Flashcards: RC Circuit Charging and Discharging — 39 cards

What components does an RC circuit contain?

An ideal voltage generator, a resistor, a capacitor, and a switch.

What is the resistance component in an RC circuit?

An ohmic conductor of resistance R.

What is the capacitance component in an RC circuit?

A previously discharged capacitor of capacitance C.

What happens when switch K is closed at t = 0 in an RC circuit?

The capacitor begins charging.

Until what time does the capacitor charge in the RC circuit?

Until the switch is opened at t = 40 ms.

What differential equation does charge satisfy in a series RC circuit?

$RC\frac{dq(t)}{dt}+q(t)=CE$

What are the differential-equation constants in the RC circuit equation?

$\alpha=RC$ and $\beta=CE$

What is the time constant $\tau$ of an RC circuit?

$\tau=RC$

What does the time constant $\tau$ characterize in an RC circuit?

The speed of capacitor charging

What is the charge $q(t)$ of an initially uncharged capacitor in an RC circuit?

$q(t)=CE\left(1-e^{-t/(RC)}\right)$

How is the charge $q(t)$ expressed using the time constant $\tau$?

$q(t)=Q_0\left(1-e^{-t/\tau}\right)$

What is the permanent-regime charge $Q_0$ in an RC circuit?

$Q_0=CE$

What is the formula for the charging current $i(t)$ in a capacitor circuit?

$i(t)=\frac{dq}{dt}=\frac{E}{R}e^{-t/(RC)}=I_0e^{-t/\tau}$

How is the initial charging current $I_0$ defined in terms of $E$ and $R$?

$I_0=\frac{E}{R}$

What is the formula for the resistor voltage $u_R(t)$ during capacitor charging?

$u_R(t)=Ri(t)=Ee^{-t/\tau}$

What is the formula for the capacitor voltage $u_C(t)$ during charging?

$u_C(t)=E\left(1-e^{-t/\tau}\right)$

When is the permanent regime not reached at t = 40 ms?

If the experimental current or resistor-voltage curve has not yet reached zero.

How is the time constant τ graphically obtained from the charging curve?

From the tangent at t = 0 intersecting the final asymptote.

What is the charge at three time constants during charging?

$q(3\tau)=Q_0\left(1-e^{-3}\right)\approx0.95Q_0$.

What is the formula for electrostatic energy stored by the capacitor in permanent regime?

$E_e=\frac{1}{2}CE^2=\frac{Q_0^2}{2C}$.

What is the time constant $\tau$ for resistors $R_0$ and $R$ in series with capacitor $C$?

$\tau = (R_0 + R)C$

What is the formula for voltage across resistor $R_0$ over time?

$u_{R_0}(t) = \frac{R_0 E}{R + R_0} e^{-t/\tau}$

What is the initial amplitude $A$ of voltage across $R_0$?

$A = \frac{R_0 E}{R + R_0}$

What is the exponent coefficient $\alpha$ in the voltage decay across $R_0$?

$\alpha = \frac{1}{\tau}$

How is the voltage across resistor $R$ related to voltage across $R_0$?

$u_R(t) = \frac{R}{R_0} u_{R_0}(t)$

What is the formula for the capacitor voltage $u_C(t)$ in terms of $E$, $u_R(t)$, and $u_{R_0}(t)$?

$u_C(t) = E - u_R(t) - u_{R_0}(t)$

What voltage does the capacitor reach at time θ if it is 99% charged?

$u_C(\theta)=0.99E$

How is the duration θ related to the time constant τ when the capacitor reaches 99% voltage?

$\theta=\tau\ln(100)\approx4.6\tau$

What formula relates θ to R in the two-resistor circuit?

$\theta=\ln(100)(R_0+R)C$

What parameters can be determined from the graph of θ versus R in the two-resistor circuit?

Capacitance C and resistance $R_0$

What is the initial current $I_0$ when the capacitor is initially charged and the circuit closes with resistance $R_1$?

$I_0=\frac{E}{R_0+R_1}$

How is the time constant $\tau$ expressed when the circuit has resistance $R_1$?

$\tau=(R_0+R_1)C$

How is the generator electromotive force obtained with a voltmeter across the resistor?

From the permanent capacitor voltage and the loop equation.

What is the initial current when an uncharged capacitor starts charging at t=0?

The initial current is $I_0=\frac{E}{R}$.

Why is the capacitor charge not instantaneous?

Because the experimental charge curve increases progressively, not immediately.

What must be increased to store the same energy while charging more slowly?

The resistance $R$ must be increased.

What is the formula for the final energy stored in the capacitor?

$E_e=\frac12CE^2$.

What is the formula for the time constant in charging a capacitor?

$\tau=RC$.

How to make charging twice less rapid while keeping E and C unchanged?

Double the resistance $R$ so $\tau' = 2\tau$.