**Problem Statement:**A 2.0 µF capacitor is charged to 12 V and then discharged through a 4.0 MΩ resistor. How long will it take for the voltage across the capacitor to drop to 3.0 V?**Hint:** The unknown time is in "log jail." Break it out with the inverse function of the exponential.**Options:**- ○ 8.0 s- ○ 24 s- ○ 11 s- ○ 22 s**Discussion:**In a discharging RC circuit, the voltage across the capacitor as a function of time is given by the equation:\[ V(t) = V_0 \cdot e^{-t/(RC)} \]Where:- \( V(t) \) is the voltage at time \( t \).- \( V_0 \) is the initial voltage (12 V).- \( R \) is the resistance (4.0 MΩ).- \( C \) is the capacitance (2.0 µF).To find the time when \( V(t) = 3.0 \) V, rearrange the equation and use the inverse function of the exponential (logarithm):\[ t = -RC \cdot \ln\left(\frac{V(t)}{V_0}\right) \]

The relation for the discharging capacitor is, V=V0e-t/RC

A 2.0 µF capacitor is charged to 12 V and then discharged through a 4.0 M2 resistor. How long will it take for the voltage across the capacitor to drop to 3.0 V? Hint: The unknown time is in "log jail". Break it out with the inverse function of the exponential. O 8.0 s O 24 s O 11 s O 22 s

A 2.0 µF capacitor is charged to 12 V and then discharged through a 4.0 M2 resistor. How long will it take for the voltage across the capacitor to drop to 3.0 V? Hint: The unknown time is in "log jail". Break it out with the inverse function of the exponential. O 8.0 s O 24 s O 11 s O 22 s

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

**Problem Statement:**

A 2.0 µF capacitor is charged to 12 V and then discharged through a 4.0 MΩ resistor. How long will it take for the voltage across the capacitor to drop to 3.0 V?

**Hint:** The unknown time is in "log jail." Break it out with the inverse function of the exponential.

**Options:**

- ○ 8.0 s
- ○ 24 s
- ○ 11 s
- ○ 22 s

**Discussion:**

In a discharging RC circuit, the voltage across the capacitor as a function of time is given by the equation:

\[ V(t) = V_0 \cdot e^{-t/(RC)} \]

Where:
- \( V(t) \) is the voltage at time \( t \).
- \( V_0 \) is the initial voltage (12 V).
- \( R \) is the resistance (4.0 MΩ).
- \( C \) is the capacitance (2.0 µF).

To find the time when \( V(t) = 3.0 \) V, rearrange the equation and use the inverse function of the exponential (logarithm):

\[ t = -RC \cdot \ln\left(\frac{V(t)}{V_0}\right) \]

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