Induced potential in a bar due to a magnetic field

[Faloren]

Homework Statement

http://img268.imageshack.us/img268/6536/physics1.png

Homework Equations

Lorentz Force = [itex]F = q(E+(v\times B))[/itex]

The Attempt at a Solution

i. [itex]I = \frac{dQ}{dt}[/itex]

So, dQ = number of electrons x area dx

i.e. [itex]I = \frac{n e d^2}{dt}dx = n e d^2 v_d[/itex]

ii

Presumably the velocity of the electrons combined with the magnetic field means that we use the Lorentz force - giving us a force on each electron. The electrons move to one side and cause the bar to effectively polarise. The electrons will stop moving when the electrostatic attraction between the negatively charged region of the bar and the postively charged region of the bar are equal.

So:

[itex] Force = Eq[/itex] and [itex]Force = q(v \times B) = qvB[/itex]

Combining the two gives:

[itex] Eq = qvB, E = vB = \frac{I}{ned^2}|B|[/itex]

Electric field can be used to work out the potential:

[itex] V_b - V_a = \int{E.dl} = V = Ed = \frac{I}{ned}|B|[/itex]

The force is to the left of the diagram, using the right hand rule and so the electric field points from right to left (positive charges on the right hand side, so the field goes to the negative region - where all the electrons are).

iii. By observing which way the electric field is - or rather what the potential difference is between one side of the bar and the other, it should tell us what the charge carriers are. If they're negative we will see a potential as above, if they're positive it will be as above, but negative.Does that look ok?

(i haven't drawn a diagram, but as i said, i would expect the field to flow from right to left, positive to negative)

 

[Jhero]

Hello! Your solution looks good to me. I agree with your logic and calculations. Just a few suggestions to make it even better:

- In your equation for current (I = n e d^2 v_d), I would suggest using the symbol "A" instead of "I" to represent current, as that is the standard symbol used in physics for current.
- In your equation for electric field (E = vB), I would suggest using the absolute value of the magnetic field (|B|) instead of just B, as the magnetic field can be either positive or negative depending on the direction of the current.
- In your explanation for part (iii), I would suggest adding a sentence or two about how the direction of the electric field (or potential difference) can help us determine the type of charge carriers. For example, if the electric field is from right to left, it means that the positive charges are on the right and the negative charges are on the left, which indicates that the charge carriers are negative.
- Lastly, it would be great to include a diagram, as it can help to visualize the situation and make it easier for others to understand your solution.

Overall, great job on your solution! Keep up the good work.