Charging and Discharging of Capacitor, Growth and Decay of current in L-R circuit, L-C circuit, Power Factor

CHARGING AND DISCHARGING OF CAPACITOR

Charging of capacitor

Consider a capacitor 'C' is connected with a source of emf V in series through resistor R as shown in figure.

Let at time t=0, the charge in the capacitor C is zero i.e. Q=0 at t=0.
Let at any instance, the charge on the capacitor is Q and the current through the series resistor R is I,then from Kirchoff's law,
we have,
\begin{align*}V-IR-\frac{Q}{C}=0 \end{align*}\begin{align*}or,\space V-\frac{dQ}{dt}R-\frac{Q}{C}=0 \end{align*}\begin{align*}or,\space \frac{dQ}{dt}=(V-\frac{Q}{RC})\rightarrow 1 \end{align*}\begin{align*}Let,\space (V-\frac{Q}{C})=x \rightarrow 2\end{align*}\begin{align*}\therefore -\frac{dQ}{dt}\frac{1}{C}=\frac{dx}{dt} \end{align*}\begin{align*}\frac{dQ}{dt}=-C.\frac{dx}{dt}\rightarrow 3 \end{align*}
From eqns 1, 2 and 3
\begin{align*}-C\frac{dx}{dt}=\frac{x}{R} \end{align*}\begin{align*}\frac{dx}{x}=-\frac{dt}{CR} \end{align*}
Integrating both sides, we get\begin{align*} \end{align*}\begin{align*}lnx=-\frac{t}{RC}+K \end{align*}\begin{align*}x=e^{k}.e^{-\frac{t}{RC}} \end{align*}\begin{align*}x=A.e^{-\frac{t}{RC}} \end{align*}
Where A is a constant
Substituting the value of x, we get
\begin{align*}V-\frac{Q}{C}=A.e^{-\frac{t}{RC}} \end{align*}
At t=0, Q=0 \(\implies \) A=V
\begin{align*}V-\frac{Q}{C}=V.e^{-\frac{t}{RC}} \end{align*}\begin{align*}\frac{Q}{C}=V[1-e^{-\frac{t}{RC}} ] \end{align*}
\begin{align*}Q=CV[1-e^{-\frac{t}{RC}} ] \end{align*}\begin{align*}Q=Q_0 [1-e^{-\frac{t}{\tau}} ] \end{align*}
Where \(Q_0=CV\) is the maximum charge on the capacitor and \(\tau =RC\) is called R-C time constant or charging time constant of the capacitor.

If \(t=\tau \) then
\begin{align*}Q=Q_0 [1-e^{-\frac{t}{\tau}} ] \end{align*}\begin{align*}Q=Q_0 [1-e^{-1} ] \end{align*}\begin{align*}Q=0.63 Q_0 \end{align*}
Thus the charging time constant for R-C ctime constant of a capacitor is defined as the time interval at which the charge on the capacitor equals the 63% of the maximum charge on the capacitor.

ENERGY STORED IN THE CAPACITOR DURING CHARGING

To derive a qualitative expression for the storage of the energy, consider an extra increase of charge dq on the capacito, the small amount of additional work will be,
\begin{align*}dU_{E}=\frac{q}{C}dq \end{align*}
This work done will increase the potential energy of the system. If process is continued until a total charge Q has been transffered, the total work done will be,
\begin{align*}U_{E}=\int dU_{E} \end{align*}\begin{align*}=\int_{0}^{Q} \frac{q}{C}dq=\frac{1}{2}\frac{Q^2}{C} \end{align*}\begin{align*}or,\space U_{E}=\frac{1}{2}\frac{Q^2}{C}=\frac{1}{2}CV^2 \end{align*}
is the total energy of the capacitor.

DISCHARGING OF A CAPACITOR

Consider a capacitor of a capacitance C with charge \(Q_0\) at time t=0. Let us connect the two plates of the capacitor throuigh a series resistor R as shown in figure.

