Charge Distributions

In principle, the smallest unit of electric charge that can be isolated is the charge of a single electron, which is

C. This is very small, and we rarely deal with electrons one at a time, so it is usually more convenient to describe charge as a quantity that is continuous over some region of space. In particular, it is convenient to describe charge as being distributed in one of three ways: along a curve, over a surface, or within a volume.

Imagine that charge is distributed along a curve

through space. Let

be the total charge along a short segment of the curve, and let

be the length of this segment. The line charge density

at any point along the curve is defined as

which has units of C/m. We may then define

to be a function of position along the curve, parameterized by

; e.g.,

. Then, the total charge

along the curve is

which has units of C. In other words, line charge density integrated over length yields total charge.

Imagine that charge is distributed over a surface. Let

be the total charge on a small patch on this surface, and let

be the area of this patch. The surface charge density

at any point on the surface is defined as

which has units of C/m

. Let us define

to be a function of position on this surface. Then the total charge over a surface

is

In other words, surface charge density integrated over a surface yields total charge.

Imagine that charge is distributed over a volume. Let

be the total charge in a small cell within this volume, and let

be the volume of this cell. The volume charge density

at any point in the volume is defined as

which has units of C/m

. Since

is a function of position within this volume, the total charge within a volume

is

In other words, volume charge density integrated over a volume yields total charge.