Inquiry into Physics
8th Edition
ISBN: 9781337515863
Author: Ostdiek
Publisher: Cengage
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PP6
![**Transcription for Educational Website**
**Problem Statement:**
6. Show that \( q(t) = q_0 \exp(-t/RC) \) is a solution of the ordinary differential equation (ODE):
\[ Ri(t) + \frac{q(t)}{C} = 0 \]
**Explanation:**
This problem involves showing that the given function \( q(t) = q_0 \exp(-t/RC) \) satisfies the provided ordinary differential equation. The ODE is a first-order linear differential equation commonly seen in the context of electrical circuits, specifically RC (resistor-capacitor) circuits. We are asked to verify if the exponential function, which represents the charge \( q \) over time \( t \), is a solution by substituting it into the equation.
**Notes:**
- \( q(t) \) denotes the charge as a function of time.
- \( q_0 \) is the initial charge.
- \( \exp \) refers to the exponential function.
- \( R \) is the resistance.
- \( C \) is the capacitance.
- \( i(t) \) implies the current as a function of time.
**Application:**
To verify, differentiate \( q(t) \) with respect to time \( t \), relate it to the current function \( i(t) \), and substitute into the ODE to check if it holds true. This problem highlights how exponential decay models the discharge of a capacitor in an RC circuit.](https://content.bartleby.com/qna-images/question/c3edbe93-d39c-43f4-8aa9-2c034a88e626/318b5abb-dcfa-47dd-b5bf-ff9ce2153517/q09ke4l_processed.jpeg)
Transcribed Image Text:**Transcription for Educational Website**
**Problem Statement:**
6. Show that \( q(t) = q_0 \exp(-t/RC) \) is a solution of the ordinary differential equation (ODE):
\[ Ri(t) + \frac{q(t)}{C} = 0 \]
**Explanation:**
This problem involves showing that the given function \( q(t) = q_0 \exp(-t/RC) \) satisfies the provided ordinary differential equation. The ODE is a first-order linear differential equation commonly seen in the context of electrical circuits, specifically RC (resistor-capacitor) circuits. We are asked to verify if the exponential function, which represents the charge \( q \) over time \( t \), is a solution by substituting it into the equation.
**Notes:**
- \( q(t) \) denotes the charge as a function of time.
- \( q_0 \) is the initial charge.
- \( \exp \) refers to the exponential function.
- \( R \) is the resistance.
- \( C \) is the capacitance.
- \( i(t) \) implies the current as a function of time.
**Application:**
To verify, differentiate \( q(t) \) with respect to time \( t \), relate it to the current function \( i(t) \), and substitute into the ODE to check if it holds true. This problem highlights how exponential decay models the discharge of a capacitor in an RC circuit.
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