Capacitor Charge Calculation
This problem involves calculating the charge redistribution between two capacitors connected in parallel after one is initially charged.
Initial Charge Calculation
- Capacitor P has capacitance $C_P = 10 \times 10^{-6} \text{ F}$ and initial potential difference $V_P = 6.0 \text{ V}$.
- The initial charge on capacitor P is calculated using the formula $Q = CV$.
- $Q_P = C_P \times V_P = (10 \times 10^{-6} \text{ F}) \times (6.0 \text{ V}) = 60 \times 10^{-6} \text{ C}$.
- Capacitor Q has capacitance $C_Q = 20 \times 10^{-6} \text{ F}$ and is initially uncharged ($Q_Q = 0$).
Charge Redistribution and Equilibrium
When the charged capacitor P is connected across capacitor Q, they form a parallel circuit. Charge flows from P to Q until they reach a common final potential difference, $V_f$. Charge is conserved during this process.
- Total initial charge = Charge on P + Charge on Q = $60 \times 10^{-6} \text{ C} + 0 = 60 \times 10^{-6} \text{ C}$.
- The equivalent capacitance in parallel is $C_{eq} = C_P + C_Q = (10 \times 10^{-6}) + (20 \times 10^{-6}) = 30 \times 10^{-6} \text{ F}$.
- The final common potential difference $V_f$ is found using $V_f = \frac{\text{Total Charge}}{C_{eq}}$.
- $V_f = \frac{60 \times 10^{-6} \text{ C}}{30 \times 10^{-6} \text{ F}} = 2.0 \text{ V}$.
Final Charge on Capacitor Q
The charge on capacitor Q at equilibrium ($Q_{Q,f}$) is calculated using its capacitance and the final common potential:
- $Q_{Q,f} = C_Q \times V_f = (20 \times 10^{-6} \text{ F}) \times (2.0 \text{ V}) = 40 \times 10^{-6} \text{ C}$.
- The question states this charge is $\alpha \times 10^{-5} \text{ C}$.
- Therefore, $40 \times 10^{-6} \text{ C} = \alpha \times 10^{-5} \text{ C}$.
- Rewriting $40 \times 10^{-6}$ as $4 \times 10^{-5}$, we get $4 \times 10^{-5} \text{ C} = \alpha \times 10^{-5} \text{ C}$.
- Comparing both sides gives $\alpha = 4$.
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 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 ______ $\Omega$.
Match the LIST-I with LIST-II
List-I: List-II: A. Radio-wave I. is produced by Magnetron valve B. Micro-wave II. due to change in the vibrational modes of atoms C. Infrared-wave III. due to inner shell electrons moving from higher energy level to lower energy level D. X-ray IV. due to rapid acceleration of electrons Choose the correct answer from the options given below:
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 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 :
