Most measuring devices, including our eyes, detect only an average over many cycles. The time average of the energy flux is the intensity I of the electromagnetic wave and is the power per unit area. It can be expressed by averaging the cosine function in Figure over one complete cycle, which is the same as time-averaging over many cycles here, T is one period :.
We can either evaluate the integral, or else note that because the sine and cosine differ merely in phase, the average over a complete cycle for is the same as for , to obtain. The intensity of light moving at speed c in vacuum is then found to be. Algebraic manipulation produces the relationship. One more expression for in terms of both electric and magnetic field strengths is useful.
Substituting the fact that the previous expression becomes. We can use whichever of the three preceding equations is most convenient, because the three equations are really just different versions of the same result: The energy in a wave is related to amplitude squared. Furthermore, because these equations are based on the assumption that the electromagnetic waves are sinusoidal, the peak intensity is twice the average intensity; that is,. A Laser Beam The beam from a small laboratory laser typically has an intensity of about.
Assuming that the beam is composed of plane waves, calculate the amplitudes of the electric and magnetic fields in the beam. Strategy Use the equation expressing intensity in terms of electric field to calculate the electric field from the intensity. Solution From Figure , the intensity of the laser beam is.
The amplitude of the magnetic field can be obtained from Figure :. Light Bulb Fields A light bulb emits 5. What are the average electric and magnetic fields from the light at a distance of 3.
Solution The power radiated as visible light is then. Significance The intensity I falls off as the distance squared if the radiation is dispersed uniformly in all directions.
Strategy The area over which the power in a particular direction is dispersed increases as distance squared, as illustrated in the figure. Then use the proportion of area A in the diagram to distance squared to find the distance that produces the calculated change in area. Solution Using the proportionality of the areas to the squares of the distances, and solving, we obtain from the diagram.
Significance The range of a radio signal is the maximum distance between the transmitter and receiver that allows for normal operation.
In the absence of complications such as reflections from obstacles, the intensity follows an inverse square law, and doubling the range would require multiplying the power by four. This can also be expressed in terms of the maximum magnetic field strength as. The three expressions for are all equivalent. When you stand outdoors in the sunlight, why can you feel the energy that the sunlight carries, but not the momentum it carries?
The amount of energy about is can quickly produce a considerable change in temperature, but the light pressure about is much too small to notice. How does the intensity of an electromagnetic wave depend on its electric field? How does it depend on its magnetic field? It has the magnitude of the energy flux and points in the direction of wave propagation. It gives the direction of energy flow and the amount of energy per area transported per second. If the beam does not diverge appreciably, how would its rms electric field vary with distance from the laser?
While outdoors on a sunny day, a student holds a large convex lens of radius 4. By what factor is the electric field in the bright spot of light related to the electric field of sunlight leaving the side of the lens facing the paper?
A plane electromagnetic wave travels northward. Hence, the energy density U of an electromagnetic wave can be expressed only in terms of either the electric field or magnetic field.
Equations 5 and 6 give the expression for the energy density of electromagnetic waves in terms of the electric field and magnetic field, respectively:. From equations 3 , 5 , and 6 , we can summarize that in a given volume, the electromagnetic energy is shared equally between the electric and magnetic fields.
The Poynting vector S is the rate at which electromagnetic energy flows through a unit surface area perpendicular to the direction of propagation of the wave. Basically, it is the rate of flow of energy in an electromagnetic wave, given by equation 7 :.
Like the electric field and magnetic field, the magnitude of the Poynting vector S also varies with time. The maximum values of E, B, and S occur at the same instant. However, the S vector is oriented towards the direction of propagation of the wave.
It is important to remember that S is time-varying. In such quantities, the average value is of great importance. The term wave intensity I gives the average time of the Poynting vector S and can be denoted as Savg. The relationship between the Poynting vector, wave intensity, and energy density can be given by the following equations:.
The energy density of an electromagnetic wave is completely dependent on the electric and magnetic fields of the wave. By using these calculations, you can determine the energy density of an electromagnetic wave. This is the amount of energy per unit volume contained within the fields. Unlike total energy, energy density can be defined easily for specific locations. To obtain it for one position, we only need to know the value of the electric and magnetic fields at that position.
This is similar to our motivation for introducing energy densities when we discussed fluids in Physics 7B. The energy density of the electric and magnetic fields are. These results are a little bit tricky to derive from what we already know, so we do not attempt a derivation here.
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