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Kurs: Elektrotechnika > Rozdział 5
Lekcja 2: Pole, potencjał i napięcie elektryczne- Natężenie pola elektrycznego pochodzącego od jednorodnie naładowanej, nieskończenie długiej, prostej nici
- Plane of charge
- Natężenie pola elektrycznego pochodzące od jednorodnie naładowanej, nieskończonej płaszczyzny (część 1)
- Natężenie pola elektrycznego pochodzące od jednorodnie naładowanej, nieskończonej płaszczyzny (część 2)
- Energia elektrostatyczna
- Electric potential, voltage
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Plane of charge
Advanced example: Electric field generated by a uniformly-charged infinite plane. Written by Willy McAllister.
Example: Electric field near a plane of charge
We investigate the next interesting charge configuration, the electric field near a plane of charge.
The result will show the electric field near an infinite plane of charge is independent of the distance away from the plane (the field does not fall off).
Imagine we have an infinite plane of charge.
The total charge on the plane is of course infinity, but the useful parameter is the amount of charge per area, the charge density, .
What is the electric field due to the plane at a location away from the plane?
We exploit the symmetry of the problem to set up some variables:
is a perpendicular line from the plane to the location of our test charge, .- Imagine a hoop of charge in the plane, centered around where
touches the plane. The radius of the hoop is , and its infinitesimal thickness is . is an infinitesimal region of charge in a section of the hoop.- Line
goes from the location of to the location of the test charge. is the electric field at point created by .
We know the field at location due to ; it's the definition of the field created by a point charge,
To solve the electric field for the whole plane, we have to do two integrations:
- first integration to sweep
around its hoop to get the field contribution from one particular hoop, and a - second integration to add up the contributions from all possible hoops (from zero radius to infinite radius).
Sweep around a hoop to get the field contribution from one particular hoop
The hoop construct cleverly allows us to duck the first integral. All parts of the hoop are the same distance away from , so each creates the same magnitude field at . Symmetry tells us the total field contribution from all 's in a hoop has to point straight away from the plane, along line . Why? Because any sideways component of the field from a particular is exactly cancelled by the on the opposite side of the hoop. The straight out " -direction" portion of the electric field is related to ,
Which gives this for , the field from a single point charge ,
Next, express the field contribution from one entire hoop ,
The electric field at the location of created by a hoop with radius , containing charge is,
Now we know the field contributed by a single hoop.
Integrate the contributions from all possible hoops
The next step is to sum up all possible hoops. Unfortunately, we can't sneak out of doing this integral. Just like we did for Line of Charge example, we perform a change of variables, from to .
After the change of variable, the diagram can be redrawn in terms of and ,
and the field equation for one hoop becomes,
which can be simplified a bit more,
Something very interesting just happened. As a result of the change of variable and cancellation, all the 's and 's vanish! Wait, What! In the resulting expression for , there is NO dependence on distance. Remarkable.
Almost home. We are ready to perform the integration,
where is the overall electric field from all hoops. Substitute for ,
What are the angle limits on the integration? The smallest possible hoop is when is zero; coincides with , and is zero. The largest hoop is when is infinite; line comes from way out at the horizon in any direction, and is or radians. So the limits on the integration run from radians.
The electric field near an infinite plane is,
Podsumowanie
This the electric field (the force on a unit positive charge) near a plane. Amazingly, the field expression contains no distance term, so the field from a plane does not fall off with distance! For this imagined infinite plane of charge, it doesn't matter if you are one millimeter or one kilometer away from the plane, the electric field is the same.
This example was for an infinite plane of charge. In the physical world there is no such thing, but the result applies remarkably well to real planes, as long as the plane is large compared to and the location is not too close to the edge of the plane.
Review
Using the notion of an electric field, the analysis technique is,
- Charge gives rise to an electric field.
- The electric field acts locally on a test charge.
Summarizing the three electric field examples worked out so far,
The field due to a | falls off at |
---|---|
point charge | |
line of charge | |
plane of charge |
These three charge configurations are a useful toolkit for predicting electric field in lots of practical situations.
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