PHYS116 Final Exam Review | Source: Chapter 25.5 lecture notes
Difficulty: Intermediate | Prerequisites: Basic understanding of electric and magnetic fields, Faraday's law of induction, and wave properties (frequency, wavelength, amplitude).
Electromagnetic waves sit at the intersection of electricity and magnetism, the payoff of everything covered so far in PHYS116. A changing magnetic field creates an electric field, and a changing electric field creates a magnetic field, and when the two sustain each other in just the right geometry, they break free and travel through space as a wave. This is the mechanism behind light, radio, microwaves, X-rays, and every other form of radiation on the EM spectrum. You should already be comfortable with Faraday's law and the basic wave equation v = λf before diving in.
Electric and magnetic fields can sustain themselves as a self-propagating transverse wave that travels at the speed of light. The wave's intensity drops with distance (inverse-square law), and a polarising filter reduces it further according to Malus's Law. Knowing the relationships between E₀, B₀, c, intensity, and power is enough to solve most exam problems in this section.
Electromagnetic (EM) wave
A transverse wave made of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of travel. Think of it as a ripple in the electric and magnetic fields that carries energy through space, no medium required.
Transverse wave
A wave whose oscillations are perpendicular to the direction the wave moves. In simple terms, the fields wiggle side-to-side while the wave travels forward.
Permittivity constant (ε₀)
A fundamental constant (8.85 × 10⁻¹² C²/N·m²) that describes how easily an electric field forms in a vacuum. It shows up in the formula for the speed of light.
Permeability constant (μ₀)
A fundamental constant (1.26 × 10⁻⁶ T·m/A) that describes how easily a magnetic field forms in a vacuum. Paired with ε₀, it sets the speed of EM waves.
Plane of polarisation
The plane that contains the oscillating electric field vectors of an EM wave. In simple terms, it tells you the direction the E-field is swinging.
Intensity (I)
The power delivered per unit area by an EM wave, measured in W/m². Think of it as how much energy the wave dumps onto each square metre of surface every second.
Malus's Law
The rule that a polarising filter transmits intensity I_trans = I_incid · cos²θ, where θ is the angle between the wave's polarisation and the filter's axis. It tells you how much light gets through a filter depending on the alignment angle.
Inverse-square law (for point sources)
Intensity falls off as I = P_source / (4πr²). In simple terms, doubling your distance from a source cuts the intensity to a quarter.
A changing magnetic field induces an electric field (even without a conducting loop), and a changing electric field induces a magnetic field. When the two fields sustain each other, they form an EM wave.
EM waves are transverse: the E-field and B-field oscillate perpendicular to each other and perpendicular to the direction of propagation.
E and B are in phase: their crests, troughs, and zeros line up.
In a vacuum, every EM wave travels at the same speed:
v_em = 1 / √(ε₀μ₀) = 3.00 × 10⁸ m/s
This is the speed of light, c. Light is an EM wave.
The wave equation applies: c = λf (same structure as v = λf for any wave).
The amplitudes of the electric and magnetic fields in an EM wave are locked together:
E₀ / B₀ = c
If you know one amplitude, you can find the other immediately.
The plane of polarisation is defined by the direction of the E-field oscillation.
A polarising filter transmits only the component of E along its axis.
Intensity measures power per unit area (W/m²) and depends on the field amplitudes:
I = ½ c ε₀ E₀² = (c / 2μ₀) B₀²
For a point source radiating equally in all directions:
I = P_source / (4πr²)
Intensity drops with the square of the distance from the source.
When polarised light passes through a polarising filter at angle θ to the polarisation plane:
I_trans = I_incid · cos²θ
At θ = 0°, all the light gets through. At θ = 90°, none does.
Speed of an EM wave in vacuum
v_em = 1 / √(ε₀μ₀) = c = 3.00 × 10⁸ m/s
Wave equation
c = λf
Field amplitude relationship
E₀ / B₀ = c
Intensity from field amplitudes
I = ½ c ε₀ E₀² = (c / 2μ₀) B₀²
Intensity from a point source
I = P_source / (4πr²)
Malus's Law
I_trans = I_incid · cos²θ
where θ is the angle between the incident polarisation direction and the polariser's transmission axis.
