Fundamentals of Physics Chapter 33 Solutions: Electromagnetic Waves

Important Topics for Halliday Resnick and Walker Fundamentals of Physics Volume 2 Solutions Chapter 33: Electromagnetic Waves

  • Electromagnetic Waves: Oscillating electric and magnetic fields make up an electromagnetic wave. Its frequencies have a broad spectrum of which visible light is a tiny part. The magnitude of an EM of an electric field E and magnetic field M, travelling along x-axis depends on x and t and given by:

Here Ea and Ma are amplitudes of the electric and magnetic waves respectively.

ω is the angular frequency of the wave = 1/√LC, LC is the oscillator

k is Angular wave number of the wave

  • Speed of an EM in a vacuum – The magnetic field is induced by the oscillating electric wave and vice versa.  Its speed in a vacuum is denoted by c and given by:

  • Poynting vector and energy flow – The pointing vector S given the rate per unit area at which energy is transported via an EM wave. The direction of energy flow and wave’s travel is perpendicular to both electric and magnetic fields.

The average rate of energy transport per unit area, over time is called the intensity I of the wave and given by:

  • Variation of EM intensity – A point source of EM emits the wave isotropically which means that it is spread with equal intensity in all the directions. We can find the intensity of the wave from a point source Ps at a distance d from it by below formula:

  • Radiation Pressure – If an EM is intercepted by a surface, it exerts pressure and force on the surface. Two scenarios arise from this:
    • Total absorption – If the surface absorbs the radiation completely then the force is given by:

F = IA/c and radiation pressure pr = I/c

    • Total reflection – If the surface reflects back the radiation completely along its original path then the force is given by:

F = 2IA/c and radiation pressure pr = 2I/c

      Here I is the intensity of the wave

      A is Area of the surface perpendicular to the direction of the wave

  • Polarization – Polarization occurs when all the electric field vectors of an EM lie in the same plane that is called the plane of oscillation. If seen from the top, all field vectors would be shown as oscillating parallel to a single axis which is perpendicular to the path of the waves. Polarization does not occur in light waves that come from common sources.
    • Polarizing sheets – When an EM passes through a ‘e-polarizing sheet’, only the electric field components which are parallel to the polarizing direction of the sheet get transmitted through. The components perpendicular to the polarizing direction of the sheet are absorbed by the sheet. The light coming out of the sheet is polarized in the direction parallel to the polarizing direction of the sheet.
    • The intensity of an unpolarized light after passing through polarized sheet becomes half of its original intensity I0

    • If the original light was polarized then its final intensity after passing through the sheet depends on the angle θ between the polarization direction of the light and that of the polarizing sheet and is given by:

  • Reflection and refraction – Treating light rays as straight-line rays for studying them is what happens in geometrical optics. When a light ray hits a transparent medium like glass, part of it is reflected and part of it gets refracted. Both these rays lie in the plane of incidence. 
  • The angle of reflection = angle of incidence
  • Angle of refraction – If the angle of incidence is θ1 and the angle of refraction is θ2 then their relationship is given by Snell’s law:

n2 Sin θ2 = n1 Sin θ1

Here n1 and n2 are dimensionless constants known as the index of refraction of the medium in which incidence and refraction are being measured.

  • Total Internal Reflection – When a wave comes in contact with a boundary in which its index of refraction decreases, then there will be an angle of incidence, called critical angle θct, where total internal reflection will happen.

  • Polarization by reflection – Any light reflected from a surface is either fully or partially polarized. When the angle of incidence is such that reflected and refracted rays are perpendicular to each other it is called the Brewster angle θB. At this point, the reflected wave is fully polarized with its electric field vectors perpendicular to the plane of incidence. Brewster angle. which is given by:

Discussion of Exercises of Resnick Halliday and Walker Fundamentals of Physics Volume 2 Solutions Chapter 33: Electromagnetic Waves

  • The first set has 12 questions on different key topics. There are problems on finding the intensity of light through polarizing sheets, the direction of electric and magnetic fields of an EM, total internal reflection, and ranking materials based on their index of refraction.
  • The 2nd set is based on general theorems of EM waves and has a total of 6 questions. The questions would need the application of formulas to find the frequency width of an EM, period of radiation emitted, wavelength of EM waves.
  • The 3rd set has 12 questions where you get to practice sums on the Poynting vector and the energy transport of EM waves. You have to calculate energy contained in a single pulse of an EM, magnetic field amplitude of EM waves, calculate the maximum value of the magnetic field, and the average intensity of light at a point near an isotropic source of EM wave.
  • The 4th set clears your concept with 13 problems on radiation pressure. Some of the highlights of the problems include calculating pressure exerted in total absorption and total reflection conditions, proving radiation pressure is equal to the energy density of the wave perpendicular to the surface, the magnitude of the force and the acceleration given by a beam of light to a sphere in its path.
  • The 5th set has 13 problems on the concept of polarization. Here you have sums like finding the percentage of initial intensity of EM transmitted by polarizing sheets, some complicated sums where there is polarization from water and sunglasses combined and one has to find what percentage of sunlight reaches the eye.
  • The 6th set is on the topic of reflection and refraction with a total of 11 questions. In these one has to find the index of refraction of the medium, the angle which an incoming and outgoing ray makes when it hits a reflecting surface, angles that different colours of a rainbow make when they pass through square drops.
  • The 7th set has 11 questions on total internal reflection. You have to apply formulas to find the critical angle for a light ray in different mediums, angle of incidence for a light to be fully polarized while crossing 2 mediums.
  • The 8th set is a mix of topics from across the modules of the chapter and has 40 questions. You are able to revise all the concepts in this set like finding electric and magnetic field components of EM waves, reflection and refraction sums, polarization problems, radiation pressure, and energy transport calculation.

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