Continuous Charge Distributions

Since macroscopic (i.e. normal-sized) objects tend to be made up of septillions of atoms, some of which are made up of dozens of protons and electrons, the number of charges to consider in everyday situations is daunting. In order to study the behavior of electric charge, will we have to keep track of the position of every charge inside the material?

The answer is no, for the same reason that we can talk about the flow of water without imagining the motion of every water molecule. We suppose that the charge is made up, not of a large number of little pieces, but as a smooth fluid (called a continuous charge distribution). This is an approximation, which works best if we don't get too close to the fluid where the positions of individual charges are most obvious.

We can define the charge density of a charge distribution by taking a small piece of the distribution, calculating the total charge inside this piece, and dividing it by its size. We will actually define three different types of charge density depending on whether the charge is spread out in one, two, or three dimensions. If the charge is spread out to fill a space (like the interior of a sphere), then the volume charge density ρ is given by the formula

$$\rho=\frac{Q}{V}=\frac{\textrm{total charge}}{\textrm{total volume}}$$
If charge is spread out over a surface (like butter on toast), then the surface charge density σ is the ratio of charge to surface area:
$$\sigma=\frac{Q}{A}=\frac{\textrm{total charge}}{\textrm{total area}}$$
and if the charge is spread out in a line (straight or curved), then we define the linear charge density λ as the ratio of charge to length:
$$\lambda=\frac{Q}{L}=\frac{\textrm{total charge}}{\textrm{total length}}$$
Why do we use three different symbols for the same idea? Mostly it's because they all have different dimensions, or sets of units:
$$[\rho]=\textrm{C/m}^3 \qquad [\sigma]=\textrm{C/m}^2 \qquad [\lambda]=\textrm{C/m}$$
and physicists prefer it when we can look at a symbol and know just what dimensions it has: it makes error checking easier. For example, is someone wrote the equation QL, where L was a length and Q a charge, we would know right away that this equation is incorrect, because the units don't match: the units on the left are coulombs, and the units on the right are C/m2.

At this point it would be useful to define a few basic terms:

The terms "shell" and "solid" can be used in conjunction with other three-dimensional shapes; for example, a paper towel roll would be a "cylindrical shell".) Notice that in each case the exponent on R matches the dimensionality of the shape: R (or R1) for the one-dimensional ring, R2 for the surfaces, and R3 for the volume. This is necessary for the dimensions to work out correctly, but can also help you remember which equation is which.

Find the appropriate charge density if 6 mC of charge is spread evenly over each of the following:
  1. A 3-meter long straight line segment
    $$\lambda=\frac{Q}{L}=\frac{6\u{mC}}{3\u{m}}=\boxed{2 \u{mC/m}}$$
  2. A ring with a 3-meter radius
    $$\lambda=\frac{Q}{2\pi R}=\frac{6\u{mC}}{2\pi(3\u{m})}=\boxed{0.32\u{mC/m}}$$
  3. A disk with a 3-meter radius
    $$\sigma=\frac{Q}{\pi R^2}=\frac{6\u{mC}}{\pi(3\u{m})^2} =\boxed{0.21\u{mC/m^2}}$$
  4. A spherical shell with radius 3 meters
    $$\sigma=\frac{Q}{4\pi R^2}=\frac{6\u{mC}}{4\pi(3\u{m})^2} =\boxed{0.053\u{mC/m^2}}$$
  5. A solid sphere with radius 3 meters
    $$\rho=\frac{Q}{{4\over3}\pi R^3} =\frac{6\u{mC}}{{4\over3}\pi(3\u{m})^3} =\boxed{0.053\u{mC/m^3}}$$

Charge density can vary as we move from place to place in an object (or on a surface or along a line), and so generally it can be thought of as a function of position, like \(\rho(x,y,z)\). If the charge density does not vary with position but is a constant throughout an object, then we say it has a uniform charge density.

A rectangle has charge spread evenly on its surface, with a surface charge density σ1. The rectangle is then split into two pieces, with surface charge density σ2 and σ3; the first piece is bigger than the second.
Which of the following is largest?
A) σ1 B) σ2 C) σ3 D) All three are equal.

D. If charge is spread evenly over a surface, then every piece of that surface will have the same charge density. The original surface has the largest area, but it also has the largest total charge, and the ratio of charge to area is the same as it is for the two pieces.