Time Constant

Q. In the given circuit for ideal diode, the current through the battery is:

1. 1 A
2. 2 A
3. 0.5 A
4. 1.5 A

Q. An inductor of $10\text{mH}$ is connected to a $20\text{V}$ battery through a resistor of $10k\Omega$ and a switch. After a long time, when maximum current is set up in the circuit, the current is switched off. The current in the circuit after $1\mu s$ is $\frac{x}{100}\text{mA}$. Then $x$ is equal to (Take $e^{-1}=0.37$)

Q. A parallel plate capacitor is charged to a potential difference of $50\text{Volts}$. It is then discharged through a resistance for $2\text{s}$ and its potential drops by $10\text{Volts}$. Calculate the fraction of energy stored in the capacitor.

1. $0.14$
2. $0.25$
3. $0.50$
4. $0.64$

Q. A charged capacitor is connected with a resistor. After how many time constants, does the energy of the capacitor become $\frac{1}{10}th$ of its initial value? $(\ln 10=2.303)$

1. $2.3$
2. $1.15$
3. $0.69$
4. $1.38$

Q. A capacitor of capacitance $C_1$ is charged by connecting it to a battery. The battery is now removed and this capacitor is connected to a second uncharged capacitor of capacitance $C_2$ . If the charge gets distributed equally on the two capacitors after connection, the ratio of the total energy stored in the capacitors after connection to the total energy stored in them before connection is

1. $1$
2. $\frac{1}{2}$
3. $\frac{1}{\sqrt{2}}$
4. $\frac{1}{3}$

Q. An uncharged capacitor of capacitance $100\mu \text{F}$ is connected to a battery of emf $20\text{V}$ at $t=0$ through a resistance $10\Omega$, then the maximum rate at which energy is stored in the capacitor is :

1. $20\text{J/s}$
2. $15\text{J/s}$
3. $10\text{J/s}$
4. $30\text{J/s}$

Q. When a certain circuit consisting of a constant emf $E$, an inductance $L$ and a resistance $R$ is closed, the current in it increases with time according to curve $1$. If one of the parameters $(E,L$ or $R)$ is changed, then the $i-t$ curve shift to the curve $2$. Which parameter is changed and how?

1. $L$ is increased
2. $L$ is decreased
3. $R$ is increased
4. $R$ is decreased

Q.

In the circuit shown, $L=1\mu H,C=1\mu F$ and $R=1k\Omega$. They are connected in series with an a.c. source $V=V_0\sin \omega t$ as shown. Which of the following options is/are correct?

1. At $\omega >>{10}^{6}rad.s^{-1}$, the circuit behaves like a capacitor

2. At $\omega \sim 0$, the current flowing through the circuit becomes nearly zero.

3. The current will be in phase with the voltage if $\omega ={10}^{4}rad.s^{-1}$

4. The frequency at which the current will be in phase with the voltage is independent of $R$

Q. A $2\mu \text{F}$ capacitor is charged as shown in figure. The percentage of its stored energy dissipated after switch $\text{S}$ is turned to position $\text{2}$ is $(8\mu F$ is the capacitance of air-filled parallel plate capacitor$)$:

1. $0\%$
2. $20\%$
3. $75\%$
4. $89\%$

Q. A coil having inductance $L$ and resistance $R$, is connected to a battery of emf $E$ and the circuit is closed at time, $t=0$. If $t_1$ and $t_2$ are time instants for $90\%$ and $99\%$ completion of current growth in the circuit, then $\frac{t_1}{t_2}$ will be equal to -

1. $1:3$
2. $3:1$
3. $2:1$
4. $1:2$

Q. A capacitor is charged and then made to discharge through a resistance. The time constant is $\tau$. In what time will the potential difference across the capacitor decreases by $10\%$ ?