The inverse-square law is why your mobile signal weakens as you walk away from a tower, and why a campfire only warms you when you are close. Malus's Law is the principle behind polarised sunglasses: they block glare by filtering out horizontally polarised light reflected off roads and water.
Students often think that increasing the intensity of light changes its speed. It does not. All EM waves travel at c in a vacuum regardless of amplitude or intensity.
Students sometimes believe that E and B in an EM wave are out of phase (one peaks when the other is zero). They are in phase: both peak and both cross zero at the same time.
A common error is applying Malus's Law with θ measured from the wrong reference. θ is the angle between the polarisation direction of the incoming light and the transmission axis of the filter, not the angle of incidence.
Students frequently forget to square the cosine in Malus's Law, writing I·cosθ instead of I·cos²θ.
⚠️ Expect problems that give you a source's power and a distance and ask for intensity (inverse-square law), then ask for the E-field amplitude from that intensity.
⚠️ Malus's Law calculations are a staple. Make sure you can work the angle correctly and remember to square the cosine.
⚠️ Know that E₀/B₀ = c. Problems may give one field amplitude and ask for the other.
⚠️ Be clear that the speed of an EM wave in vacuum is fixed at c regardless of frequency, wavelength, or intensity.
True or false: An EM wave requires a medium to propagate. False. EM waves travel through a vacuum.
Fill in the blank: The E-field and B-field in an EM wave are ______ to each other and ______ to the direction of travel. Perpendicular; perpendicular.
True or false: Doubling the distance from a point source reduces the intensity to one-half. False. It reduces to one-quarter (inverse-square law).
Fill in the blank: In Malus's Law, the transmitted intensity depends on cos²____. θ, the angle between the polarisation direction and the filter axis.
True or false: If E₀ = 300 N/C, then B₀ = 300 T. False. B₀ = E₀/c = 300 / (3.00 × 10⁸) = 1.00 × 10⁻⁶ T.
Q: A cell phone emits 0.500 W of 1.80 GHz radio waves. What is the intensity of this wave at a distance of 15 cm from the phone?
A: I = P / (4πr²) = 0.500 / (4π(0.15)²) = 1.77 W/m²
Q: Using the intensity from the previous question, what is the amplitude of the electric field of the EM wave emitted by the phone?
A: I = ½ c ε₀ E₀². Rearranging: E₀ = √(2I / cε₀) = √(2 × 1.77 / (3.00 × 10⁸ × 8.85 × 10⁻¹²)) = 36.5 N/C
Q: The intensity of a polarised EM wave is 10.0 W/m². What will the intensity be after passing through a polarising filter whose axis makes a 30° angle with the plane of polarisation?
A: I_trans = I_incid · cos²θ = 10.0 × cos²(30°) = 10.0 × (0.866)² = 7.5 W/m²
Q: An EM wave has a magnetic field amplitude of B₀ = 2.0 × 10⁻⁶ T. What is the electric field amplitude?
A: E₀ = c × B₀ = (3.00 × 10⁸)(2.0 × 10⁻⁶) = 600 N/C
This connects to the photon model of EM waves (Ch. 25.6), where the same waves are described as streams of massless particles called photons, each carrying energy E = hf. The intensity and field-amplitude relationships here underpin how photon energy and photon count are calculated.
It also connects to the photoelectric effect (Ch. 28.2), where the frequency of an EM wave determines whether it can eject electrons from a metal surface. Understanding that EM wave speed is fixed and independent of intensity is critical there.
The EM spectrum (Ch. 25.7) expands on the idea that all EM waves travel at c, differing only in frequency and wavelength.
Tags: electromagnetic wave, EM wave, transverse wave, speed of light, permittivity, permeability, E-field amplitude, B-field amplitude, polarisation, plane of polarisation, polarising filter, Malus's Law, intensity, inverse-square law, point source, W/m², PHYS116, general physics, University of Washington, Chapter 25.5