Question

During a summer internship as an electronics technician, you are asked to measure the self inductance $L$ of a solenoid. You connect the solenoid in series with a 10.0-$\Omega$ resistor, a battery that has negligible internal resistance, and a switch. Using an ideal voltmeter, you measure and digitally record the voltage $v_L$ across the solenoid as a function of the time $t$ that has elapsed since the switch is closed. Your measured values are shown in $\textbf{Fig. P30.67}$, where $v_L$ is plotted versus $t$. In addition, you measure that $v_L = 50.0$ V just after the switch is closed and $v_L = 20.0$ V a long time after it is closed. (a) Apply the loop rule to the circuit and obtain an equation for $v_L$ as a function of t. [$Hint:$ Use an analysis similar to that used to derive Eq. (30.15).] (b) What is the emf $\varepsilon$ of the battery? (c) According to your measurements, what is the voltage amplitude across the 10.0-$\Omega$ resistor as $t\rightarrow\infty$? Use this result to calculate the current in the circuit as $t\rightarrow\infty$. (d) What is the resistance $R_L$ of the solenoid? (e) Use the theoretical equation from part (a), Fig. P30.67, and the values of $\varepsilon$ and $R_L$ from parts (b) and (d) to calculate $L$. ($Hint:$ According to the equation, what is $v_L$ when $t = \tau$, one time constant? Use Fig. P30.67 to estimate the value of $t = \tau$.)

          During a summer internship as an electronics technician, you are asked to measure the self inductance $L$ of a solenoid. You connect the solenoid in series with a 10.0-$\Omega$ resistor, a battery that has negligible internal resistance, and a switch. Using an ideal voltmeter, you measure and digitally record the voltage $v_L$ across the solenoid as a function of the time $t$ that has elapsed since the switch is closed. Your measured values are shown in $\textbf{Fig. P30.67}$, where $v_L$ is plotted versus $t$. In addition, you measure that $v_L = 50.0$ V just after the switch is closed and $v_L = 20.0$ V a long time after it is closed. (a) Apply the loop rule to the circuit and obtain an equation for $v_L$ as a function of t. [$Hint:$ Use an analysis similar to that used to derive Eq. (30.15).] (b) What is the emf $\varepsilon$ of the battery? (c) According to your measurements,
what is the voltage amplitude across the 10.0-$\Omega$ resistor as $t\rightarrow\infty$? Use this result to calculate the current in the circuit as $t\rightarrow\infty$. (d) What is the resistance $R_L$ of the solenoid? (e) Use the theoretical equation from part (a), Fig. P30.67, and the values of $\varepsilon$ and $R_L$ from parts (b) and (d) to calculate $L$. ($Hint:$ According to the equation, what is $v_L$ when $t = \tau$, one time constant? Use Fig. P30.67 to estimate the value of $t = \tau$.)

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