Total energy and power of electromagnetic fields

In summary, the conversation discusses the problem of a charging capacitor, where a fat wire with a constant current I is connected to a narrow gap of wire to form a parallel-plate capacitor. The electric and magnetic fields in the gap are found as functions of distance and time, along with the energy density and Poynting vector. The total energy in the gap is determined by integrating the energy density, and the power is calculated by integrating the Poynting vector over the appropriate surface. The solution involves solving integrals using cylindrical coordinates, and the length of the gap (w) is an important factor in the calculations.
  • #1
RawrSpoon
18
0

Homework Statement


Consider the charging capacitor in problem 7.34
(A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap of wire, of width w, w<<a, forms a parallel-plate capacitor)
a) Find the electric and magnetic fields in the gap, as functions of the distance s from the axis and the time t (Assume the charge is zero at t=0)
b) Find the energy density uem and the Poynting vector S in the gap.
c) Determine the total energy in the gap, as a function of time. Calculate the total power flowing into the gap by integrating the Poynting vector over the appropriate surface. Check that the power input is equal to the rate of increase of energy in the gap.

Homework Equations


I've solved a and b, the electric field is
[tex]E= \frac{It}{\pi a^2 \epsilon_0}\hat{z}[/tex]
the magnetic field is
[tex]B= \frac{\mu_0 I s}{\pi a^2}\hat{\phi}[/tex]
the energy density is
[tex]u_{em}= \frac{I^2 \mu_0}{8 \pi^2 a^4}[4c^2t^2+s^2][/tex]
the Poynting vector is
[tex]S= -\frac{I^2 t s}{2 \epsilon_0 \pi^2 a^4 }\hat{s}[/tex]

However, I don't know how to solve c. How would I find the total energy?

The Attempt at a Solution


First I figured I could solve this via an addition of work done for each magnetic and electric fields
[tex]W_{total}=\frac{\epsilon_0}{2}\int E^2 d\tau + \frac{1}{2 \mu_0}\int B^2 d\tau[/tex]
However, this didn't give me a solution in the solutions manual.

I then figured that since the energy density uem is the energy per unit volume, I could integrate via
[tex]U_{em}=\int u_{em} d\tau[/tex]

Alternatively, as the Poynting vector is the energy per unit time, per unit area, I figure maybe I could integrate twice, once as a surface integral and the other with respect to time?

I'm kind of lost, because while I've done just fine without the solutions manual, my attempt at the work haven't been fruitful. Additionally, the solutions manual states the solution is
[tex]U_{em}=\frac{\mu_0 w I^2 b^2}{2 \pi a^4}[(ct)^2+\frac{b^2}{8}][/tex]
in which I have no idea what w is supposed to represent.

I'd greatly appreciate it if anyone could point me in the right direction!
 
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  • #2
Nevermind, I solved it after all! I used my original method by finding the sum of the two components of work and realized that the volume element in cylindrical coordinates is [tex]s ds d\phi dz[/tex] so I solved the integrals that way, which gave me a variation of the solution. I had completely forgotten that w was the length of the gap, and when evaluating the integrals I realized that the z component is from 0 to w, which is why the answer made sense.

The power was simple after that.

Nevertheless, hopefully this post will help anyone stuck on this problem anyway :P
 

FAQ: Total energy and power of electromagnetic fields

What is total energy and power of electromagnetic fields?

Total energy and power of electromagnetic fields refer to the combined amount of energy and power carried by all electromagnetic waves in a given space. This includes all forms of electromagnetic radiation, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

How is total energy and power of electromagnetic fields measured?

The total energy of electromagnetic fields is measured in joules (J) and represents the amount of energy carried by the waves. The power of electromagnetic fields is measured in watts (W) and represents the rate at which the energy is being transferred. Both are measured using specialized instruments, such as power meters and spectrometers.

What factors affect the total energy and power of electromagnetic fields?

The total energy and power of electromagnetic fields are affected by several factors, including the amplitude (or strength) of the waves, the frequency or wavelength of the waves, and the size and shape of the source emitting the waves. The medium through which the waves travel can also impact the total energy and power.

How does the total energy and power of electromagnetic fields relate to the electromagnetic spectrum?

The electromagnetic spectrum is a range of all possible frequencies of electromagnetic radiation. The total energy and power of electromagnetic fields increase as the frequency of the waves increases. This means that higher frequency waves, such as X-rays and gamma rays, have higher total energy and power than lower frequency waves, such as radio waves and microwaves.

What are some practical applications of understanding total energy and power of electromagnetic fields?

Understanding the total energy and power of electromagnetic fields has many practical applications, including in the fields of telecommunications, medicine (such as X-rays and MRI scans), and energy production (such as solar panels). It is also important for understanding the potential effects of electromagnetic radiation on living organisms and the environment.

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