1. The data of \( I \) and \( t \) measurements during the charging of the capacitor will be given 3. Find the experimental time constant \( \tau \) of the by the Table (1). circuit from the \( I \) vs \( t \) graphs. Find it from both the charging and discharging graphs. Table-1: The data values during the charging of the Then, compare experimental time constant capacitor. \( \tau \) with its theoretical value obtained by \begin{tabular}{|c|c||c|c|} \hline \multirow{2}{*}{\( t(\mathrm{sec}) \)} & Measured & \multirow{2}{*}{\( t(\mathrm{sec}) \)} & Measured \\ \cline { 4 - 4 } & \( I(\mu A) \) & & \( I(\mu A) \) \\ \hline 0 & \( \cdots \cdots \) & 50 & \\ \hline 5 & & 65 & \\ \hline 10 & & 80 & \\ \hline 15 & & 100 & \\ \hline 20 & & 120 & \\ \hline 25 & & 140 & \\ \hline 30 & & 160 & \\ \hline 35 & & 180 & \\ \hline \end{tabular} \( \tau=R C \). Fill the experimental and theoretical time constant in the Table-(3). Table-3: Time constant \( \tau \) of the circuit from the I vs \( t \) graphs. \begin{tabular}{|l||l|} \hline Comparison & Time constant \( \tau(\mathrm{sec}) \) \\ \hline Charging Curve & \( \cdots \cdots \) \\ \hline Discharging Curve & \( \cdots \cdots \) \\ \hline Theoretical Value & \( \cdots \cdots \) \\ \hline \end{tabular} 2. The data values obtained during the 4. Find and report the charge \( q \) on the discharging of the capacitor will be filled in capacitor at the time you finished charging. the data Table-(2). Compare this value with the \( q=V C \) by using the data Table-(4).
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Charging and discharging a capacitor. A 1.50$\mu \mathrm{F}$ capacitor is charged through a 125$\Omega$ resistor and then discharged through the same resistor by short-circuiting the battery. While the capacitor is being charged, find (a) the time for the charge on its plates to reach $1-1 / e$ of its maximum value and (b) the current in the circuit at that time. (c) During the discharge of the capacitor, find the time for the charge on its plates to decrease to 1/e of its initial value. Also, find the time for the current in the circuit to decrease to 1$/ e$ of its initial value.
The following data describe the charge $Q$ remaining on the capacitor (measured in microcoulombs, $\mu C )$ at time $t$ t measured in seconds). $$\begin{array} { | c | c | c | c | c | c | c | } \hline t & { 0.00 } & { 0.02 } & { 0.04 } & { 0.06 } & { 0.08 } & { 0.10 } \\ \hline Q & { 100.00 } & { 81.87 } & { 67.03 } & { 54.88 } & { 44.93 } & { 36.76 } \\ \hline \end{array}$$ (a) Use a graphing calculator or computer to find an exponential model for the charge. (b) The derivative $Q ( t )$ represents the electric current (measured in microamperes, $\mu$ A) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when $t = 0.04$ s. Compare with to estimate the current when Section $2.1 .$
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A $2.0-\mu \mathrm{F}$ capacitor is charged through a $30-\mathrm{M} \Omega$ resistor by a 45-V battery. Find $(a)$ the charge on the capacitor and $(b)$ the current through the resistor, both determined $83 \mathrm{~s}$ after the charging process starts. The time constant of the circuit is $R C=60 \mathrm{~s}$. Also, $$ q_{\infty}=V_{\infty} C=(45 \mathrm{~V})\left(2.0 \times 10^{-6} \mathrm{~F}\right)=9.0 \times 10^{-5} \mathrm{C} $$ (a) $q=q_{\infty}\left(1-e^{-t / R C}\right)=\left(9.0 \times 10^{-5} \mathrm{C}\right)\left(1-e^{-83 / 60}\right)$ But $e^{-83 / 60}=e^{-1.383}=0.25$ Then substitution gives $$ q=\left(9.0 \times 10^{-5} \mathrm{C}\right)(1-0.25)=67 \mu \mathrm{C} $$ (b) $i=i_{0} e^{-t / R C}=\left(\frac{45 \mathrm{~V}}{30 \times 10^{6} \Omega}\right)\left(e^{-1.383}\right)=0.38 \mu \mathrm{A}$
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