1. Separate the differential equation by: a. Combining the two fractions on the right-hand side into one term (the common denominator is RC) b. Multiplying both sides by dt c. Dividing both sides q - Ce d. You can tell that the differential equation has been separated because all the q's are on the dq side and all the t's (there weren't any) are on the dt side.
1. Separate the differential equation by: a. Combining the two fractions on the right-hand side into one term (the common denominator is RC) b. Multiplying both sides by dt c. Dividing both sides q - Ce d. You can tell that the differential equation has been separated because all the q's are on the dq side and all the t's (there weren't any) are on the dt side.
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![**Title: Solving Differential Equations for Charging and Discharging Capacitors in RC Circuits**
We will begin by deriving the equations on the first page for charging and discharging capacitors in RC circuits. This requires solving a simple differential equation, but we will break it into simple steps in case you haven’t had differential equations yet. It can be shown using Kirchhoff’s rules (we will study them in a future lab) that for a charging capacitor in a simple RC circuit with a battery of \( \epsilon \) voltage, the rate of change of charge \( q(t) \) on the positive capacitor plate is given by:
\[
\frac{dq}{dt} = \frac{\epsilon}{R} - \frac{q}{RC}
\]
We don’t write \( q(t) \) in the above equation because it becomes too cluttered and it is implied by the derivative anyway.
\( \epsilon \) is the voltage \( \Delta V \) of the battery.
Our goal is to solve this equation for the **function** \( q(t) \).
Notice that the right-hand side becomes smaller as the charge on the plate \( q \) increases. This means it becomes increasingly harder to add charge to the plate; less charge is added as time increases.
This equation is known as a **first-order, ordinary, linear, and separable differential equation** for the function \( q(t) \). Quite a mouthful! **First order** means the equation only contains a first derivative, \( dq/dt \). **Ordinary** means there are only full derivatives involved and one independent variable, instead of partial derivatives and several independent variables. **Linear** means there are no powers of our function or trigonometry functions involving our function, for example. **Separable** means it separates which is the first step to solving it!
1. **Separate** the differential equation by:
a. **Combining** the two fractions on the right-hand side into one term (the common denominator is \( RC \))
b. **Multiplying** both sides by \( dt \)
c. **Dividing** both sides \( q - C\epsilon \)
d. You can tell that the differential equation has been separated because all the \( q \)’s are on the \( dq \) side and all the \( t \)’s (there weren’t any) are on the \( dt \) side.
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Transcribed Image Text:**Title: Solving Differential Equations for Charging and Discharging Capacitors in RC Circuits**
We will begin by deriving the equations on the first page for charging and discharging capacitors in RC circuits. This requires solving a simple differential equation, but we will break it into simple steps in case you haven’t had differential equations yet. It can be shown using Kirchhoff’s rules (we will study them in a future lab) that for a charging capacitor in a simple RC circuit with a battery of \( \epsilon \) voltage, the rate of change of charge \( q(t) \) on the positive capacitor plate is given by:
\[
\frac{dq}{dt} = \frac{\epsilon}{R} - \frac{q}{RC}
\]
We don’t write \( q(t) \) in the above equation because it becomes too cluttered and it is implied by the derivative anyway.
\( \epsilon \) is the voltage \( \Delta V \) of the battery.
Our goal is to solve this equation for the **function** \( q(t) \).
Notice that the right-hand side becomes smaller as the charge on the plate \( q \) increases. This means it becomes increasingly harder to add charge to the plate; less charge is added as time increases.
This equation is known as a **first-order, ordinary, linear, and separable differential equation** for the function \( q(t) \). Quite a mouthful! **First order** means the equation only contains a first derivative, \( dq/dt \). **Ordinary** means there are only full derivatives involved and one independent variable, instead of partial derivatives and several independent variables. **Linear** means there are no powers of our function or trigonometry functions involving our function, for example. **Separable** means it separates which is the first step to solving it!
1. **Separate** the differential equation by:
a. **Combining** the two fractions on the right-hand side into one term (the common denominator is \( RC \))
b. **Multiplying** both sides by \( dt \)
c. **Dividing** both sides \( q - C\epsilon \)
d. You can tell that the differential equation has been separated because all the \( q \)’s are on the \( dq \) side and all the \( t \)’s (there weren’t any) are on the \( dt \) side.
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