From 5d13ae430c6ea6e14ab31127a6e9aa65abce2c25 Mon Sep 17 00:00:00 2001 From: Miles Dai Date: Sun, 20 May 2018 16:46:00 -0400 Subject: [PATCH] Clarified solution to 1.15 --- Chapter1.pdf | Bin 194788 -> 194836 bytes Chapter1.tex | 6 +++--- 2 files changed, 3 insertions(+), 3 deletions(-) diff --git a/Chapter1.pdf b/Chapter1.pdf index 2c2114ffe20c1b92b12a3057c6d4d104fc27db3e..3fbaa2ba1a7a1e22ccae58dc63189823904d289a 100644 GIT binary patch delta 17690 zcmZsDV{~9$(`9Ts>DcPnw$-t1+sTcMj_r(;r4xA4h#@O1@Z5FAY0#AylNL8+3a#9)Bg+LK8zJ5%l7Z>Yp+^ep{Q(U4sx zwn)-QnfB{g3&@JH{4?)ws`XNlXX?GSf9m>TUA+fA{)l*kt2UbJMO!upW45Q-bO8-7-F-GX2w(GAAj1gI~ zC&~Q=!yPHi5w7Jw!md*{CVoskcQR)t_x+8){lNOeg=;^Cig7YwE&*fuQIAghJ*3G@ z=_7tmW3|nmdq)YADw~RS@W;&=fxt*@OLf)3qK3z z2w}py2e3BkY(to?MhyvD*Jrnae$M}yLgQlOTU1AwGFf>%mPkx;&jDYE_u<%BXWsw= zU|_yyO1cbvdHM8g96Yvs`KKZ~^6GB=p7k~UnFZw+xD-%sV}4!Q(&dPUVT?})0bdl7 z-|t1pPO3%rw30GpxQ1i zQ6Ywr`tZ%BN#RMOUJ@iD*2*$T^?@(4AQmi57{bA9)k*ZaK8j{o46H`F9Sy2to*Nx2Dos 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zv$FmJ<6`|E4$%L`4`60yXZ|bxe_?F@2lmg1SpM7``J03V$npXLHa%fUYwE7xDy z0y1*~|1Q%Xj1>U-tL}eXX8i}rpFINjmoq?CR!;W6R|G2?fa|aH1OMa?_%{a|+n;+{ z|Lzaa|G!WDDexb+|GQ~`?3_U6zcCIr_P=WkiZ#^c! zpKS;5&-MPNssBzP*Z((Q_OzH@_JheT!N|qP(Z$@{#mUst+|beq7>M$@Bo>ua6s4wd0gDq0Qv)tlRabvEE&y6Eb%6i? diff --git a/Chapter1.tex b/Chapter1.tex index 92a6c03..6b60519 100644 --- a/Chapter1.tex +++ b/Chapter1.tex @@ -205,12 +205,12 @@ \[C_\text{total} = \frac{Q_\text{total}}{V_\text{total}}\] Notice that $V_\text{total}$ is the sum of the voltages across $C_1$ and $C_2$. We can get each of these voltages using the definition of capacitance. \[V_\text{total} = V_1 + V_2 = \frac{Q_1}{C_1} + \frac{Q_2}{C_2}\] - The key observation now is that because the right plate of $C_1$ is connected to the left plate of $C_2$, the charge stored on both plates must be equal. \footnotemark Therefore, we have $Q_1 = Q_2$. Let us call this charge stored $Q$ (i.e. $Q = Q_1 = Q_2$). Now, we know that the total charge stored is also $Q$. \footnotemark Therefore, we know that $Q_\text{total} = Q$. Now, we have + The key observation now is that because the right plate of $C_1$ is connected to the left plate of $C_2$, the charge stored on both plates must be of equal magnitude.\footnotemark Therefore, we have $Q_1 = Q_2$. Let us call this charge stored $Q$ (i.e. $Q = Q_1 = Q_2$). Now, we know that the total charge stored is also $Q$.\footnotemark Therefore, we know that $Q_\text{total} = Q$. Now, we have \[C_\text{total} = \frac{Q_\text{total}}{V_\text{total}} = \frac{Q}{Q_1/C_1 + Q_2/C_2} = \frac{Q}{Q/C_1 + Q/C_2} = \frac{1}{1/C_1 + 1/C_2}\] - \footnotetext{If this were not true, then there would a difference of charge between these two plates which means there would be a voltage across these two connected plates. This violates the idea that two points that are electrically connected must have the same voltage.} + \footnotetext{If this were not true, then there would be a net charge on these two plates and the wire between them. Because we assume that the capacitors started out with no net charge and there is no way for charge to leave the middle wire or the two plates it connects, this is impossible. } - \footnotetext{If you are having trouble seeing this, suppose we apply a positive voltage to the left plate of $C_1$ relative to the right plate of $C_2$. Suppose this causes the left plate of $C_1$ to charge to some charge $q$. Thus we must have the right plate of $C_1$ has a charge of $-q$ by pushing $q$ units of charge onto the left plate of $C_2$. Now the left of $C_2$ has $q$ units of charge which causes a corresponding $-q$ charge on the right side of $C_2$. Thus the overall total charge separated across these two capacitors is $q$. + \footnotetext{If you are having trouble seeing this, suppose we apply a positive voltage to the left plate of $C_1$ relative to the right plate of $C_2$. Suppose this causes the left plate of $C_1$ to charge to some charge $q$. We now must have a charge of $-q$ on the right plate of $C_1$ because $q$ units of charge are now pushed onto the left plate of $C_2$. Now the left of $C_2$ has $q$ units of charge which causes a corresponding $-q$ charge on the right side of $C_2$. Thus the overall total charge separated across these two capacitors is $q$. } \ex{1.16}