Merge pull request milesdai#4 from arturombug/master

Added 1.3

Chapter1.tex

@@ -19,15 +19,27 @@
     \ex{1.3}
     Consider a simple series resistor circuit.
     \begin{circuit}{fig:1.3.1}{A basic series circuit.}
-        (0,0) to[V=$V_\in$] (4,0)
-            to (4,2)
-            to[R=$R_2$] (2,2)
-            to[R=$R_1$] (0,2)
+        (0,0) to[V=$V$,invert] (0,4)
+            to[short,i=$I$] (2,4)
+            to[R=$R_1$,v=$V_{1}$] (2,2)
+            to[R=$R_2$,v=$V_{2}$] (2,0)
             to (0,0)
     \end{circuit}
-    By 
-    \todo{Solve this problem}
-
+	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.}
+		(0,0) to[V=$V$,invert] (0,3)
+		to[short,i=$I$] (2,3)
+		to[R=$R_1$,i>^=$I_{1}$] (2,0);
+		\draw (2,3) to[short] (4,3)
+		to[R=$R_2$,i>^=$I_{2}$] (4,0)
+		to (0,0)
+	\end{circuit}
+	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.
     \ex{1.4}
     \todo{Solve this problem}