A capacitor $C_1 = 1.0 \mu F$ is charged up to a voltage $V = 60$ V by connecting it to battery B through switch (1). Now $C_1$ is disconnected from battery and connected to a circuit consisting of two uncharged capacitors $C_2 = 3.0 \mu F$ and $C_3 = 6.0 \mu F$ through switch (2), as shown in the figure. The sum of final charges on $C_2$ and $C_3$ is :
To solve this problem, we need to calculate the total charge on the capacitors \(C_2\) and \(C_3\) after they are connected to the charged capacitor \(C_1\). Here are the steps to solve the problem:
Step 1: Calculate the initial charge on \(C_1\).
The charge \(Q_1\) on the capacitor \(C_1\) can be determined using the formula:
\(Q = CV\)
Where:
- \(C = 1.0 \, \mu \text{F}\)
- \(V = 60 \, \text{V}\)
Thus,
\(Q_1 = 1.0 \times 60 = 60 \, \mu \text{C}\)
Step 2: Determine the equivalent capacitance when \(C_1\) is connected to \(C_2\) and \(C_3\).
The capacitors \(C_2\) and \(C_3\) are connected in series, so the equivalent capacitance \(C_{\text{eq}}\) is given by:
\(\frac{1}{C_{\text{eq}}} = \frac{1}{C_2} + \frac{1}{C_3}\)
Substituting the given values:
\(\frac{1}{C_{\text{eq}}} = \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\)
Thus, \(C_{\text{eq}} = 2 \, \mu \text{F}\)
Step 3: Calculate the final charge distribution.
When \(C_1\) is connected to the equivalent capacitance \(C_{\text{eq}}\), the total charge is conserved. The initial charge on \(C_1\) is now distributed over \(C_1\) and the series combination \(C_2\) and \(C_3\).
The new combined capacitance becomes:
\(C_{\text{total}} = C_1 + C_{\text{eq}} = 1 + 2 = 3 \, \mu \text{F}\)
The voltage across the capacitors is:
\(V_{\text{final}} = \frac{Q_1}{C_{\text{total}}} = \frac{60}{3} = 20 \, \text{V}\)
Hence, the charge on the series combination \((Q_2 = Q_3)\) is:
\(Q_2 = C_{\text{eq}} \times V_{\text{final}} = 2 \times 20 = 40 \, \mu \text{C}\)
Conclusion: The sum of the final charges on \(C_2\) and \(C_3\) is \(40 \, \mu \text{C}\).
Similar Questions
- The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by,
$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(where x, t and other values have S.I. units). The dielectric constant of the medium is _________.
(speed of light in free space is $3 \times 10^8 \text{ m/s}$) Figure shows the circuit that contains three resistances ($9 \, \Omega$ each) and two inductors (4 mH each). The reading of ammeter at the moment switch K is turned ON, is _________ A.
- A capacitor $P$ with capacitance $10 \times 10^{-6} \text{ F}$ is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor $P$ is connected across another capacitor $Q$ with capacitance $20 \times 10^{-6} \text{ F}$. The charge on capacitor $Q$ when equilibrium is established will be $\alpha \times 10^{-5} \text{ C}$ (assume capacitor $Q$ does not have any charge initially), the value of $\alpha$ is _________.
- A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$ - A cylindrical conductor of length 2 m and area of cross-section $0.2 \text{ mm}^2$ carries an electric current of 1.6 A when its ends are connected to a 2 V battery. Mobility of electrons in the conductor is $\alpha \times 10^{-3} \text{ m}^2\text{/V.s}$. The value of $\alpha$ is :
(electron concentration $= 5 \times 10^{28} \text{/m}^3$ and electron charge = $1.6 \times 10^{-19} \text{ C}$) - There are three co-centric conducting spherical shells $A$, $B$ and $C$ of radii $a$, $b$ and $c$ respectively ($c > b > a$) and they are charged with charge $q_1$, $q_2$ and $q_3$ respectively. The potentials of the spheres $A$, $B$ and $C$ respectively, are :
For the series $LCR$ circuit connected with 220 V, 50 Hz a.c source as shown in the figure, the power factor is $\frac{\alpha}{10}$. The value of $\alpha$ is ______.
- Two resistors of $100\, \Omega$ each are connected in series with a 9 V battery. A voltmeter of $400\, \Omega$ resistance is connected to measure the voltage drop across one of the resistors. The voltmeter reading is ______ V.
- The electrostatic potential in a charged spherical region of radius $r$ varies as $V = ar^3 + b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $\alpha \times \pi a \epsilon_0$. The value of $\alpha$ is ______.
(permittivity of vacuum is $\epsilon_0$) Two resistors $2\, \Omega$ and $3\, \Omega$ are connected in the gaps of bridge as shown in figure. The null point is obtained with the contact of jockey at some point on wire $XY$. When an unknown resistor is connected in parallel with $3\, \Omega$ resistor, the null point is shifted by 22.5 cm toward $Y$. The resistance of unknown resistor is ______ $\Omega$.
Important Questions from Electricity and Magnetism
- The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by,
$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(where x, t and other values have S.I. units). The dielectric constant of the medium is _________.
(speed of light in free space is $3 \times 10^8 \text{ m/s}$) Figure shows the circuit that contains three resistances ($9 \, \Omega$ each) and two inductors (4 mH each). The reading of ammeter at the moment switch K is turned ON, is _________ A.
- A capacitor $P$ with capacitance $10 \times 10^{-6} \text{ F}$ is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor $P$ is connected across another capacitor $Q$ with capacitance $20 \times 10^{-6} \text{ F}$. The charge on capacitor $Q$ when equilibrium is established will be $\alpha \times 10^{-5} \text{ C}$ (assume capacitor $Q$ does not have any charge initially), the value of $\alpha$ is _________.
- A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$ - A cylindrical conductor of length 2 m and area of cross-section $0.2 \text{ mm}^2$ carries an electric current of 1.6 A when its ends are connected to a 2 V battery. Mobility of electrons in the conductor is $\alpha \times 10^{-3} \text{ m}^2\text{/V.s}$. The value of $\alpha$ is :
(electron concentration $= 5 \times 10^{28} \text{/m}^3$ and electron charge = $1.6 \times 10^{-19} \text{ C}$)
