A series combination of $10$ capacitors, each of value ' $C_1$ ' is charged by a source of potential difference ' $4\text{ V}$ '. When another parallel combination of $8$ capacitors, each of value ' $C_2$ ' is charged by a source of potential difference ' $V$ ', it has the same total energy stored in it as in the first combination. The value of ' $C_2$ ' is
Show Hint
- $\frac{C_1}{5}$
- $\frac{8}{5} C_1$
- $\frac{64}{5} C_1$
- $\frac{C_1}{40}$
The Correct Option is A
Solution and Explanation
Case 1: 10 identical capacitors, each of capacitance \(C_1\), connected in series
For capacitors connected in series,
\[ \frac{1}{C_{\text{eq1}}} = \frac{1}{C_1} + \frac{1}{C_1} + \cdots \text{ (10 terms)} = \frac{10}{C_1} \]
Hence,
\[ C_{\text{eq1}}=\frac{C_1}{10} \]
The combination is connected across a potential difference of \(4\,\text{V}\).
Energy stored in a capacitor is:
\[ U=\frac{1}{2}CV^2 \]
Therefore,
\[ U_1 = \frac{1}{2}\left(\frac{C_1}{10}\right)(4)^2 \]
\[ U_1 = \frac{1}{2}\times\frac{C_1}{10}\times16 \]
\[ U_1 = \frac{16C_1}{20} = \frac{4C_1}{5} \]
\[ U_1=0.8C_1 \]
Case 2: 8 identical capacitors, each of capacitance \(C_2\), connected in parallel
For capacitors connected in parallel,
\[ C_{\text{eq2}} = C_2+C_2+\cdots \text{ (8 terms)} = 8C_2 \]
If this combination is connected across a potential difference \(V\), then the energy stored is:
\[ U_2 = \frac{1}{2}C_{\text{eq2}}V^2 \]
\[ U_2 = \frac{1}{2}(8C_2)V^2 \]
\[ U_2 = 4C_2V^2 \]
According to the question, both combinations store the same energy.
So,
\[ U_1=U_2 \]
\[ \frac{4C_1}{5}=4C_2V^2 \]
Dividing both sides by \(4\),
\[ \frac{C_1}{5}=C_2V^2 \]
Therefore,
\[ C_2=\frac{C_1}{5V^2} \]
If \(V=1\),
\[ C_2=\frac{C_1}{5} \]
Final Answer:
\[ \boxed{\frac{C_1}{5}} \]
Top MHT CET Physics Questions
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- A particle moves along a circle of radius ?$r$? with constant tangential acceleration. If the velocity of the particle is ?$?$? at the end of second revolution, after the revolution has started then the tangential acceleration is
- If $M_z$ = magnetization of a paramagnetic sample, $B$ = external magnetic field, $T$ = absolute temperature, $C$ = curie constant then according to Curie�s law in magnetism, the correct relation is
- In a fundamental mode, the time required for the sound wave to reach upto the closed end of a pipe filled with air is $'t'$ second. The frequency of vibration of air column is
- In a parallel plate capacitor, the capacity increases if
Top MHT CET Capacitors and Capacitance Questions
- A $ 20\,mu F $ capacitor is connected to $ 45\,V $ battery through a circuit whose resistance is $ 2000\, \Omega $ . What is the final charge on the capacitor ?
- Energy stored in a capacitor is given by the equation \[ E = \frac{1}{2} C V^2 \] where: - \( C \) is the capacitance, - \( V \) is the voltage, - \( E \) is the energy stored. Given the values of \( C \), \( V \), and \( E \), determine the energy stored.}
- Determine the equivalent capacitance of an infinite circuit formed by repeating identical capacitors of capacitance \(C\).
- Calculate the equivalent capacitance of an infinite circuit formed by repeating identical capacitors of capacitance \(C\).
- Find the energy stored in a capacitor of \(10\,\mu F\) charged to a potential of \(50\,V\).
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?