The figure shows a circuit containing an electromotive...

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of $C$ farads $(\mathrm{F}),$ and a resistor with a resistance of $R$ ohms $(\Omega) .$ The voltage drop across the capacitor is $Q / C,$ where $Q$ is the charge (in coulombs), so in this case Kirchhoff's Law gives $$R I+\frac{Q}{C}=E(t)$$ But $I=d Q / d t($ see Example 3 in Section 3.7$),$ so we have $$R \frac{d Q}{d t}+\frac{1}{C} Q=E(t)$$ Suppose the resistance is $5 \Omega,$ the capacitance is 0.05 $\mathrm{F}$ , a battery gives a constant voltage of $60 \mathrm{V},$ and the initial charge is $Q(0)=0 \mathrm{C} .$ Find the charge and the current at time $t$.

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of $C$ farads $(\mathrm{F}),$ and a resistor with a resistance of $R$ ohms $(\Omega) .$ The voltage drop across the capacitor is $Q / C,$ where $Q$ is the charge (in coulombs), so in this case Kirchhoff's Law gives
$$R I+\frac{Q}{C}=E(t)$$
But $I=d Q / d t($ see Example 3 in Section 3.7$),$ so we have
$$R \frac{d Q}{d t}+\frac{1}{C} Q=E(t)$$
Suppose the resistance is $5 \Omega,$ the capacitance is 0.05 $\mathrm{F}$ , a battery gives a constant voltage of $60 \mathrm{V},$ and the initial charge is $Q(0)=0 \mathrm{C} .$ Find the charge and the current at time $t$.

Key Concepts

Capacitor Behavior in Circuits

In electrical circuits, capacitors store energy in the electric field created between their plates. The voltage across a capacitor is proportional to the stored charge (given by Q/C), and its current is the time derivative of this charge (dQ/dt). This relationship is fundamental in analyzing how capacitors charge or discharge over time.

Kirchhoff's Voltage Law

This principle states that the sum of all voltage drops and rises around any closed loop in an electrical circuit must equal zero. In circuit analysis, it is used to set up equations that relate the voltage contributions from different circuit elements, allowing one to derive the governing differential equations for the circuit's behavior.

Resistor and Ohm's Law

Ohm’s Law relates the voltage across a resistor to the current flowing through it and its resistance (V = IR). In circuit analysis, this law is applied to determine the voltage drops across resistive elements, which is essential when using Kirchhoff's Voltage Law to develop the circuit's differential equation.

First-Order Linear Differential Equations

Many circuit problems, especially those involving RC circuits, lead to first-order linear differential equations. These equations capture how the charge or current changes over time in response to applied voltages, and standard solution techniques, such as integrating factors, are used to solve them.

Time Constant in RC Circuits

The time constant, denoted by ? (tau) and defined as the product of resistance and capacitance (? = RC), characterizes the rate at which a capacitor charges or discharges. It represents the time required for the charge or voltage to reach a significant portion of its final value, playing a crucial role in the transient analysis of RC circuits.

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