Lecture 1

Question 1

Write the equation for a plane electromagnetic wave propagating in the z-direction.

Answer 1

E(z,t)=E0​cos(kz−ωt)

Question 2

What does the term kk (wave number) represent in the wave equation, and how does it relate to the wavelength?

Answer 2

k represents the number of wave cycles per unit distance, and it is related to the wavelength λλ by k=2πλk=λ2π​.

Question 3

Visualize the oscillations of the electric field (E) and magnetic field (B). Which directions do they oscillate relative to each other and to the wave’s propagation?

Answer 3

The electric field oscillates in the x-direction, the magnetic field oscillates in the y-direction, and both are perpendicular to each other and the z-direction of wave propagation.

Question 4

What does it mean if the divergence ∇⋅F∇⋅F is zero?

Answer 4

It means there is no net flow in or out at that point, indicating a neutral point with neither a source nor a sink.

Question 5

How does positive divergence (∇⋅F>0∇⋅F>0) differ from negative divergence (∇⋅F<0∇⋅F<0) in terms of field behavior?

Answer 5

Positive divergence indicates a source, where the field “flows” outward. Negative divergence indicates a sink, where the field “flows” inward.

Question 6

What happens to a vector field at a point where the divergence is very large and positive?

Answer 6

The field expands outward rapidly from that point, indicating a strong source.

Question 7

Explain why a field with zero divergence cannot be a source or a sink.

Answer 7

A zero divergence means no net increase or decrease in field magnitude; thus, the field neither originates from (source) nor converges into (sink) that point.

Question 8

Describe the difference between linear and circular polarization.

Answer 8

In linear polarization, the electric field oscillates along a single axis. In circular polarization, the electric field rotates in a helical pattern, making one full rotation per cycle.

Question 9

What happens when horizontally polarized light passes through a vertically oriented polarizing filter?

Answer 9

The light is blocked since the horizontal oscillations cannot pass through a vertical filter.

Question 10

If the angle between an incoming light’s polarization direction and the filter axis is 45 degrees, how much light passes through?

Answer 10

Half the light passes through. When light polarization and filter axis are at 45 degrees, the transmitted intensity is reduced to 50%.

Question 11

Define a cavity and explain why only certain frequencies create standing waves within it.

Answer 11

A cavity is a closed or semi-closed space that traps electromagnetic waves. Only specific frequencies create standing waves because the wave’s length must fit an integer number of half-wavelengths within the cavity dimensions.

Question 12

The stationary electric field E in a cavity depends on the cavity’s size and boundary conditions. How does increasing the cavity’s size affect the possible modes?

Answer 12

Increasing the cavity’s size allows more wavelengths to fit, leading to a higher number of possible modes.

Question 13

Describe the equation for the stationary modes and explain how the frequency is determined by the cavity dimensions.

Answer 13

The stationary modes equation relates frequency, cavity dimensions, and mode indices. The larger the cavity, the lower the minimum frequency that can create a standing wave.

Question 14

What does the diagonal line on the spectral mode density graph represent?

Answer 14

The line represents a direct relationship between frequency and mode density. Higher frequencies allow more standing wave modes within the cavity, increasing the mode density with frequency.

Question 15

Write the Rayleigh-Jeans law for spectral energy density

Answer 15

u(ν,T)=c38πν2kT​

Question 16

If the frequency ν is very low, how does this affect the energy density u(ν,T)?

Answer 16

At low frequencies, u(ν,T) is also low, as it depends on ν2.

Question 17

What happens to u(ν,T) as temperature T increases?

Answer 17

The energy density u(ν,T) increases with temperature, as it is directly proportional to T.

Question 18

Describe how Planck’s law corrected the Rayleigh-Jeans law at high frequencies.

Answer 18

Planck’s law introduced quantized energy levels, which limited energy density at high frequencies and resolved the “ultraviolet catastrophe” predicted by the Rayleigh-Jeans law.