

[Fermat1]

1)A meter long metal rod of mass m is falling with a constant velocity of 10 m/s. The rod is attached to two conducting rails. Determine the emf if there is a uniform magnetic field directed perpendicular to the motion of the bar. The resistance has a value of 105 ohms.

I don't know how I can do it when the magnetic field is not given

2)A loop of wire is put in a changing magnetic field. The magnetic flux through the loop is given by $4t(t+2)$. The loop is connected to a parallel plate capacitor that has a plate separation of 15mm. Determine the electic field between the plates at time $t=3$ s.

Can I assume that potential difference equal emf if the wire has no resistance?

3)Let $V=(xyt^3T)i+(x^4tT)j$. Find the curl of the magnetic field and the electric field. I've found the curl.
The curl I found to be $-(4x^3t)j+(yt^3)k$ It's sensible to leave out the units T right?

How do I find the electric field?

 

[I like Serena]

1)A meter long metal rod of mass m is falling with a constant velocity of 10 m/s. The rod is attached to two conducting rails. Determine the emf if there is a uniform magnetic field directed perpendicular to the motion of the bar. The resistance has a value of 105 ohms.

I don't know how I can do it when the magnetic field is not given

It seems to me that if there is no current in the bar, and if there is not change in flux, there won't be an emf.

2)A loop of wire is put in a changing magnetic field. The magnetic flux through the loop is given by $4t(t+2)$. The loop is connected to a parallel plate capacitor that has a plate separation of 15mm. Determine the electic field between the plates at time $t=3$ s.

Can I assume that potential difference equal emf if the wire has no resistance?

That is what I would assume with no more information given.

3)Let $V=(xyt^3T)i+(x^4tT)j$. Find the curl of the magnetic field and the electric field. I've found the curl.
The curl I found to be $-(4x^3t)j+(yt^3)k$ It's sensible to leave out the units T right?

How do I find the electric field?

Since the formula for $V$ contains the unit T, it makes sense to also specify it in an answer. Anyway, since it is an SI unit, it won't hurt much to leave it out.

How did you find the curl of the magnetic field?

And what is being asked exactly? The electric field? Or the curl of the electric field?

 

[topsquark]

1)A meter long metal rod of mass m is falling with a constant velocity of 10 m/s. The rod is attached to two conducting rails. Determine the emf if there is a uniform magnetic field directed perpendicular to the motion of the bar. The resistance has a value of 105 ohms.

I don't know how I can do it when the magnetic field is not given

Lenz's Law says that if the flux through the loop is changing then there will be a force to counteract the changing flux. In this case we know that the rod is falling with a constant speed...that is to say that there must be a force on the rod, equal to the weight of the rod, in the upward direction. What does that tell you about the magnetic field?

-Dan

 

[jacobi1]

It seems to me that if there is no current in the bar, and if there is not change in flux, there won't be an emf.

There is a current in the bar-otherwise the bar will accelerate as it falls.
The magnetic force is given by \(\displaystyle iL \times B\), where i is current, L is the bar's length, and B is the magnetic field. The magnetic field is perpendicular to the motion, so the magnitude, \(\displaystyle iLB \sin \theta \), becomes \(\displaystyle iLB\), and the direction, by the right-hand rule, is opposite to the direction of the gravitational force. Since the bar falls with constant velocity, this force is equal in magnitude to the gravitational force. Thus \(\displaystyle iLB=mg\).
But \(\displaystyle i= \frac{\epsilon}{R} \), where \(\displaystyle \epsilon\) is emf. So, Fermat, what do you think you should do now?
Faraday's law, maybe?

 

[CellCoree]

1) To calculate the emf (electromotive force) in this scenario, we can use Faraday's law of induction, which states that the emf induced in a closed loop is equal to the rate of change of magnetic flux through the loop. In this case, the magnetic flux is given by the product of the magnetic field strength and the area of the loop, which is equal to the length of the metal rod in this case. So, we can write the equation as: emf = B * L * v, where B is the magnetic field strength, L is the length of the rod, and v is the velocity of the rod. Since the velocity is constant, the emf will also be constant. Therefore, we can calculate the emf by simply multiplying the magnetic field strength by the length of the rod and the velocity.

2) Yes, you can assume that the potential difference is equal to the emf if the wire has no resistance. In this case, the electric field between the plates can be calculated by dividing the potential difference by the distance between the plates. So, in this scenario, the electric field at time t=3s can be calculated by dividing the emf by the plate separation of 15mm.

3) To find the electric field, we can use the relationship between electric field and potential: E = -∇V, where ∇ is the gradient operator and V is the potential. In this scenario, the electric field can be calculated by taking the gradient of the potential given in the question. Since the potential is a function of position, we can use the partial derivative with respect to each coordinate to find the electric field components. So, the electric field can be written as: E = -(∂V/∂x)i -(∂V/∂y)j -(∂V/∂z)k. To find the electric field at a specific point, we can substitute the values of x, y, and z into this equation. Additionally, it is not necessary to include the units of T in the curl calculation as the units will cancel out in the final result.