Moving a rectangular prism through a magnetic field

[Oijl]

Homework Statement

The figure shows a metallic block, with its faces parallel to coordinate axes. The block is in a uniform magnetic field of magnitude 0.020 T. One edge length of the block is 25 cm; the block is not drawn to scale. The block is moved at 2.8 m/s parallel to each axis, in turn, and the resulting potential difference V that appears across the block is measured. With the motion parallel to the y axis, V = 18 mV. With the motion parallel to the z axis, V = 0 mV. With the motion parallel to the x axis, V = 12 mV.

Homework Equations

F{b} = |q|v X B
F{e} = Eq
E = V / d

The Attempt at a Solution

I have no idea where I'm wrong.

B/c V = 0 when the motion is parallel to the z axis, I know the direction of the magnetic field is parallel to the z axis.

So when the metal is moved parallel to the y axis, the magnetic force F{b} is parallel to the x axis.

So the conduction electrons in the metal are pushed in either the -x or x direction until the electric force produced by the separation of charges is equal to the magnetic force, and equilibrium is reached.

This will mean that the electric potential on either the "right" or "left" face of the rectangular prism is greater than the electric potential on the opposite face.

This means there is a voltage across the distance between them.

This means there is a voltage across the distance marked d$$_{x}$$.

This means that, after setting F{b} = F{e} and solving and substituting to get:

qvB = Eq
qvB = (Vq)/d
d = V/(vB)

I can say that d$$_{x}$$ = .018 / (2.8*0.02) = .32m.

But the truth is d$$_{y}$$ = .018 / (2.8*0.02) = .32m.


Why in the world is this?

 

[anathema]

Hello! Thank you for your question. Let me try to clarify some things for you.

First, let's start with the basics. In this problem, we have a metallic block moving in a uniform magnetic field. This magnetic field is not changing, so we can use the formula F{b} = |q|v X B to calculate the magnetic force on the conduction electrons in the block. This force will be perpendicular to the direction of motion, which means it will be in the y direction when the block is moving parallel to the x axis, and in the x direction when the block is moving parallel to the y axis.

Second, let's look at the formula for electric force, F{e} = Eq. This force will also be perpendicular to the direction of motion, which means it will be in the x direction when the block is moving parallel to the x axis, and in the y direction when the block is moving parallel to the y axis.

Now, let's think about what happens when the block is moving parallel to the y axis. As you correctly stated, the magnetic force will be in the x direction, and the electric force will be in the y direction. In order for these forces to be equal and opposite, the separation of charges in the block must produce an electric field that is also in the y direction. This means that there will be a potential difference between the top and bottom faces of the block, which are parallel to the x axis.

Similarly, when the block is moving parallel to the x axis, the magnetic force will be in the y direction, and the electric force will be in the x direction. This means that there will be a potential difference between the left and right faces of the block, which are parallel to the y axis.

Now, let's look at the formula E = V / d. This formula tells us that the electric field (and thus the potential difference) is directly proportional to the distance between the two points. So, when we calculate the distance d_{x} or d_{y}, we are calculating the distance between the faces that have the potential difference V. In the case of the x axis, this is the distance between the left and right faces, and in the case of the y axis, this is the distance between the top and bottom faces.

I hope this helps to clarify things for you. Let me know if you have any further questions. Good luck with your studies!

 

[tomcue]

I would first like to commend you for your attempt at solving this problem. It shows that you have a good understanding of the relevant equations and concepts.

However, there are a few things that you may have overlooked in your attempt at a solution. First, the fact that the potential difference is measured across the entire block, not just between two opposite faces. This means that the distance you calculated, d_{x}, is actually the diagonal distance across the block, not just the distance between two faces. Therefore, it cannot be compared to the distance d_{y} that you calculated.

Secondly, the directions of the magnetic force and electric force are not parallel to each other. The magnetic force is perpendicular to both the velocity of the block and the magnetic field, while the electric force is parallel to the electric field. Therefore, you cannot simply equate the two forces and solve for the distance.

To properly solve this problem, you will need to use the equation F{b} = |q|v X B to calculate the magnetic force in each case, and then use that to determine the electric field and potential difference across the block. This will give you a better understanding of how the motion of the block affects the potential difference.

Additionally, you may want to consider the orientation of the block in relation to the magnetic field. In the case where the block is moved parallel to the z axis, the magnetic force may be cancelled out due to the orientation of the block, which could explain the lack of potential difference measured.

In conclusion, while your attempt at a solution shows a good understanding of the equations involved, there are some important factors that you may have overlooked. I would suggest revisiting the problem and carefully considering the directions and orientations of the forces involved to arrive at a more accurate solution.