- #1
maverick280857
- 1,789
- 4
Hello.
My textbook says that a Gaussean surface must be carefully chosen so that a point charge (or point charges constituting a discrete charge distribution) does not lie ON it, as otherwise the electric field at the location of the charge would be infinite and hence, it would not be possible to compute the flux through such a surface. In contrast, it says, that a Gaussean surface can well pass through a continuous charge distribution (a volume charge distribution) as the field everywhere is defined for such a distribution.
The simplest example I can think of which justifies this proposition is that of a uniform solid nonconducting sphere with uniform volume charge: the field inside grows linearly with radial distance and falls of as the inverse square of radial distance outside. So, clearly, any spherical Gaussean surface drawn in the interior of the solid sphere passes through a continuous charge distribution yet the field and flux are well defined there.
But now suppose we build our continuous charge distribution by somehow piling discrete charges on the solid material so that when this process has been carried out, the charge distribution has become continuous. At any intermediate stage, we could identify the distribution as discrete and so point charges may well reside on our Gaussean surface but at the end of the process, there is no problem because now, the field is well defined everywhere!
Yet, we do regard a continuous charge distribution as made up of infinitely many point charges somehow stacked together though they would--when distinguishable--fly off due to mutual repulsion. Then, what is the fundamental explanation for this apparent inconsistency: that the Gaussean surface does not pose problems though infinitely many point charges lie on it..but it does pose a problem when countably finite charges lie on it?
According to the book, the reason is that the field for a volume charge is continuous and defined everywhere as opposed to that of a discrete distribution. Perhaps this is the reason but I am not convinced so if you have a better fundamental explanation to give for this, please do post your views here.
Cheers
Vivek
My textbook says that a Gaussean surface must be carefully chosen so that a point charge (or point charges constituting a discrete charge distribution) does not lie ON it, as otherwise the electric field at the location of the charge would be infinite and hence, it would not be possible to compute the flux through such a surface. In contrast, it says, that a Gaussean surface can well pass through a continuous charge distribution (a volume charge distribution) as the field everywhere is defined for such a distribution.
The simplest example I can think of which justifies this proposition is that of a uniform solid nonconducting sphere with uniform volume charge: the field inside grows linearly with radial distance and falls of as the inverse square of radial distance outside. So, clearly, any spherical Gaussean surface drawn in the interior of the solid sphere passes through a continuous charge distribution yet the field and flux are well defined there.
But now suppose we build our continuous charge distribution by somehow piling discrete charges on the solid material so that when this process has been carried out, the charge distribution has become continuous. At any intermediate stage, we could identify the distribution as discrete and so point charges may well reside on our Gaussean surface but at the end of the process, there is no problem because now, the field is well defined everywhere!
Yet, we do regard a continuous charge distribution as made up of infinitely many point charges somehow stacked together though they would--when distinguishable--fly off due to mutual repulsion. Then, what is the fundamental explanation for this apparent inconsistency: that the Gaussean surface does not pose problems though infinitely many point charges lie on it..but it does pose a problem when countably finite charges lie on it?
According to the book, the reason is that the field for a volume charge is continuous and defined everywhere as opposed to that of a discrete distribution. Perhaps this is the reason but I am not convinced so if you have a better fundamental explanation to give for this, please do post your views here.
Cheers
Vivek
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