Cylindrical electromagnetic cavity.

[miew]

Homework Statement

A cylindrical electromagnetic cavity 4.8cm in diameter and 7.3 cm long is oscillating.

a) Assume that, for points on the axis of the cavity Em=13kV/m. The frequency of oscillation is 2.4 GHz.
For such axial points, what is the maximun rate (dE/dt)m, at which E changes?

b) Assume that the average value of (dE/dt)m for all points over a cross section of the cavity, is one-half the value found above for axial points. On this assumption, what is the maximum value of B at the cylindrical surface of the cavity?

Homework Equations

The Attempt at a Solution

I got a), which is 1.96 *10^14 V/ms

This is what I did for b): (dE/dt)cavity =1/2 (dE/dt)m = 9.8 10^13 V/ms.
So Ecav= 9.8 10^13/(2pi 2.4 10^9) = 6.5 10^3.

So Bm=Em/c = 2.16 10^-5. This is wrong but I don't know what else to do.

Thanks!

 

[anathema]

Thank you for your post. I am a scientist and I would like to help you with your questions.

For part a), you are correct in calculating the maximum rate of change of E, which is 1.96*10^14 V/ms. This value represents the maximum electric field strength, which is changing at the fastest rate at points on the axis of the cavity.

For part b), you are correct in assuming that the average value of (dE/dt)m for all points over a cross section of the cavity is one-half the value found for axial points. However, in order to calculate the maximum value of B at the cylindrical surface of the cavity, we need to use the equation B = (μ0/μr)H, where μ0 is the permeability of free space and μr is the relative permeability of the medium (in this case, air). In order to find the maximum value of B, we need to find the maximum value of H.

To find the maximum value of H, we can use the equation H = E/η, where η is the intrinsic impedance of the medium. In free space, η = √(μ0/ε0) = 377 Ω. Therefore, H = Em/η = 13*10^3/377 = 34.5 A/m.

Now, we can calculate the maximum value of B using the equation B = (μ0/μr)H = (4π*10^-7/1)(34.5) = 4.33*10^-6 T.

I hope this helps! Let me know if you have any further questions.