Example: A Tilted Segment of Charge
Suppose we have a segment of uniformly distributed charge stretching from the point to , which has total charge . We also have a point . Define a convenient for the segment, and between a point on the segment to the point . Also, give appropriate limits on an integration over (you don't have to write any integrals, just give appropriate start and end points). First, do this for the given coordinate axes. Second, define a new set of coordinate axes to represent and in a simpler way and redo.
Facts
- The segment stretches from to .
- The segment has a charge , which is uniformly distributed.
- .
Lacking
- and
- A new set of coordinate axes
Approximations & Assumptions
- The thickness of the segment is infinitesimally small, and we can approximate it as a line segment.
- The total charge is a constant - not discharging.
Representations
- For the first part, we can draw a set of coordinate axes using what we already know. The first part of the example involves the following orientation:
- We can represent and for our line as follows:
- For the second part when we define a new set of coordinate axes, it makes sense to line up the segment along an axis. We choose the -axis. We could have chosen the -axis, and arrived at a very similar answer. Whichever you like is fine!
Solution
In the first set of axes, the segment extends in the and directions. A simple calculation of the Pythagorean theorem tells us the total length of the segment is , so we can define the line charge density . When we define , we want it align with the segment, so we can have . Since along the segment, we can simplify a little bit. . Note, that we chose to express in terms of , instead of . This is completely arbitrary, and the solution would be just as valid the other way. Now, we can write an expression for :
The units here might look a little weird, since distance was defined without dimensions in the example statement. Next, we need . We will put it in terms of , not , just as we did for . (Note that while the initial variable picking was arbitrary we now have to match) We know , and . Again, , so we can rewrite . We now have enough to write :
In the second set of axes, the segment extends only in the direction. This problem is now very similar to the examples in the notes. The length of the segment is still , so we can define the line charge density . When we define , we want it align with the segment, which is much simpler this time: . Now, we can write an expression for :