Find a relationship between C & E

[seekingaeolus]

Homework Statement

Given a coaxial cylindrical capacitor with a dialectric constant of 87.9, what field strength is needed to oscillate a dipole within the field at 175,000 hz...?

What dimensions would the capacitor have in order to produce the field strength needed for that oscillation...?

Homework Equations

Given that Torque = P (dipole moment) X E(field strength)

T= P X E

& T=I (moment of Inertia) * a (alpha, angular accelleration)

Therefore P X E = I * a

rearranging a = P X E/I

For simple harmonic motion, T (period in seconds) = 2*pi*sqrt(I/P X E)

T= 1/f

P= 6.24E-30 C m

I = (z axis) = 1.92E-47 Kg m^2

The Attempt at a Solution

My result yields the E (field strength) of 2.67E5 N/C

If this is correct, what capacitor design will provide this given the following constraints?

Radius no greater than approximately 7 cm for the outer electrode

Electrode length no longer than approximately 30 cm for both electrodes

What voltage will be applied?

 

[seekingaeolus]

I did come up with something else...

Taking the Capacitance of charged coaxial cylinders to be 2*Pi* (Epsilon Naut) * L (length of the capacitor) ? ln(b/a), where

Epsilon Naut= 8.85E-12

L = .1 meters

b= radius of outer electrode = 3.81 cm

and a = radius of inner electrode = 1.91 cm

I found that C = 8.08E-12 N/C

Multiplied by the dielectric constant 87.9 gave me 6.474E-10

Then I set Q = C*V where V= 12 Volts, giving me a total charge of 7.77E-9 C

After that, I used Guassian symmetry to determine the radius of the field from one electrode to the field strength of 5.87E5 N/C

E = Q/L*2*Pi*Epsilon Naut*r

rearranging gives Q/E*L*2*Pi*Epsilon Naut = radius

Thus the distance from the cylinder to a point where the field strength (E) is 5.23 mm. At this point, a dipole at the forementioned moment for both interia and dipole will oscillate at a frequency of 175,000 hz.

Unless my Field Strength is incorrect from the first problem. Which is possible because my physics professor included a "2" to be multiplied with P & E in the first equation.

Does anyone know if this is correct?

Thanks for your help!

 

[baba89]

I would first check the calculations provided to ensure they are accurate and relevant to the given scenario. I would also consider the units used to ensure they are consistent throughout the equations.

Once I have confirmed the accuracy of the calculations, I would then analyze the relationship between C (capacitance) and E (field strength). From the equations provided, it can be seen that the field strength is directly proportional to the dipole moment and inversely proportional to the moment of inertia. This means that as the field strength increases, the dipole moment should also increase while the moment of inertia decreases.

To determine the dimensions of the capacitor needed to produce the desired field strength, I would first consider the given constraints. The outer electrode should have a radius no greater than 7 cm, which would determine the maximum radius of the capacitor. The electrode length should be no longer than 30 cm, which would determine the maximum length of the capacitor. From these constraints, I would then use the formula for capacitance (C = ε0εrA/d) to determine the necessary dimensions for the capacitor.

Once the dimensions of the capacitor have been determined, I would then consider the voltage that needs to be applied. This can be calculated using the formula V = Ed, where V is the voltage, E is the field strength, and d is the distance between the electrodes.

In conclusion, the relationship between C and E in this scenario is determined by the dimensions of the capacitor and the voltage applied. By understanding this relationship and using the relevant equations, we can determine the necessary dimensions and voltage for a coaxial cylindrical capacitor to produce the desired field strength for oscillating a dipole at a frequency of 175,000 Hz.