33.10
Intensity Of Electromagnetic Waves
The energy transport per unit area per unit time, or the Poynting vector, gives the energy flux of an electromagnetic wave at any specific time. For a…
When a capacitor is connected to an alternating current, the direction of the electric field between the plates changes with time, which induces a varying magnetic field.
The energy flow given by the Poynting vector is parallel to the plate. It varies rapidly with time.
Taking the time average of the Poynting vector gives the wave's intensity.
Consider a plane electromagnetic wave with the electric field oscillating along the y-axis and the magnetic field along the z-axis. Then, the Poynting vector points in the positive x-direction.
By replacing the fields with wave functions, the expression for the Poynting vector can be rewritten.
As the fields are mutually perpendicular, the Poynting vector magnitude equals the product of the peak amplitudes of the fields and a square of cosine term. By integrating the Poynting vector magnitude over a wave period, the expression for the intensity can be obtained.
Using the relationship between fields and the speed of light, the equivalent expressions for intensity can be defined in terms of the magnitudes of the electric and magnetic fields alone.
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Q1: What is the Poynting vector and how does it relate to electromagnetic wave intensity?
The Poynting vector represents the energy flux of an electromagnetic wave, indicating power per unit area at any instant. It points perpendicular to both the electric and magnetic fields. The time average of the Poynting vector gives the wave's intensity, which is the measurable power per unit area that detectors and our eyes actually observe.
Q2: Why do we need to calculate the time average of the Poynting vector instead of using instantaneous values?
Electromagnetic waves oscillate at extremely high frequencies, such as visible light at 10^14 Hz, causing the energy flux to vary rapidly with time. Most measuring devices and human eyes detect only the average over many cycles. Time-averaging the Poynting vector over one complete period yields the intensity that we can actually measure.
Q3: How is electromagnetic wave intensity expressed in terms of electric and magnetic field amplitudes?
Intensity can be expressed using the peak electric field E0 and peak magnetic field B0 amplitudes. By integrating the Poynting vector magnitude over a wave period and using the relationship between fields and the speed of light, equivalent expressions for intensity are derived in terms of these field magnitudes alone.
Q4: What role does the cosine function play in calculating electromagnetic wave intensity?
The Poynting vector magnitude contains a cosine-squared term that oscillates with time. When averaged over one complete period, the average of cos²θ equals 1/2. This factor is crucial for converting the time-varying energy flux into the constant intensity value that represents the average power per unit area.
Q5: How do perpendicular electric and magnetic fields determine the direction of energy flow?
In a plane electromagnetic wave, the electric field and magnetic field are mutually perpendicular to each other and to the direction of propagation. The Poynting vector, calculated as the cross product of these fields, points in the direction of energy propagation. For example, if E oscillates along the y-axis and B along the z-axis, the Poynting vector points along the x-axis.
Q6: What happens to the electromagnetic field when an alternating current is applied to a capacitor?
When alternating current connects to a capacitor, the electric field direction between the plates changes with time. This time-varying electric field induces a corresponding varying magnetic field. The resulting energy flow, described by the Poynting vector, is parallel to the capacitor plates and varies rapidly with time.
Q7: How does the speed of light relate to electromagnetic wave intensity calculations?
The relationship between electric and magnetic field amplitudes depends on the speed of light in the medium. Using this fundamental relationship, equivalent expressions for intensity can be derived that express the wave's power per unit area solely in terms of field magnitudes and the propagation speed electromagnetic waves.