Now we want to calculate the electrostatic potential due to a continuous charge distribution, and from that the electric field. As an example, consider a rod in which total charge Q is uniformly distributed over its length L. We will first consider the case of axial points - those points which lie along the same line as the charge.
To calculate the potential at point P at distance x from the origin, we first find dV, the potential due to a small charge element within a segment ds, and all such contributions over the entire length L.
| (22) |
where K = = 9
x 109 N m2/C2 and l
= linear charge density = Q/L.
The electric field is
| (23) |
The potential and the field are calculated for L = 1 m and Q = 0.1 C in this spreadsheet and the results are shown graphically below.
Notice that both the potential and the field approach the results for a point charge (yellow points) as x becomes larger than the length of the charge distribution (1 meter in this case). The point charge used in this example has the same charge as our line charge, but is concentrated at the point x=0.
Now that you understand on-axis points, how about off-axis points?