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--- 184_notes:examples:week4_two_segments [2018/02/03 21:10] – [Example: Two Segments of Charge] tallpaul
+++ 184_notes:examples:week4_two_segments [2021/05/25 14:27] – schram45
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+[[184_notes:dq|Return to $dQ$]]
 ===== Example: Two Segments of Charge =====
 Suppose we have two segments of uniformly distributed charge, one with total charge $+Q$, the other with $-Q$. The two segments each have length $L$, and lie crossed at their endpoints in the $xy$-plane. The segment with charge $+Q$ lies along the $y$-axis, and the segment with charge $-Q$ lies along the $x$-axis. See below for a diagram of the situation. Create an expression for the electric field $\vec{E}_P$ at a point $P$ that is located at $\vec{r}_P=r_x\hat{x}+r_y\hat{y}$. You don't have to evaluate integrals in the expression.
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 {{ 184_notes:4_two_segments.png?350 |Axes with Two Segments}}
-===Approximations & Assumptions===
+====Solution====
+<WRAP TIP>
+=== Approximation ===
+We begin with an approximation, which will make our calculations simpler, and makes sense based on our representation:
   * The thicknesses of both segments are infinitesimally small, and we can approximate them as line segments.
+</WRAP>
-====Solution====
+This example is complicated enough that it's worthwhile to make a plan.
-Because we know that electric fields add through superposition, we can treat of the charges separately, find the electric field, then add the fields together at $P$ at the end. We can begin with the electric field due to the segment along the $y$-axis. We start by finding $\text{d}Q$ and $\vec{r}$. The charge is uniformly distributed so we have a simple line charge density of $\lambda=Q/L$. The segment extends in the $y$-direction, so we have $\text{d}l=\text{d}y$. This gives us $\text{d}Q$: $$\text{d}Q=\lambda\text{d}l=\frac{Q\text{d}y}{L}$$
+<WRAP TIP>
+=== Plan ===
+We will use integration to find the electric field from each segment, and then add the electric fields together using superposition. We'll go through the following steps.
+  * For the first segment, find the linear charge density, $\lambda$.
+  * Use $\lambda$ to write an expression for $\text{d}Q$.
+  * Assign a variable location to the $\text{d}Q$ piece, and then use that location to find the separation vector, $\vec{r}$.
+  * Write an expression for $\text{d}\vec{E}$.
+  * Figure out the bounds of the integral, and integrate to find electric field at $P$.
+  * Repeat the above steps for the other segment of charge.
+  * Add the two fields together to find the total electric field at $P$.
+</WRAP>
+Because we know that electric fields add through superposition, we can treat each of the charges separately, find the electric field, then add the fields together at $P$ at the end. We can begin with the electric field due to the segment along the $y$-axis. We start by finding $\text{d}Q$ and $\vec{r}$. The charge is uniformly distributed so we have a simple line charge density of $\lambda=Q/L$. The segment extends in the $y$-direction, so we have $\text{d}l=\text{d}y$. This gives us $\text{d}Q$: $$\text{d}Q=\lambda\text{d}l=\frac{Q\text{d}y}{L}$$
+<WRAP TIP>
+===Assumption===
+The charge is evenly distributed along each segment of charge. This allows each little piece of charge to have the same value for charge density along each line.
+</WRAP>
 {{ 184_notes:4_two_segments_pos_dq.png?450 |dQ for Segment on y-axis}}