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\frac{V}{R_{1}} + \frac{V}{R_{2}} = \left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\cdot V \] + solving for V as a function of I we get + \[V = \dfrac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}\cdot I = \frac{R_{1}R_{2}}{R_{1}+R_{2}}\cdot I = R\cdot I \] + where \[\mans{R = \dfrac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}}\] is the resistance of $R_{1}$ and $R_{2}$ in parallel. \ex{1.4} \todo{Solve this problem} From 5178c1cfbca94efbc4a6a8e147d0e800778ae435 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Arturo=20Morales=20Bugue=C3=B1o?= Date: Mon, 10 Aug 2020 03:50:59 -0400 Subject: [PATCH 2/2] Added 1.3 --- Chapter1.pdf | Bin 244016 -> 244035 bytes Chapter1.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/Chapter1.pdf b/Chapter1.pdf index 5af23aedbe97f230a166a2ab0858470830bc779b..d01da2d7bb9c84ad8415d883da78f496f1c198c5 100755 GIT binary patch delta 6134 zcmai02{e@L`?f`?4_UG=B`QnJ`@XYAlNQ8>kZf5hjqS4zS*tgpB3qWEQG~KL80(CZ 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    mHN=j9AH5CM)45+AqO3JE$8dgb3MFoigQ1Sw@|8Iz;fV@1Dm-kgq WZ+}ladnU^DXaLD1EUc!h&h&p|P#?hn diff --git a/Chapter1.tex b/Chapter1.tex index a5e0a7e..fa1634a 100644 --- a/Chapter1.tex +++ b/Chapter1.tex @@ -25,7 +25,7 @@ to[R=$R_2$,v=$V_{2}$] (2,0) to (0,0) \end{circuit} - By LVK and Ohm's law \[ V = V_{1} + V_{2} = R_{1}\cdot I + R_{2} \cdot I = (R_{1}+R_{2}) \cdot I = R \cdot I \] + By KVL and Ohm's law \[ V = V_{1} + V_{2} = R_{1}\cdot I + R_{2} \cdot I = (R_{1}+R_{2}) \cdot I = R \cdot I \] where \[\mans{R = R_{1} + R_{2}}\] is the resistance of $R_{1}$ and $R_{2}$ in series. Now, consider a simple parallel resistor circuit. \begin{circuit}{fig:1.3.2}{A basic parallel circuit.} @@ -36,7 +36,7 @@ to[R=$R_2$,i>^=$I_{2}$] (4,0) to (0,0) \end{circuit} - By LCK and Ohm's law \[ I = I_{1} + I_{2} = \frac{V}{R_{1}} + \frac{V}{R_{2}} = \left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\cdot V \] + By KCL and Ohm's law \[ I = I_{1} + I_{2} = \frac{V}{R_{1}} + \frac{V}{R_{2}} = \left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\cdot V \] solving for V as a function of I we get \[V = \dfrac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}\cdot I = \frac{R_{1}R_{2}}{R_{1}+R_{2}}\cdot I = R\cdot I \] where \[\mans{R = \dfrac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}}\] is the resistance of $R_{1}$ and $R_{2}$ in parallel.