Notes about Electric Fields

G19-2. Notes about Electric Fields

Read these notes in conjuction with sections 19-4,5 of the text.

The electric field is a concept invented primarily to help deal with collections of charges. This could be a collection of individual charges in a particular arrangement in space or a distribution of charges on a solid such as a sphere, cylinder, or plane. If one knows the electric field of a collection or distribution of charges, then one can use that field to calculate the electric force that it creates on a point charge q placed in the field. If E is a vector representing the field, then the electric force on the charge is given by:

F_el = qE

The equation above is equation 19-9 in the text. We've added the 'el' subscript to the force to identify the force as an electrical force. Note that the above equation is a vector equation. Note also that q is not shown between absolute value marks. That's because in the above equation, q is substituted with sign as well as value. This tells you that if the charge is positive, the electric force on the charge is in the same direction as the electric field. If the charge is negative, the electric force is in the opposite direction as the field.

There are two basic ways to determine the electric field due to a collection of charges.

Method 1 -- Operational. In this method, one places a tiny, positive test charge q₀ at a particular location in the field. The direction and magnitude of the electric force on the test charge is measured. The electric field at that point is then given by the following equation:

E = F_el/ q₀

This equation is actually the definition of electric field. Note that the charge q₀ used to test the field is positive; this is simply convention. With a positive charge, the direction of the field is the same as the direction of the electric force. The test charge is assumed to be so small that its presence doesn't alter the field of the other charges.

Method 2 -- Calculation for particular cases. For particular distributions of charge, equations can be developed to calculate the field of the distribution. The simplest distribution is just a single, point charge. We can use Coulomb's Law to calculate the magnitude of the field of point charge q.

E = F_el/ q₀ = (k|q₀||q|/r²) / q₀ = k|q|/r²

The above is equation 19-10 in the text. This formula is used for calculating the field of a point charge or of a collection of point charges. When there's more than one point charge, one simply finds the superposition (vector sum) of the fields of all the charges. Note that the direction of the field of each charge is the direction of the electric force that the charge would exert on a positive test charge. The textbook gives a number of examples of the calculation of fields of collections of point charges.

Formulas can be derived for uniform distributions of charge on a plane or the surface of a sphere. For a uniform, distribution of charge on the surface of a sphere, the electric field is given by the same formula as for a point charge. That is, the spherical distribution acts as if all the charge were concentrated at the center of the sphere. For a plane of charge, the electric field is proportional to the surface charge density. You aren't responsible for the derivations of these formulas. They are done in optional section 19.7 if you're interested.

Besides the above, one other important thing to remember in doing problems involving forces and fields is to know what you're calculating. Don't confuse forces with fields, and don't confuse notation either.