Chapter 22 Tutorials

Chapter 22
The Giancoli onLine Tutor

Electromagnetic Waves

I. Displacement Current

Like the previous few chapters this chapter covers very advanced material at a very low level. As a result most problems simply involve "plugging" into the appropriate equation the correct physical values.

One basic equation is

lf = c

And typical problems involve giving a frequency, f, find l, or vice versa. Simple and straight forward. Make sure take into account metric multipliers like MHz (Mega Hertz = 10⁶ Hz).

Other problems involve the so called Displacement Current, I_D. A more discriptive term would be virtual current. The defining equation is

I_D = e_oDF_E/Dt,

where F_E is the Electric Flux equal to ExA (the electric field multiplied by the area). In practice the easiest way to find the displacement current inside a capacitor is to find the conduction current in the circuit connected to it, they are always equal.

From an earlier chapter we know that the voltage, V, on the capacitor builds up as

V = E(1-e^-t_/RC),

where E represents the emf. Initially the current flows at its maximum value E/R, as the capacitor charges, the falling current is given by

I_c = (E/R)e^-t_/RC

This then gives at the same time the displacement current in the capacitor. The electric field in the capacitor is given by V/d and the flux is ExA.

Interactive Example 1

If R= 100 W, C= 40 mF and E = 10 V, what is the displacement current at 2 ms.
Enter your answer and click the clover for good luck (and the answer) .
What is the voltage on the capacitor at 2 ms.
Enter your answer and click the clover for good luck .
If the plate separation is 1 mm what is the Electric Field at 2 ms.
Enter your answer and click the clover for good luck .
If the plate area is .1 m² what is the Electric Flux at 2 ms.
Enter your answer and click the clover for good luck .

II. Energy Density in an Electric Field

II-a. Energy Density in a Capacitor

The energy density, u , in a capacitor at any time is

u = e_oE²/2

Interactive Example 2

In Interactive Example 1 what is the energy density stored in the electric field (in mJ/m³) in the capacitor at 2 ms.
Enter your answer and click the clover for good luck .

II-b. Energy Density in an Electromagnetic Wave

In electromagnetic radiation (visible light, infrared, etc.) energy is carried, in equal amounts, by both the electric field and the magnetic field and the energy density is twice that above. The result

u = e_oE² = B²/m_o
remember B = E/c
and c = SQRT(e_om_o)

The above is of little practical value since the field is oscillating and moving though space and thus the energy value at any point is continually changing. Of more significance is the Intensity ( or Pointing vector),

S = e_ocE² = cB²/m_o.

The units are watts/meter² and it represents the energy transmited across 1 square meter in 1 second. While this is also time dependment a useful parameter is its average value S_ave,

S_ave = e_ocE_o²/2 = cB_o²/2m_o,

where B_o and E_o are the maximum values of the magnetic and electric vector, respectively. The rms (root-mean-square) value of B and E are each equal to their maximum value divided by the squareroot of 2, so

S_ave = e_ocE_rms² = cB_rms²/m_o,

Interactive Example 3

The solar constant, (the energy reaching the earth per meter squared per second), is approximately 1400 W/m², what is the maximum electric vector? Use S_ave = e_ocE_o²/2 , to find E_o to the nearest V/m and check your answer below.
Use either S_ave = cB_o²/2m_o , to find B_o or more simply B_o = E_o/c to the nearest nearest tenth of a mT and check your answer below.

finis

Electromagnetic Waves

I. Displacement Current

lf = c

ID = eoDFE/Dt,

V = E(1-e-t/RC),

Ic = (E/R)e-t/RC