• Electromagnetics I
  • Ch 5: Electrostatics
  • Loc 5.3
  • Electromagnetics I
  • Ch 5
  • Loc 5.3

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.

Line Charge Distribution

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

(5.3.1)

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

(5.3.2)

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

Surface Charge Distribution

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

(5.3.3)

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

(5.3.4)

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

Volume Charge Distribution

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

(5.3.5)

which has units of C/m

. Since

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

is

(5.3.6)

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

 

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0

 
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