An Example in One Dimension
Three charges are equally spaced along a line. The distance between neighboring charges is a. From left to right the charges are:
q1 = –Q
q2 = +Q
q3 = +Q
What is the magnitude of the force experienced by q2, the charge in the center?
Let's define positive to the right.
The net force on q2 is the vector sum of the forces from q1 and q3.
F2 = F21 + F23
| F2 |
= |
| –kQ2
|  |
| a2
|
|
– |
| kQ2
| |
| a2
|
|
= |
| –2kQ2
| |
| a2
|
|
In other words, the force has a magnitude of 2kQ2/a2 and points to the left.
Handling the signs correctly is a critical part of any vector addition problem. The negative signs in each of the terms above come from the direction of each of the forces (both to the left) and not from the signs of the charges. I generally drop the signs of the charges and get any signs off the diagram by drawing in the forces.
Rank the charges according to the magnitude of the net force they experience, from largest to smallest.
- F1 = F2 > F3
- F1 > F2 > F3
- F2 > F1 = F3
- F2 > F1 > F3
- None of the above.
You should be able to show that:
| F1 |
= |
| +kQ2
| |
| a2
|
|
+ |
| kQ2
| |
| 4a2
|
|
= |
| +5kQ2
| |
| 4a2
|
|
| F2 |
= |
| –2kQ2
| |
| a2
|
|
| F3 |
= |
| +kQ2
| |
| a2
|
|
– |
| kQ2
| |
| 4a2
|
|
= |
| +3kQ2
| |
| 4a2
|
|
The correct ranking by magnitude is choice 4 above, F2 > F1 > F3