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184_notes:examples:week4_tilted_segment [2018/02/05 17:29] – tallpaul | 184_notes:examples:week4_tilted_segment [2018/06/12 18:49] (current) – [Solution] tallpaul | ||
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===== A Tilted Segment of Charge ===== | ===== A Tilted Segment of Charge ===== | ||
Suppose we have a segment of uniformly distributed charge stretching from the point $\langle 0,0,0 \rangle$ to $\langle 1 \text{ m}, 1 \text{ m}, 0 \rangle$, which has total charge $Q$. We also have a point $P=\langle 2 \text{ m},0,0 \rangle$. Define a convenient $\text{d}Q$ for the segment, and $\vec{r}$ between a point on the segment to the point $P$. Also, give appropriate limits on an integration over $\text{d}Q$ (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 $\text{d}Q$ and $\vec{r}$ in a simpler way and redo. | Suppose we have a segment of uniformly distributed charge stretching from the point $\langle 0,0,0 \rangle$ to $\langle 1 \text{ m}, 1 \text{ m}, 0 \rangle$, which has total charge $Q$. We also have a point $P=\langle 2 \text{ m},0,0 \rangle$. Define a convenient $\text{d}Q$ for the segment, and $\vec{r}$ between a point on the segment to the point $P$. Also, give appropriate limits on an integration over $\text{d}Q$ (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 $\text{d}Q$ and $\vec{r}$ in a simpler way and redo. | ||
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It will also be helpful to see how the dimensions of $\text{d}Q$ break down. Here is how we choose to label it: | It will also be helpful to see how the dimensions of $\text{d}Q$ break down. Here is how we choose to label it: | ||
- | {{ 184_notes:4_q.png?500 |Tilted Segment dQ Representation}} | + | {{ 184_notes:4_dq_dimensions.png?200 |Tilted Segment dQ Representation}} |
The segment extends in the $x$ and $y$ directions. A simple calculation of the Pythagorean theorem tells us the total length of the segment is $\sqrt{2} \text{ m}$, so we can define the line charge density $\lambda=Q/ | The segment extends in the $x$ and $y$ directions. A simple calculation of the Pythagorean theorem tells us the total length of the segment is $\sqrt{2} \text{ m}$, so we can define the line charge density $\lambda=Q/ |