The equivalent capacitance of three capacitors of capacitance ${C_1},{C_2}$ and ${C_3}$ are connected in parallel is $12$ units and product ${C_1}.{C_2}.{C_3} = 48$. When the capacitors ${C_1}$ and ${C_2}$ are connected in parallel, the equivalent capacitance is $6$ units. Then the capacitance are
Medium
Download our app for free and get started
(d) ${C_1}+{C_2}+{C_3} = 12$....$(i)$
${C_1}.{C_2}.{C_3} = 48$....$(ii)$
$C_1 + C_2 = 6$ ....$(iii)$
From equation $(i)$ and $(iii)$
$C_3 = 6$ ....$(iv)$
From equation $(ii)$ and $(iv)$ $C_1C_2 = 8$
Also ${({C_1} - {C_2})^2} = {({C_1} + {C_2})^2} - 4{C_1}{C_2}$
${({C_1} - {C_2})^2} = {(6)^2} - 4 \times 8 = 4$
$==>$ $ C_1 -C_2 = 2$.....$(v)$
On solving $(iii)$ and $(v)$ $C_1 = 4,\,\, C_2 = 2 $
${C_1}.{C_2}.{C_3} = 48$....$(ii)$
$C_1 + C_2 = 6$ ....$(iii)$
From equation $(i)$ and $(iii)$
$C_3 = 6$ ....$(iv)$
From equation $(ii)$ and $(iv)$ $C_1C_2 = 8$
Also ${({C_1} - {C_2})^2} = {({C_1} + {C_2})^2} - 4{C_1}{C_2}$
${({C_1} - {C_2})^2} = {(6)^2} - 4 \times 8 = 4$
$==>$ $ C_1 -C_2 = 2$.....$(v)$
On solving $(iii)$ and $(v)$ $C_1 = 4,\,\, C_2 = 2 $
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1In the following circuit, the resultant capacitance between $A$ and $B$ is $1\,\mu F$. Then value of $C$ isView Solution
- 2View SolutionAssertion : Two equipotential surfaces cannot cut each other.
Reason : Two equipotential surfaces are parallel to each other.
- 3In the $R-C$ circuit shown in the figure the total energy of $3.6 \times 10^{-3}\ J$ is dissipated in the $10$ $\Omega$ resistor when the switch $S$ is closed. The initial charge on the capacitor is.....$\mu C$View Solution
- 4A parallel plate capacitor having a separation between the plates $d$ , plate area $A$ and material with dielectric constant $K$ has capacitance $C_0$. Now one-third of the material is replaced by another material with dielectric constant $2K$, so that effectively there are two capacitors one with area $\frac{1}{3}\,A$ , dielectric constant $2K$ and another with area $\frac{2}{3}\,A$ and dielectric constant $K$. If the capacitance of this new capacitor is $C$ then $\frac{C}{{{C_0}}}$ isView Solution
- 5If $50$ joule of work must be done to move an electric charge of $2 \,C$ from a point, where potential is $-10$ volt to another point, where potential is $V$ volt, the value of $V$ is ......... $V$View Solution
- 6Effective capacitance between $A$ and $B$ in the figure shown is (all capacitance are in $\mu F$)View Solution
- 7An electric dipole of moment $p$ is placed in an electric field of intensity $E.$ The dipole acquires a position such that the axis of the dipole makes an angle $\theta$ with the direction of the field. Assuming that the potential energy of the dipole to be zero when $\theta = 90^o$ , the torque and the potential energy of the dipole will respectively beView Solution
- 8A capacitor stores $60\ \mu C$ charge when connected across a battery. When the gap between the plates is filled with a dielectric , a charge of $120\ \mu C$ flows through the battery , if the initial capacitance of the capacitor was $2\ \mu F$, the amount of heat produced when the dielectric is inserted.......$\mu J$View Solution
- 9View SolutionA small conducting sphere is hanged by an insulating thread between the plates of a parallel plate capacitor as shown in figure. The net force on the sphere is
- 10Two identical thin rings each of radius $R$ meters are coaxially placed at a distance $R$ meters apart. If $Q_1$ coulomb and $Q_2$ coulomb are respectively the charges uniformly spread on the two rings, the work done in moving a charge $q$ from the centre of one ring to that of other isView Solution