Chapter 4: Electric field and work

Interactive 1

The work done by external forces W in moving a test charge Q from point  to  is given by

, J

where E is the electric field vector.

If a positive charge is moved against the field lines, the external forces are doing positive work, thereby increasing the potential energy of the charged particle, and, thus, of the whole electrostatic system. The electrostatic system receives energy from the external system.

On the contrary, if a positive charge is moved along the field lines, it is in fact the electric field, which expends energy in doing the work—the external forces are doing negative work, i.e., the external system receives energy from the electric system. The potential energy of the charged particle decreases, thereby bringing down the energy of the whole electrostatic system. The work done by the external forces is always opposite in sign to the work done by the electric field itself.

Here, we illustrate the simplest way to compute W. If the line integral from  to  is taken along the field streamline, i.e., E is always collinear with dL, the dot product of the two vectors conveniently reduces to the product of their magnitudes: . In fact, this calculation reflects physical reality best: when a test charge is introduced in an electric field, it moves along the respective streamline, just like a cork in a river being carried down by the stream.

We can now represent any path of integration (the path of the test charge) in a piecewise-linear manner, i.e., we approximate it with small linear segments. In Figure 1, the streamline from  to  is approximated by fourteen straight-line segments. The work done in moving the charge along such a straight-line segment is simple to compute:

.

Here,  is the length of the segment, and  is the field intensity at point . The work done in moving the charge from point M to point N is simply the sum of all  over all segments making up the path between the two points: .

Figure 2 shows the field magnitude all the way along the arc streamline from point 1 to point 15. The arc length between two neighboring points is  mm ().

We have now everything necessary to compute the work W: (1) Choose the initial and end points for the journey of your test charge along the arc depicted in Figure 1. The electric-field work per unit charge, , is computed and displayed numerically as well as geometrically in Figure 2. (2) Choose the value of your charge—it may be positive or negative. The numerical value of W for this charge is shown at the bottom.

 
Interactive Applet

Electric field and work