1. $\tau \ln \left(0.1\right)$
2. $\tau \ln \left(0.9\right)$
3. $\tau \ln \left(\frac{10}{9}\right)$
4. $\tau \ln \left(\frac{11}{10}\right)$

Q. An uncharged capacitor of capacitance $5\mu \text{F}$ is connected to a battery of EMF $50\text{V}$ at $t=0$ through a resistance of $2\text{M}\Omega$, then the time at which rate of energy stored is maximum is -

1. $\left(10\ln 2\right)\text{s}$
2. $(\ln 2\text{)}\text{s}$
3. $\left(5\ln 2\right)\text{s}$
4. $\left(0.5\ln 2\right)\text{s}$

Q. A closed circuit consists of a source of constant emf $xi$ and a choke coil of inductance $L$ connected in series. The active resistance of the whole circuit is equal to $R$. It is in steady state. At the moment $t=0$ the choke coil inductance was decreased abruptly $4$ times its initial value. The current in the circuit as a function of time $t$ is $\xi /R[1+x{e}^{-4tR/L}]$. Find out value of $x$_______

Q. An inductor, a resistor and a battery are connected in series as shown in the figure. Find the time elapsed before the current reaches $99\%$ of the maximum value.

1. $3\ln \left(10\right)\text{ms}$
2. $2.5\ln \left(10\right)\text{ms}$
3. $\ln \left(10\right)\text{ms}$
4. $0.4\ln \left(10\right)\text{ms}$

Q. A capacitor $C$ is charged by a battery of $V$ volts . Then it is connected to an uncharged capacitor $2C$ as shown in figure. No match the following two columns.

Column-1 Column-2

$\text{(a)}$ After closing the switch energy stored in $C$.	$\text{(p)}\frac{1}{9}C{V}^2$
$\text{(b)}$ After closing the switch energy stored in $2C$.	$\text{(q)}\frac{1}{6}C{V}^2$
$\text{(c)}$ After closing the switch loss of energy during redistribution of charge.	$\text{(r)}\frac{1}{18}C{V}^2$
$\text{(s)}$ None of these

1. $\text{(a)}\to \text{(p)};\text{(b)}\to \text{(r)};\text{(c)}\to \text{(s)}$
2. $\text{(a)}\to \text{(r)};\text{(b)}\to \text{(p)};\text{(c)}\to \text{(s)}$
3. $\text{(a)}\to \text{(s)};\text{(b)}\to \text{(q)};\text{(c)}\to \text{(r)}$
4. $\text{(a)}\to \text{(r)};\text{(b)}\to \text{(q)};\text{(c)}\to \text{(r)}$

Q. The figure shows a series LCR circuit with $L=54H,C=80\mu F,R=40\Omega$ connected to a variable frequency $240V$ source, calculate
(i) the angular frequency of the source which drives the circuit at resonance,
(ii) the current at the resonating frequency,
(iii) the rms potential drop across the inductor at resonance.

Q. The instantaneous charges on capacitor in two discharging $RC$ circuits are plotted with respect to time as shown in figure below. Choose the correct statement(s) (where $E_1$ and $E_2$ are emfs of two DC sources in two different charging circuits and capacitors are fully charged initially).

1. $R_1{C}_1>R_2{C}_2$
2. $\frac{R_1}{R_2}<\frac{C_2}{C_1}$
3. $R_1>R_2\text{if}{E}_1=E_2$
4. $C_2>C_1\text{if}{E}_1=E_2$

Q. In the electrical network at $t<0$, key was placed on $\left(1\right),$ until the capacitor becomes fully charged at $t=0$.
Position of key is changed to $\left(2\right)$ at $t=0$. The energy in the capacitor and the inductor will be same for the first time is

1. $\frac{\pi }{4}\sqrt{LC}$
2. $\frac{3\pi }{4}\sqrt{LC}$
3. $\frac{\pi }{3}\sqrt{LC}$
4. $\frac{2\pi }{3}\sqrt{LC}$

Q.

