PHY385 Module 8: Superposition Principle and Energy Density

 

   Activity 8.1  – Superposition principle

               8.1.1.  You will watch the superposition of two disturbances with animations created by Prof. Daniel Russell from Pennsylvania State University. The two pulses are travelling 

1. in the opposite direction;

2. in the same direction 

What is the amplitude of the resulting disturbance in any location of space in both  cases (1) and (2)?

Give your estimate of a speed of propagation of the disturbance in the cases (1) and (2)?

 

The two waves are traveling

     3.  in the opposite direction;

     4.  in the same direction with an observer at rest in the reference frame of one of the wavefronts; and

     5.  in the same direction with slightly different frequency

Treating the amplitude as the maximum possible magnitude of the disturbance (displacement), what is

• the amplitude of the resulting wave in the cases (3), (4) and (5)?
• the wavelength of the resultant in the cases (3) - (5)?  
• frequency of the resulting disturbance in the cases (3) - (5)?

 

               8.1.2.  The superposition principle states that the superposition of the waves is also a wave. 

1. Determine which of the following describe traveling waves: 

\(\Psi(y.t) = e^{-(a^2y^2+b^2t^2-2abty)}\)

\(\Psi(z,t)=A\sin(Az^2-bt^2)\)

\(\Psi(x,t)=A\cos^2[2\pi(t-x)]\)

       2. For the traveling wave(s) from the above list, write the direction of propagation and the speed of the wave.

       3. Choose any one of the traveling waves determined above. Prove that the superposition of the two of such waves is also a wave.

 

 

   Activity 8.2  – Electromagnetic Wave

 

A 550-nm harmonic EM wave whose electric field is in the z-direction is traveling in the y-direction in vacuum.

1. What is the frequency of the wave?
2. Determine ω and k of the wave.
3. If the electric field amplitude is 600 V/m, what is the amplitude of the magnetic field?
4. Write an expression for E(t) and B(t) given that each is zero at x = 0 and t = 0. Put in the appropriate units

 

   Activity 8.3  – Complex form of the wave function

 

An electromagnetic wave is specified (in SI units) by the following equation:

\(E = (-6\hat{i}+3\sqrt{5}\hat{j})\bigg(10^4\frac{V}{m}\bigg)e^{i\bigg[\frac{1}{3}(\sqrt{5}x+2y)10^7\pi - 9.42\times10^{15}t\bigg]}\)

Find:

1. the direction along which the electric field oscillates;
2. the scalar value of amplitude of the electric field;
3. the direction of propagation of the wave
4. the propagation number;
5. the frequency and the angular frequency; and
6. the speed of the wave.

 

 

   Activity 8.4  – Energy Density & Poynting Vector

 

Consider a linearly polarized plane electromagnetic wave traveling in the + x direction in free space having as its plane of vibration the xy- plane. Given that its frequency is10 MHz and its amplitude is E₀ = 0.08 V/m,

1. Find the period and wavelength of the wave.
2. Write an expression for E(t) and B(t).
3. Find the expression for the Poynting vector S.
4. The flux density of the wave is the time average of the magnitude of the Poynting vector S. Find the flux density <S> of the wave.
5. Find the energy density of the wave.
6. Prove that the energy densities of the electric and magnetic fields are equal ( u_E =u_B ) for electromagnetic field.