If you think about a basketball player standing in one spot and dribbling a basketball you can get a picture of whatʼs being represented here. The basketball is going up and down. Itʼs “vibrating” and the charge –q also vibrates up and down in the radiating element of an antenna. Imagine that the inhabitants of the Doodle Galaxy, 1000 light years distant, have invented a fantastic telescope that lets them observe day-to-day activities here on earth. On some particular day, 100 years in the future, one of their astronomers is very excited to report that they have spotted an earthman demonstrating how to dribble a basketball. Of course, it took the light from Earth 1000 years to reach the Doodle Galaxy so the Doodle astronomers are seeing an event that took place 1000 years in their past. If they tried to perform calculations on the motion of the basketball they would be assessing an event that happened 1000 years earlier. In the meantime the basketball player would have grown old and died. The ratio rʼ/c adjusts the calculation to account for the fact that the –q charge was actually measured in retarded time. But this is simply a ratio and weʼre going to use this ratio to add an adjustment factor to Coulombʼs result. The thing thatʼs going to be adjusted is the second part of the equation.
The Derivative of the Energy With Respect To Time
Continuing our examination of the Electric Field Strength equation (Figure 5.2) weʼre going to consider the erʼ/rʼ2 in the third term. Remember that erʼ is the unit vector representation of the –q chargeʼs position relative to the point of measurement and rʼis the apparent distance that –q is from (x,y,z) (since –q may have moved since the signal originated and weʼre only seeing it in retarded time).
To understand whatʼs happening here we need to understand that the term erʼ/rʼ2 term is being differentiated with respect to time. The second derivative of the final term erʼ is being taken.
The first derivative represents the rate of change of the expanding field and the second derivative represents the rate at which the rate of change is, itself, changing. Multiplying the rate of change of the field (the first derivative of erʼ/rʼ2) by the adjustment required for retarded time compensates for the fact that weʼre not seeing the charge “now”. The final term adds the product of 1/c2 and the second derivative of the unit vector erʼ.
Perhaps if youʼre unable to sleep some night and youʼre diligently working through this fundamental formula for determining the strength of an electric charge –q at location (x,y,z) you may encounter a challenge that is not yet totally solved by the physicists of today. You see, the charge, when viewed as a point particle, has an electric energy E that applies to itself. That is “What is the value of E when P is at the point of the charge itself?” The problem with this is that rʼ= 0 is not allowed as a denominator and that would be the case if you tried to plug values into the field formula when (x,y,z) is at the point charge itself.
Effective Radiated Power
Determining the energy of an electromagnetic field at some particular point involves some complicated math and introduces some surprises. Weʼll discover that there are two different regions of influence related to the radiated power and they are ultimately defined relative to the field equation.
To understand this we first consider some level of input power applied to the transmitting antenna. This power (a value of E and I across the resisting antenna element) is converted into an
electromagnetic field that radiates outward from the antenna. The power output of the antenna is called the Effective Radiated Power (ERP) and is typically measured in milliwatts.
Math and Physics for the 802.11 Wireless LAN Engineer |
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Copyright 2003 - Joseph Bardwell