If Q be the charge on the capacitor and I be the current through the resistor R at time 't' then from kirchoff's law,
\begin{align*}\frac{Q}{C}+IR=0 \end{align*}\begin{align*}or,\space \frac{Q}{C}=-\frac{dQ}{dt}R \end{align*}\begin{align*}or,\space \frac{dQ}{Q}=-\frac{dt}{RC} \end{align*}
Integrating both sides, we get
\begin{align*}lnQ=-\frac{t}{RC}+K(Constant) \end{align*}\begin{align*}Q=A.e^{-\frac{t}{RC}} \end{align*}\begin{align*}where,\space A=e^{K}\space is\space a\space constant \end{align*}\begin{align*}At\space t=0,\space Q=Q_0,\space A=Q_0 \end{align*}\begin{align*}\therefore Q=Q_0 e^{-\frac{t}{RC}} \end{align*}
This is the value of charge on the capacitor at any instant of time t.

\begin{align*}Q=Q_0 e^{-t/\tau} \end{align*}
where \(\tau \)=RC is called the discharging time constant. If t=\(\tau \) then
\begin{align*}Q=Q_0 e^{-1} \end{align*}\begin{align*}Q=0.37 Q_0 \end{align*}
Thus the discharging time constant of a capacitor is defined as the time interval at which the charge on the capacitor is equal to the 37% of the initial charge on the capacitor.

Growth and Decay of Current in L-R circuit

GROWING CURRENT IN L-R CIRCUIT

Consider an inductor 'L' and resistor 'R' are connected in series witrh a source of emf E as shown in figure.

Let at any time t=0, current in the circuit I=0. If I be the current in the circuit at any instant of time t, applying Kirchoff's law, we get,
\begin{align*}E=L\frac{dI}{dt}+IR \end{align*}\begin{align*}E-IR=L\frac{dI}{dt}\rightarrow 1 \end{align*}\begin{align*}Let\space x=E-IR \rightarrow 2 \end{align*}\begin{align*}\frac{dx}{dt}=-\frac{dI}{dt}R\rightarrow 3 \end{align*}
From 1,2 and 3
\begin{align*}x=-\frac{L}{R}.\frac{dx}{dt} \end{align*}\begin{align*}or,\space \frac{dx}{x}=-\frac{R}{L}dt \end{align*}
Integrating both sides, we get \begin{align*}ln=-\frac{Rt}{L}+K \end{align*}\begin{align*}or,\space x=A.e^{-\frac{t}{L/R}} \end{align*}
where A is constant\(=e^{K}\)
\begin{align*}\frac{E}{R}-I=A.e^{-\frac{t}{\tau}} \end{align*}
where \(\tau=\frac{L}{R}\) is the time constant for the L-R circuit.
At t=0, I=0 and E/R=A
\begin{align*}\frac{E}{R}-I=\frac{E}{R}e^{-\frac{t}{\tau}} \end{align*}\begin{align*}I=I_0 (1-e^{-\frac{t}{\tau}}) \end{align*}
where \(I_0=\frac{E}{R}\) is the maximum current in the circuit.

We have,
At \(t=\tau \),\begin{align*}I=I_0 (1-e^{-1}) \end{align*}\begin{align*}I=0.63 I_0 \end{align*}
Thus the growing time constant of L-R circuit is defined as the time interval at which the current in the circuit is equalto 63% of the maximum current \(I_0 \) in the circuit.

DECAY CURRENT IN L-R CIRCUIT

Let \(I_0 \) be the current in the L-R circuit when the source is just removed from the circuit. i.e.
\begin{align*}I=I_o\space at\space t=0 \end{align*}
Let I be the current in the circuit after time t.
Now, applying Kirchoff's law, we get
\begin{align*}L\frac{dI}{dt}+IR=0 \end{align*}\begin{align*}or,\space \frac{dI}{I}=-\frac{dt}{L/R} \end{align*}\begin{align*}\implies lnI=-\frac{t}{L/R}+K(constant) \end{align*}\begin{align*}I=Ae^{-\frac{t}{L/R}} \end{align*}
where A is a constant
\begin{align*}If\space t=0,\tau=0\space and\space A=I_0 \end{align*}\begin{align*}so\space I=I_0.e^{-t/\tau } \end{align*}
where \(\tau \)=L/R is the decay time constant L-R circuit.

If \(t=\tau \), \(I=I_0.e^{-1}\)\begin{align*}I=0.37 I_0 \end{align*}
Thus, the decay time constant of L-R circuit is defined as the time interval at which the current in the circuit is equal to the 37% of the maximum current in the circuit.

L-C CIRCUIT

Consider a circuit containing an inductor L and a capacitor C as shown in figure.

