Current Electricity Concept Page - 15

Example

RC circuits at steady state

In the given circuit, the steady state voltage drop across the capacitor C isIn steady state C′′=0$C_{{}^{\prime }}^{{}^{\prime }}=0$
V=I[r1+r2]$V=I[r_1+r_2]$
VC=Ir1$V_C=I{r}_1$
         =Vr1r1+r2$={\displaystyle \frac{V{r}_1}{r_1+r_2}}$

Formula

Discharging RC circuit

The differential equation of RC circuit is
dqq=−∫1CRdt${\displaystyle \frac{dq}{q}}=-\int {\displaystyle \frac{1}{CR}}dt$

Formula

Charging of an RC circuit

Rdqdt=ϵC−qC$R{\displaystyle \frac{dq}{dt}}={\displaystyle \frac{ϵC-q}{C}}$ where ϵ$ϵ$ is the emf of the cell

Definition

Time constant in an RC circuit

The time constant of an RC circuit is the time required to charge the capacitor, through the resistor, by 63.2 percent of the difference between the initial value and final value or discharge the capacitor to 36.8 percent.
τ=RC$\tau =RC$ where τ$\tau$ is the time constant and R and C are the values of resistance and capacitance respectively 

Formula

General solution of RC circuit for a capacitor

General solution of RC circuit for a capacitor after time t is given by :
q(t)=q(∞)−(q(∞)−q(0))e−tRC${\displaystyle q(t)=q(\mathrm{\infty })-(q(\mathrm{\infty })-q(0))e^{-{\displaystyle \frac{t}{RC}}}}$
where:
q(t):$q(t):$ Charge on the capacitor after time t
q(0):$q(0):$ Initial charge on the capacitor
q(∞):$q(\mathrm{\infty }):$ Charge on the capacitor at steady state
RC:$RC:$ Time constant
Note:
The solution remains valid for other parameters of capacitors like voltage, current, etc.

Definition

General solution of RC circuit for a resistor

General solution of RC circuit for a resistor after time t is given by :
i(t)=i(∞)−(i(∞)−i(0))e−tRC${\displaystyle i(t)=i(\mathrm{\infty })-(i(\mathrm{\infty })-i(0))e^{-{\displaystyle \frac{t}{RC}}}}$
where:
i(t):$i(t):$ Current in the resistor after time t
i(0):$i(0):$ Initial current in the resistor
i(∞):$i(\mathrm{\infty }):$ Current in the resistor at steady state
RC:$RC:$ Time constant
Note:
The solution remains valid for other parameters of resistors like voltage.

Formula

Charge on a capacitor in a charging RC circuit

Charge on a capacitor in a charging circuit is given by the following equation.
q=q0(1−e−tτ)$q=q_0(1-e^{{\displaystyle \frac{-t}{\tau }}})$
where q0$q_0$ is the initial charge of the capacitor and q$q$ is the charge at a time t$t$.

Formula

Voltage in a charging/discharging RC circuit

Charging  V(t)=V0(1−e−t/τ)$V(t)=V_0(1-e^{-t/\tau })$              Vr(t)=V0(e−t/τ)$V_r(t)=V_0(e^{-t/\tau })$
 Discharging  Vc(t)=V0(e−t/τ)$V_c(t)=V_0(e^{-t/\tau })$           Vr(t)=−V0(e−t/τ)$V_r(t)=-V_0(e^{-t/\tau })$
where τ=RC$\tau =RC$ is the time constant.

Formula

Current in a charging/discharging RC circuit

Charging: i=VRe−t/τ$i={\displaystyle \frac{V}{R}}{e}^{-t/\tau }$
Discharging i=−V0Re−t/τ$i=-{\displaystyle \frac{V_0}{R}}{e}^{-t/\tau }$
where V0$V_0$ is the initial voltage.

BookMarks

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