Return to Answer

The microscopic electric field is what you see if you really zoom in. For example, consider your every day table. If you zoom in really closely, you will actually see a wildly varying electric field. For example, at a point really close to an electron the field might be massive, but at another point close by the field might point in another direction or be much smaller.

Now one can see that dealing with such electric fields would get really messy!

The concept of a macroscopic electric field is to average the wildly varying microscopic electric fields into a macroscopic electric field that is smoothly varying. You go from microscopic to macroscopic via an averaging procedure.

To my understanding, the averaging procedure is not well defined. Griffiths explains it as such:

[The macroscopic field] is defined as the average field over regions large enough to contain many thousands of atoms (so that the uninteresting microscopic fluctuations are smoothed over), and yet small enough to ensure that we do not wash out any significant large-scale variations in the field. (In practice, this means we must average over regions much smaller than the dimensions of the object itself.)

The microscopic electric field is the electric field at a very small scale. This electric field is 1) wildly varying in space, and 2) wildly varying in time. Dealing with such a field is really messy!

We introduce the concept of a macroscopic electric field to average out the insignificant wild variations of the microscopic electric field without losing large scale variations in the electric field. It is in effect averaging out the microscopic noise.

Griffiths explains it as such:

[The macroscopic field] is defined as the average field over regions large enough to contain many thousands of atoms (so that the uninteresting microscopic fluctuations are smoothed over), and yet small enough to ensure that we do not wash out any significant large-scale variations in the field. (In practice, this means we must average over regions much smaller than the dimensions of the object itself.)

The microscopic electric field is what you see if you really zoom in. For example, consider your every day table. If you zoom in really closely, you will actually see a DISCONTINUOUS charge density (just consider the chargeless space between atoms). Now one can see that dealing with such charge densities would get really messy! Griffiths in his Introduction to Electrodynamics actually states that it is also usually unnecessary.

The concept of a macroscopic electric field is to average the discontinuous microscopic electric fields into a continuous one, which is much easier to perform accurate enough calculations with. This is done by an averaging procedure: take the average of the microscopic electric field over a small sphere, replace the microscopic electric fields with the macroscopic, and do this over all space.

To my understanding, the averaging procedure is not well defined. Griffiths explains it as such:

[The macroscopic field] is defined as the average field over regions large enough to contain many thousands of atoms (so that the uninteresting microscopic fluctuations are smoothed over), and yet small enough to ensure that we do not wash out any significant large-scale variations in the field. (In practice, this means we must average over regions much smaller than the dimensions of the object itself.)

The microscopic electric field is what you see if you really zoom in. For example, consider your every day table. If you zoom in really closely, you will actually see a wildly varying electric field. For example, at a point really close to an electron the field might be massive, but at another point close by the field might point in another direction or be much smaller. 

Now one can see that dealing with such electric fields would get really messy!

The concept of a macroscopic electric field is to average the wildly varying microscopic electric fields into a macroscopic electric field that is smoothly varying. You go from microscopic to macroscopic via an averaging procedure.

To my understanding, the averaging procedure is not well defined. Griffiths explains it as such:

[The macroscopic field] is defined as the average field over regions large enough to contain many thousands of atoms (so that the uninteresting microscopic fluctuations are smoothed over), and yet small enough to ensure that we do not wash out any significant large-scale variations in the field. (In practice, this means we must average over regions much smaller than the dimensions of the object itself.)

The microscopic electric field is what you see if you really zoom in. For example, consider your every day table. If you zoom in really closely, you will actually see a DISCONTINUOUS charge density (just consider the chargeless space between atoms). Now one can see that dealing with such charge densities would get really messy! Griffiths in his Introduction to Electrodynamics actually states that it is also unnecessary.

The concept of a macroscopic electric field is to average the discontinuous microscopic electric fields into a continuous one, which is much easier to perform accurate enough calculations with. This is done by an averaging procedure: take the average of the microscopic electric field over a small sphere, replace the microscopic electric fields with the macroscopic, and do this over all space.

The microscopic electric field is what you see if you really zoom in. For example, consider your every day table. If you zoom in really closely, you will actually see a DISCONTINUOUS charge density (just consider the chargeless space between atoms). Now one can see that dealing with such charge densities would get really messy! Griffiths in his Introduction to Electrodynamics actually states that it is also usually unnecessary.

The concept of a macroscopic electric field is to average the discontinuous microscopic electric fields into a continuous one, which is much easier to perform accurate enough calculations with. This is done by an averaging procedure: take the average of the microscopic electric field over a small sphere, replace the microscopic electric fields with the macroscopic, and do this over all space.

To my understanding, the averaging procedure is not well defined. Griffiths explains it as such:

[The macroscopic field] is defined as the average field over regions large enough to contain many thousands of atoms (so that the uninteresting microscopic fluctuations are smoothed over), and yet small enough to ensure that we do not wash out any significant large-scale variations in the field. (In practice, this means we must average over regions much smaller than the dimensions of the object itself.)