Electromagnetic Waves

Sources and Nature of EM Waves

  • Stationary charges produce only electric field.
  • Moving charges produce both electric field \tt \overline{E} and magnetic field \tt \overline{B}
  • Only Accelerated charges radiates energy in the form of em waves.
  • An oscillating charge produces a time varying magnetic field.
  • The time varying magnetic field is a source of time varying electrical field.
  • The oscillating electric and magnetic fields regenerate each other and propagate through space as waves. These waves are called electromagnetic waves.
  • An electric charge oscillating harmonically with frequency v produces electromagnetic waves of the same frequency v.
  • Frequency, \tt v = \frac{1}{2 \pi \sqrt{LC}}
  • \tt \overline{E}= E_{y} \widehat{j} = E_{0} \sin [Kx - Wt]\widehat{j} = E_{0} \sin 2 \pi \left(\frac{x}{\lambda} - vt\right)\widehat{j} = E_{0} \sin 2 \pi \left(\frac{x}{\lambda} - \frac{t}{T}\right)\widehat{j} \Rightarrow E_{x} = E_{z} = 0
  • \tt \overline{B}= B_{z} \widehat{K} = B_{0} \sin [Kx - Wt]\widehat{K} = B_{0} \sin 2 \pi \left(\frac{x}{\lambda} - vt\right)\widehat{K} = B_{0} \sin 2 \pi \left(\frac{x}{\lambda} - \frac{t}{T}\right)\widehat{K} \Rightarrow B_{x} = B_{y} = 0
  • The magnetic of the wave vector \tt \overline{K} is given by \tt K = \frac{2 \pi}{\lambda}
  • The magnitudes of \tt \overline{E} and \tt \overline{B} are related by \tt \frac{E}{B} = C (or) \tt \frac{E_{0}}{B_{0}} = C
  • Electromagnetic waves travel through vacuum with the speed of light c, \tt C = \frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}} = 3 × 108 m/s
  • \tt C_{med} = \frac{1}{\sqrt{\mu{E}}} = \frac{1}{\sqrt{\mu_{r}\mu_{0} \varepsilon_{1} \varepsilon_{0}}} = \frac{C_{0}}{\sqrt{\mu_{r} \varepsilon_{r}}}
  • \tt \sqrt{\mu_{r} \varepsilon_{r}} = \frac{C_{0}}{C_{med}} = n (n = refractive index)
  • The Poynting vector \tt \overrightarrow{S} = \overrightarrow{E} \times \overrightarrow{H} = \frac{\overrightarrow{E} \times \overrightarrow{B}}{\mu_{0}} represents the energy flow direction per unit area per second along the direction of wave propagation.
  • Wave Impedance (z) value in a medium is \tt Z = \frac{E}{H} = \sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{\mu_{c}}{\varepsilon_{r}}} \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}
  • Z wave Impedance for vacuum or free space. \tt Z = \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} = 376.6 \Omega
  • Z value for a conducting medium \tt Z_{e} \sqrt{\frac{\mu_{\omega}}{\sigma}}
  • In free space the E and B are related as \tt \frac{E}{B} = \frac{E}{\mu_{0}H} \Rightarrow \frac{E}{B} = \frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}} = C
  • Electric field intensity E = BC                    B = Magnetic field intensity
  • Total average energy density = \tt \frac{1}{2} \varepsilon_{0} E_{0}^{2}
  • Electric Energy Density = \tt \frac{1}{2} \varepsilon E^{2}
  • Magnetic Energy Density = \tt \frac{B^{2}}{2 \mu}
  • Total Energy Density = \tt \frac{1}{2} \varepsilon E^{2} + \frac{B^{2}}{2 \mu}
  • The intensity of radiation is defined as the amount of energy passing through unit area in unit time.

\tt I = \frac{Energy / time}{Area} = \frac{Power}{Area}

\tt I = U C = \varepsilon_{0} C E^{2}_{rms}

  • \tt P = \frac{U}{C}       P = Momentum delivered

                      U = Energy transferred to the surface.

  • When the radiation incident on a surface is entirely reflected back along its original path, the magnitude of momentum delivered to the surface is \tt P = \frac{2U}{C}
  • When radiation is incident on a surface, Radiation pressure, \tt P_{r} = \frac{I}{C} (total absorption)
  • Total reflection back along the incident path is \tt P_{r} = \frac{2I}{C}

View the Topic in this video From 01:02 To 53:19

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1. The equation of an electric field along x-axis is given as Ex = E0 sin (kz − ωt)

2. Here k is propagation constant and is equal to \frac{2\pi}{\lambda}
ω is the angular frequency = \frac{2\pi}{T}

3. The speed of electromagnetic wave in free space is
c = \frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}

4. The speed of electromagnetic wave in a medium is
v = \frac{1}{\sqrt{\mu \varepsilon}}

5. The amplitudes of electric and magnetic fields in free space, in electromagnetic waves are related by
E_{0} = cB_{0} \ {\tt or} \ B_{0} = \frac{E_{0}}{c}