Let Q be the charge on capacitor at any instant of time and I be the current in the inductor at the same instant of time. Then from Kirchoff's law:
\begin{align*}L\frac{dI}{dt}+\frac{Q}{C}=0\rightarrow 1 \end{align*}
we have\begin{align*}I=\frac{dQ}{dt}\rightarrow 2 \end{align*}
From eqn 1 and 2
\begin{align*}L\frac{d^2Q}{dt^2}+\frac{Q}{C}=0 \end{align*}
\begin{align*}\implies \frac{d^2Q}{dt^2}+\frac{Q}{LC}=0\rightarrow 3 \end{align*}
From eqn 3, it is seen that L-C circuit is oscillatory in nature. The oscillation of L-C circuit is similar to the oscillation in simple harmonic motion.\begin{align*}[i.e.\space \frac{d^2x}{dt^2}+\omega^2x=0] \end{align*}\begin{align*}where\space \omega=\sqrt{\frac{1}{LC}} \end{align*}
The frequency of oscillation of L-C circuit is \begin{align*}f=\frac{1}{2\pi}\sqrt{\frac{1}{LC}} \end{align*}
The charge on the capacitor at any instant of time.
\begin{align*}Q=Asin\omega t,\space where\space \omega =2\pi f \end{align*}

CAPACITIVE CIRCUIT

Consider a capacitor of capacitance 'C' connected in series with A.C. source of emf \(E=E_0 sin\omega t\) as shown in figure.

Now applying Kirchoff's Law, we get
\begin{align*}E=\frac{Q}{C} \end{align*}\begin{align*}Q=CE_0 sin\omega t \end{align*}
Diff. both sides with respect to 't' we get
\begin{align*}\frac{dQ}{dt}=\omega CE_0 cos(\omega t) \end{align*}\begin{align*}I=\frac{E_0}{\frac{1}{\omega C}}sin(\omega t+\pi /2) \end{align*}\begin{align*}I=I_0 sin(\omega t+\pi /2) \end{align*}
Where \(I=\frac{E_0}{\frac{1}{\omega C}} \) is the maximum current in the circuit. The term \(\frac{1}{\omega C}\) behaves as the resistance, is known as reactance of the capacitor i.e. \(X_{C}=\frac{1}{\omega C}\)
The nature of current 'I' and emf 'E' in capacitive circuit is shown in figure.

From this figure,it is seen that in capacitive circuit, current leads emf by \(\pi /2\)

INDUCTIVE CIRCUIT

Consider an inductor of inductance 'L' is connected in series with a source \(E=E_0 sin\omega t\) as shown in figure.

Applying Kirchoff's law, we get
\begin{align*}L\frac{dI}{t}=E_0sin\omega t \end{align*}\begin{align*}or,\space dI=\frac{E_0}{L}sin\omega t.dt \end{align*}
Integrating on both sides, we get
\begin{align*}I=\frac{E_0}{L\omega }(-cos\omega t)=I_0 sin(\omega t-\pi /2) \end{align*}
where \(I_0=\frac{E_0}{L \omega }\) is the maximum current in the inductive circuit and the term \(L\omega \) behaves as a resistance, is reactance of inductor i.e. \(X_{L}=L\omega \). The nature of current I and emf E in inductive circuit is as shown in figure.

From this figure, it is seen that emf leads to the current by \(\pi /2\) in purely inductive circuit.

Power Factor

Consider an emf \(E=E_0 sin(\omega t) \) is applied to the circuit. Let \(I=I_0 sin(\omega t +\phi ) \) be the current in the circuit. The amount of work done to pass down the current I in the interval of time dt in the circuit.

\begin{align*}dw=IEdt \end{align*}\begin{align*}=I_0 sin(\omega t +\phi ).E_0 sin(\omega t)dt \end{align*}

The average work done (W) is given by \begin{align*}W=\frac{1}{2\pi }\int_{0}^{2\pi }I_0E_0 sin\omega t.sin(\omega t +\phi )d(\omega t) \end{align*}\begin{align*}=\frac{1}{2\pi }I_0E_0\int_{0}^{2\pi }[sin^2\omega t.cos\phi +sin\omega t.cos\omega t.sin\phi ]d(\omega t) \end{align*}\begin{align*}=\frac{1}{2\pi }I_0E_0\int_{0}^{2\pi }sin^2\omega t.cos\phi.d(\omega t) \end{align*}\begin{align*}=\frac{1}{4\pi }I_0E_0\int_{0}^{2\pi }(1-cos2\omega t)d(\omega t) \end{align*}\begin{align*}=\frac{I_0E_0}{2}cos\phi\end{align*}\begin{align*}[\because \int_{0}^{2\pi } cos2\omega td(\omega t)=0] \end{align*}\begin{align*}W=\frac{E_0}{\sqrt{2}}\frac{I_0}{\sqrt{2}}cos\phi \end{align*}

where the term \(cos\phi \) is known as power factor of the circuit which gives the energy loss of circuit per cycle of A-C source. The value of power factor of a circuit is always less than unity.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Griffiths, D.J., Introduction to Electrodynamics, PHI Learning Private Limited, 2013