Equation (22-7) gives the electric field caused by a single point charge. But in most realistic situations that involve electric fields and forces, we encounter charge that is distributed over space. The charged plastic and glass rods in Fig. 22-1 have electric charge distributed over their surfaces, as do the drum and toner brush of an electrostatic photocopy machine. In this section we’ll learn to calculate electric fields caused by various distributions of electric charge. Calculations of this kind are of tremendous importance for technological applications of electric forces. To determine the trajectories of electrons in a TV tube, of atomic nuclei in an accelerator for cancer radiotherapy, or of charged particles in a semiconductor electronic device, you have to know the detailed nature of the electric field acting on the charges.
To find the field caused by a charge distribution, we imagine the distribution to be made up of many point charges ql’ q2′ q3′ …. (This is actually quite a realistic description, since we have seen that charge is carried by electrons and protons that are so small as to be almost point like.) At any given point P, each point charge produces its own electric field E” E2′ E3′ … , so a test charge qo placed at P experiences a force F, – qclf, from charge q” a force F2 – qclf2 from charge q2′ and so on. From the principle of superposition of forces discussed in Section 22-5, the total force Fo that the charge distribution exerts on qo is the vector sum of these individual forces:
The total electric field at P is the vector sum of the fields at P due to each point charge in the charge distribution. This is the principle of superposition of electric fields.
When charge is distributed along a line, over a surface, or through a volume, a few additional terms are useful. For a line charge distribution (such as a long, thin, charged plastic rod) we use A. (“lambda”) to represent the linear charge density (charge per unit length, measured in C/m). When charge is distributed over a surface (such as the surface of the imaging drum of a photocopy machine), we use a (“sigma”) to represent the surface charge density (charge per unit area, measured in C/m\ And when charge is distributed through a volume, we use p (“rho”) to represent the volume charge density (charge per unit volume),
Some of the calculations in the following examples may look fairly intricate; in electric-field calculations a certain amount of mathematical complexity is in the nature of things. After you’ve worked through the examples one step at a time, the process will seem less formidable. Many of the calculation techniques in these examples will be used again in Chapter 29 to calculate the magnetic fields caused by charges in motion.