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35. Special Relativity2h 10m

32. Electromagnetic Waves

Intro to Electromagnetic (EM) Waves

Physics

32. Electromagnetic Waves

Intro to Electromagnetic (EM) Waves

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2:55 minutes

Problem 25

Textbook Question

(I) The $\vec{E}$ field in an EM wave has a peak of 23.5 mV/m. What is the average rate at which this wave carries energy across unit area per unit time?

Verified step by step guidance

The average rate at which an electromagnetic wave carries energy per unit area per unit time is given by the Poynting vector's time-averaged magnitude, denoted as ⟨S⟩. This is related to the electric field amplitude (E₀) by the formula: ⟨S⟩ = (1/2) * ε₀ * c * E₀², where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m), c is the speed of light in a vacuum (3.00 × 10⁸ m/s), and E₀ is the peak electric field.

Substitute the given value of the peak electric field, E₀ = 23.5 mV/m, into the formula. First, convert E₀ to volts per meter: E₀ = 23.5 × 10⁻³ V/m.

Square the electric field amplitude: E₀² = (23.5 × 10⁻³)² V²/m².

Multiply the squared electric field by ε₀ and c, and then divide by 2: ⟨S⟩ = (1/2) * (8.85 × 10⁻¹²) * (3.00 × 10⁸) * E₀².

Simplify the expression to find the average energy transfer rate per unit area, ⟨S⟩, in units of W/m².

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field in Electromagnetic Waves

The electric field (E) in an electromagnetic (EM) wave represents the force per unit charge experienced by a charged particle in the wave. Its peak value indicates the maximum strength of the field, which is crucial for determining the energy carried by the wave. In this context, the electric field is measured in volts per meter (V/m) or millivolts per meter (mV/m).

Poynting Vector

The Poynting vector describes the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic wave. It is calculated as the cross product of the electric field (E) and the magnetic field (B) vectors, divided by the permeability of free space. The magnitude of the Poynting vector gives the average power carried by the wave across a unit area.

Average Power Density

Average power density refers to the average amount of energy carried by an electromagnetic wave per unit area over time. It can be calculated using the formula P = (1/2) * ε₀ * c * E², where ε₀ is the permittivity of free space, c is the speed of light, and E is the peak electric field. This concept is essential for understanding how much energy is transmitted through a given area by the wave.

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Master Introduction to Electromagnetic (EM) Waves with a bite sized video explanation from Patrick

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Estimate how long an AM antenna would have to be if it were (a) ½ ⋋ or (b) ¼ ⋋ AM radio is roughly 1 MHz (530 kHz to 1.7 MHz).

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Textbook Question

(I) Electromagnetic waves and sound waves can have the same frequency. (a) What is the wavelength of a 1.00-kHz electromagnetic wave? (b) What is the wavelength of a 1.00-kHz sound wave? (The speed of sound in air is 341 m/s.) (c) Can you hear a 1.00-kHz electromagnetic wave?

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(II) How long would it take a message sent as radio waves from Earth to reach Mars when Mars is (a) nearest Earth, (b) farthest from Earth? Assume that Mars and Earth are in the same plane and that their orbits around the Sun are circles (Mars is 230 x 10⁶ km from the Sun).

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(II) Pulsed lasers used for science and medicine produce very brief bursts of electromagnetic energy. If the laser light wavelength is 1062 mm (Neodymium–YAG laser), and the pulse lasts for 32 picoseconds, how many wavelengths are found within the laser pulse? How brief would the pulse need to be to fit only one wavelength?

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Textbook Question

(II) The magnetic field in a traveling EM wave has an rms strength of 22.5 nT. How long does it take to deliver 415 J of energy to 1.00 cm² of a wall that it hits perpendicularly?

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Textbook Question

What are E₀ and B₀ at a point 2.50 m from a light source whose output is 5.0 W? Assume the bulb emits radiation of a single frequency uniformly in all directions.

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(I) The E→\overrightarrow{E} field in an EM wave has a peak of 23.5 mV/m. What is the average rate at which this wave carries energy across unit area per unit time?

Key Concepts

Electric Field in Electromagnetic Waves

Poynting Vector

Average Power Density

Watch next

(I) The $\vec{E}$ field in an EM wave has a peak of 23.5 mV/m. What is the average rate at which this wave carries energy across unit area per unit time?