Force Between Cylindrical Capacitor Plates

[maxmax1]

Homework Statement

Capacitor plates are length L with radii a,b a<b are co-axial and intially covering each other with line charge density of the inner plate Q/L.

determine the force between plates as the inner plate is partially withdrawn along the axis.

i) if cylinders charged to voltage V and then discnnected
ii) if cylinders remain connected and maintained at voltage V

Homework Equations

u=(1/2)cv^2=(1/2)qv F=-du/dL

voltage between plates = Qln(b/a)/2Lpi*e0

capacitance between plates = 2Lpi*e0/ln(b/a)

The Attempt at a Solution

using partial derivitives. for

i) hold q constant differentiate for V

du=(1/2)qdv

-du/dl=(-1/2)Q(d/dL)(Qln(b/a)/2Lpi*e0)

= (1/4)(Q^2)ln(b/a)/2(L^2)pi*e0)

and is directed same direction to one being removed..


using partial derivitives. for

i) hold v constant differentiate for q

du=(1/2)(v^2)dc

-du/dl=(-1/2)(v^2)(d/dL)(2Lpi*e0/ln(b/a))

= (-1/2)(v^2)(2pi*e0/ln(b/a))


and is directed in opposite direction to one being removed



Are this working OK?

Many thanks!

 

[anathema]

I would like to provide some feedback on your attempt at solving this problem. It seems like you have a good understanding of the relevant equations and how to use partial derivatives. However, I would recommend being more clear and organized in your approach. It would also be helpful to explicitly state your assumptions and any simplifications you are making in your calculations. Additionally, I would suggest using consistent units throughout your solution.

For part i), it is not clear what you are holding constant and what you are differentiating with respect to. It seems like you are trying to hold the charge constant and differentiate for the voltage, but your notation is confusing. Also, you should state explicitly that you are using the equation for the energy stored in a capacitor (u = 1/2 CV^2). Your final result for the force is also missing units.

For part ii), it is not clear what you are holding constant and what you are differentiating with respect to. It seems like you are trying to hold the voltage constant and differentiate for the charge, but your notation is confusing. Also, you should state explicitly that you are using the equation for the energy stored in a capacitor (u = 1/2 QV). Your final result for the force is also missing units.

Overall, your approach seems to be on the right track, but I would recommend clarifying and organizing your calculations to make them easier to follow. It would also be helpful to include a diagram or visual representation of the problem to aid in understanding. Keep up the good work!

 

[baba89]

I would first like to commend you for your attempt at solving this problem using partial derivatives. Your approach is generally correct, but there are a few things that could be improved upon.

Firstly, your notation for the charge and voltage is a bit confusing. It would be clearer to use Q1 and Q2 for the charges on the inner and outer plates, and V1 and V2 for the voltages on the inner and outer plates.

Secondly, in your first attempt at a solution, you used the equation for energy (u) instead of the equation for force (F). The correct equation for force between cylindrical capacitor plates is F = -dU/dL, where U is the potential energy.

Thirdly, in both of your attempts at a solution, you only considered the force on the inner plate. However, in this problem, we are interested in the force between the two plates, so you should also consider the force on the outer plate.

Finally, your notation for the partial derivatives is a bit unclear. It would be more correct to write them as dQ/dL and dV/dL, since you are differentiating with respect to L. Also, in your second attempt, you should include the negative sign when taking the derivative of the voltage equation.

Overall, your approach is on the right track, but I would suggest revising your notation and making sure to consider the force on both plates. Additionally, it would be helpful to explicitly state the direction of the forces in your final answers.