Study sheet: RC Circuit Charging and Discharging

📋 Course Outline

1. RC Circuit Components and Charging
2. Differential Equation of Charge
3. Current and Capacitor Voltage
4. Graphical Study of Charging
5. Series Resistors and Time Constant
6. Experimental Determination of Parameters
7. Energy and Charging Speed

📖 1. RC Circuit Components and Charging

📝 Essential Points

• The RC circuit contains an ideal voltage generator of electromotive force E, an ohmic conductor of resistance R, a previously discharged capacitor of capacitance C, and a switch K.

• When switch K is closed at t = 0, the capacitor undergoes charging until the switch is opened at t = 40 ms.

💡 Memory Hook

Close K → current flows → capacitor charges → permanent regime

📖 2. Differential Equation of Charge

🔑 Key Concepts & Definitions

• Time constant : $\tau=RC$ and characterizes the speed of capacitor charging.

📝 Essential Points

📐 Formula — For a series RC circuit, the charge satisfies $RC\frac{dq(t)}{dt}+q(t)=CE$, so the differential-equation constants are $\alpha=RC$ and $\beta=CE$.

📐 Formula — For an initially uncharged capacitor, the charge is $q(t)=CE\left(1-e^{-t/(RC)}\right)=Q_0\left(1-e^{-t/\tau}\right)$, where $Q_0=CE$ is the permanent-regime charge.

💡 Memory Hook

Resistance limits current → charging slows → charge approaches a maximum

📖 3. Current and Capacitor Voltage

📝 Essential Points

📐 Formula — The charging current is $i(t)=\frac{dq}{dt}=\frac{E}{R}e^{-t/(RC)}=I_0e^{-t/\tau}$, where $I_0=\frac{E}{R}$.

📐 Formula — The resistor voltage during charging is $u_R(t)=Ri(t)=Ee^{-t/\tau}$, while the capacitor voltage is $u_C(t)=E\left(1-e^{-t/\tau}\right)$.

💡 Memory Hook

Current is initially maximum, whereas capacitor voltage is initially zero

📖 4. Graphical Study of Charging

★ Must-know

📌 The permanent regime is not reached at t = 40 ms if the experimental current or resistor-voltage curve has not yet reached zero.

📌 Graphically, the time constant τ is obtained from the tangent at t = 0: its intersection with the final asymptote gives the time τ.

📐 Formula — The electrostatic energy stored by the capacitor in the permanent regime is $E_e=\frac{1}{2}CE^2=\frac{Q_0^2}{2C}$.

Further detail

📐 Formula — At three time constants, the charge is $q(3\tau)=Q_0\left(1-e^{-3}\right)\approx0.95Q_0$.

💡 Memory Hook

The tangent at t = 0 meets the final horizontal asymptote at one time constant

📖 5. Series Resistors and Time Constant

★ Must-know

📐 Formula — With resistors R₀ and R in series, the time constant is $\tau=(R_0+R)C$.

📐 Formula — The resistor voltages satisfy $u_R(t)=\frac{R}{R_0}u_{R_0}(t)$ and the capacitor voltage is $u_C(t)=E-u_R(t)-u_{R_0}(t)$.

Further detail

📐 Formula — The voltage across R₀ is $u_{R_0}(t)=\frac{R_0E}{R+R_0}e^{-t/\tau}$, with initial amplitude $A=\frac{R_0E}{R+R_0}$ and exponent coefficient $\alpha=\frac{1}{\tau}$.

💡 Memory Hook

R increases the time constant, whereas R₀ sets the initial resistor voltage division

📖 6. Experimental Determination of Parameters

★ Must-know

📐 Formula — If the capacitor reaches 99% of its maximum voltage after a duration θ, then $u_C(\theta)=0.99E$ and $\theta=\tau\ln(100)\approx4.6\tau$.

📐 Formula — When the capacitor is initially charged and the circuit is closed again with resistance R₁, the initial current is $I_0=\frac{E}{R_0+R_1}$ and the time constant is $\tau=(R_0+R_1)C$.

Further detail

📐 Formula — For the two-resistor circuit, measuring θ as a function of R gives $\theta=\ln(100)(R_0+R)C$, allowing C and R₀ to be determined from the graph.

💡 Memory Hook

Read τ and E → determine C → determine R or R₁

📖 7. Energy and Charging Speed

★ Must-know

📐 Formula — At t = 0, an initially uncharged capacitor has q(0)=0, so the current is $I_0=\frac{E}{R}$.

📌 The capacitor charge is not instantaneous because the experimental charge curve increases progressively rather than jumping immediately to its final value.

📌 To store the same final energy while charging more slowly, E and C must remain unchanged and the resistance R must be increased, because the final energy is $E_e=\frac12CE^2$ whereas the time constant is $\tau=RC$.

Further detail

• With a voltmeter across the resistor indicating a constant voltage U = 6 V, the generator electromotive force is obtained from the permanent capacitor voltage and the loop equation.

📐 Formula — To make charging twice less rapid while keeping E and C unchanged, the resistance R must be doubled so that the time constant becomes $\tau'=2\tau$.

💡 Memory Hook

Increase RC → slower charging, while unchanged E and C preserve final energy

📊 Synthesis Tables

Main RC quantities

Quantity	Expression	Behavior during charging
Charge	$q(t)=Q_0(1-e^{-t/\tau})$	Increases toward Q₀
Current	$i(t)=I_0e^{-t/\tau}$	Decreases toward zero
Capacitor voltage	$u_C(t)=E(1-e^{-t/\tau})$	Increases toward E
Time constant	$\tau=RC$	Sets charging speed