Unit 1 - Electric Field

The electric field at a point can be determined directly if there is a charge at that point, by the statement (definition of E) that the electrical force on the charge is given by the product of the charge and the field. That is,

F_{elec, on Q} = Q E_{at Q}
We distinguish an electrical force on Q from other forces on it (such as gravitational or magnetic) by specifying that the electrical force is that part of the force which is proportional to the electrical charge (instead of the mass), which is independent of Q's velocity, and which therefore is present when Q's velocity equals zero.
We should note that the electric field determined by such a test is actually the electric field due to all the other charges and effects; notice that the contribution of any charge to an electric field at its own location would involve the force contribution by the charge on itself, which would be undefined due to the zero distance denominator.

If there is no charge at the point, then we can consider placing a hypothetical charge there. We can then determine what the electrical force on that hypothetical charge would be, and apply the previous relationship. This also provides, in principle, an experimental method of determining electric field: bring a "test charge" to the location in question and measure the force. (In practice, this won't work effectively; charge is difficult to control and force is difficult to measure.) The test charge should be "small", lest its presence distort what we are trying to determine (but if it is hypothetical this is not so important), and for mathematical convenience I recommend you consider only positive hypothetical test charges.

Since Coulomb's Law gives the force exerted between two electrical charges at rest, we can use it directly to determine the electric field due to one charge.

Since F_{on Q at r} = 1/(4pe_o) Q Q_by (r - r_by)/|r - r_by|³
we read off E(r) = 1/(4pe_o) Q_by (r - r_by)/|r - r_by|³
where I have kept the 'by' subscripts. More often, this would use a different labeling:
E(r)_{due to Q} = 1/(4pe_o) Q (r - r_Q)/|r - r_Q|³

Of course, since electrical forces on a single charge obey superposition, so does the electric field. Hence if the charge causing the field is not a single charge, or is not pointlike, then one has respectively a sum or an integral (a fancy sum) over Q.

Two things I see beginners do wrong when a sum or especially an integral is involved. One, they evaluate the denominator as position from a center, and thus a constant. The correct denominator is distance from the charge, and the charge is not all at the same place. (If it were, you wouldn't need the sum or integral in the first place.) The only way the denominator can be the same for two such contributions is if your field point is exactly in the middle: you should be so lucky. (And since the denominator depends on a variable, you will have a nontrivial integral.) Two, they put a total charge value and in addition write an integration. But getting the total charge is part of the process of adding up the contributions, so putting both total charge and an integration is redundant.

There is also often confusion over what kind of integration it will be, and that depends on what the charge distribution is. If it's along a line or curve, then you get a 'line integral', a single integration (but more than one variable will be involved unless the line is parallel to an axis). If it's spread over a surface, you get an 'area integral', a double integration. (If the surface is curved or tilted, all three variables may be involved.) If the charge is spread through a three-dimensional region, you get a 'volume integral', a triple integration. In each case, you need to express charge in terms of density times geometric 'little bit of region'. If the distribution is uniform, sometimes you will be given the density and sometimes the total charge; in the second case density (which is what you need) is (total charge)/(size of region). If the distribution is not uniform, then you need density as a function (which you can not treat as a constant).

For curved lines or surfaces, you will have extra coordinates; this does not mean extra integrations, you just need to relate the extra variables to whichever you are taking as independent variables, using the equation(s) of the line or surface.

E-M Home Page/Unit 1/

Format for this page adapted with permission from pages
constructed by Dr. Lionel D. Hewett for his course
Modern Physics 1.