An inductor-coil carries a steady-state current of 2.0 A when connected across an ideal battery of emf 4.0 V. If its inductance is 1.0 H, find the time constant of the circuit.

Q. A charged capacitor is connected with a resistor. After how many time constants, does the energy stored in the capacitor becomes ${\left(\frac{1}{10}\right)}^{\text{th}}$ of its initial value?
$[\text{Take}\log {{}_e}^{10}=2.303\log {{}_{10}}^{10}]$

1. $2.3$
2. $1.15$
3. $0.69$
4. None of the above

Q. A capacitor of capacitance $4\mu \text{F}$ is charged upto $80\text{V}$ and another capacitor of capacitance $6\mu \text{F}$ is changed upto $30\text{V}$. When they are connected together the energy lost by the $4\mu \text{F}$ capacitor is

1. $2.5\text{mJ}$
2. $3.2\text{mJ}$
3. $4.6\text{mJ}$
4. $7.8\text{mJ}$

Q.

How many time constants will elapse before the power delivered by the battery drops to half of its maximum value in an RC circuit ?

Q. A capacitor discharges through a resistance. The stored energy $U_0$ in one capacitive time constant falls to

1. $\frac{U_0}{e^2}$
2. $e{U}_0$
3. $\frac{U_0}{e}$
4. None of these

Q. In the figure shown below, switch $S$ is closed at $t=0$ with inductor fully charged. Find the charge $\left(q\right)$ on the capacitor as a function of time $\left(t\right)$.

$i_0$ is the maximum value of current in the circuit.

1. $q=\sqrt{LC}{i}_0\sin \left(\frac{2t}{\sqrt{LC}}\right)$
2. $q=\sqrt{LC}\sin \left(\frac{t}{\sqrt{LC}}\right)$
3. $q=\sqrt{LC}{i}_0\sin \left(\frac{t}{\sqrt{LC}}\right)$
4. $q=2\sqrt{LC}{i}_0\sin \left(\frac{t}{\sqrt{LC}}\right)$

Q. The circuit shown in the figure is a part of a big network. If at a certain instant of time, the current $\left(i\right)$ through it is $5\text{A}$ and is decreasing at the rate of ${10}^3\text{A/s}$, then $V_B-V_A$ at that instant is -

1. $50\text{V}$
2. $10\text{V}$
3. $15\text{V}$
4. $20\text{V}$

Q. A $5\mu \text{F}$ capacitor is charged fully by a $220\text{V}$ supply. It is then disconnected from the supply and is connected across another uncharged $2.5\mu \text{F}$ capacitor. If the energy change during the charge redistribution is $\frac{X}{100}\text{J}$ then the value of $X$ to the nearest integer is

Q. An inductor coil carries a steady state current of $2.0\text{A}$ when connected across an ideal battery of emf $4.0\text{V}$. If its inductance is $1.0\text{H}.$ Find the time constant of the circuit.

1. $1\text{s}$
2. $\frac{1}{2}\text{s}$
3. $\frac{3}{2}\text{s}$
4. $2\text{s}$

Q. The turns ratio ($r$) for a step-up transformer is

1. $r<1$
2. $r=1$
3. $r>1$
4. r $=$ 0

Q. Assertion : Current flowing in the circuit is $i=2t-8$.

At $t=1\text{s}$, $V_a-V_b=+4\text{V}$

Reason : $V_a-V_b$ is $+4\text{V}$ all the time.

For the above two statements, the correct alternative is -

1. Both assertion and reason are true, and the reason is the correct explanation of assertion
2. Both assertion and reason are true, and the reason is not the correct explanation of assertion
3. The assertion is true, but the reason is false
4. The assertion is false, but the reason is true

Q. A network consisting of three resistor, three batteries and a capacitor is shown in the figure. Charge on the capacitor C in steady state is

1. $10\mu c$
2. $12\mu c$
3. $12\mu c$
4. $12\mu c$