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--- 184_notes:examples:week6_node_rule [2017/09/27 20:45] – [Solution] tallpaul
+++ 184_notes:examples:week6_node_rule [2018/02/03 22:33] – [Solution] tallpaul
@@ Line 1: / Line 1: @@
 =====Example: Application of Node Rule=====
-Suppose you have the circuit below. Nodes are labeled for simplicity of discussion. you are given a few values: $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$. Determine all other currents in the circuit, using the [[184_notes:current#Current_in_Different_Parts_of_the_Wire|Current Node Rule]]. Draw the direction of the current as well.
+Suppose you have the circuit below. You are given a few values: $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$. Determine all other currents in the circuit, using the [[184_notes:current#Current_in_Different_Parts_of_the_Wire|Current Node Rule]]. Draw the direction of the current as well.
+{{ 184_notes:6_nodeless.png?300 |Circuit}}
-{{ 184_notes:6_nodes.png?300 |Circuit with Nodes}}
 ===Facts===
   * $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$.
   * $I_1$, $I_2$, and $I_3$ are directed as pictured.
+  * The Node Rule is $I_{in}=I_{out}$, for any point along the current.
-===Lacking===
+===Goal===
-  * All other currents (including their directions).
+  * Find all the currents in the circuit and their directions.
-===Approximations & Assumptions===
-  * The current is not changing (circuit is in steady state).
-  * All current in the circuit arises from other currents in the circuit.
-  * No resistance in the battery (approximating the battery as a mechanical battery)
 ===Representations===
-  * We represent the situation with diagram given.
+For simplicity of discussion, we label the nodes in an updated representation:
-  * We represent the Node Rule as $I_{in}=I_{out}$.
+{{ 184_notes:6_nodes.png?300 |Circuit with Nodes}}
 ====Solution====
+Okay, there is a lot going on with all these nodes. Let's make a plan to organize our approach.
+<WRAP TIP>
+=== Plan ===
+Take the nodes one at a time. Here's the plan in steps:
+  * Look at all the known currents attached to a node.
+  * Assign variables to the unknown currents attached to a node.
+  * Set up an equation using the Node Rule. If not sure about whether a current is going in or coming out of the node, guess.
+  * Solve for the unknown currents.
+  * If any of them are negative, then we guessed wrong two steps ago. We can just flip the sign now.
+  * Repeat the above steps for all the nodes.
+</WRAP>
 Let's start with node $A$. Incoming current is $I_1$, and outgoing current is $I_2$. How do we decide if $I_{A\rightarrow B}$ is incoming or outgoing? We need to bring it back to the Node Rule: $I_{in}=I_{out}$. Since $I_1=8 \text{ A}$ and $I_2=3 \text{ A}$, we need $I_{A\rightarrow B}$ to be outgoing to balance. To satisfy the Node Rule, we set
 $$I_{A\rightarrow B} = I_{out}-I_2 = I_{in}-I_2 = I_1-I_2 = 5 \text{ A}$$