Combination of four parallel plate air capacitors is shown in figure. Separation between the plate P1 & P2 is $\frac{d}{3}$ and separation between the plates P2 and P3 is d. The equivalent capacitance of the configuration is _____________$\mu$F.
(Given $\frac{\epsilon_0 A}{d} = 4\mu$F . Where A=Area of plates)
To find the equivalent capacitance of the configuration shown in the figure, follow these steps:
1. Analyze the plate arrangement:
- The configuration is divided into two main gaps: the gap between P1 and P2 (separation $d/3$) and the gap between P2 and P3 (separation $d$).
- Plate P1 is composed of three area segments: $A/6$, $A/3$, and $A/2$. The total area of P1 is $\frac{A}{6} + \frac{A}{3} + \frac{A}{2} = A$.
- Middle plate P2 and bottom plate P3 also span the total area $A$.
2. Identify the series and parallel combinations:
- The three capacitive regions between P1 and P2 are in parallel with each other. Their combined capacitance ($C_{12}$) is:
$C_{12} = \frac{\epsilon_0 (A/6)}{d/3} + \frac{\epsilon_0 (A/3)}{d/3} + \frac{\epsilon_0 (A/2)}{d/3}$
$C_{12} = \frac{\epsilon_0 A}{d/3} = 3 \left(\frac{\epsilon_0 A}{d}\right)$ - The capacitive region between P2 and P3 ($C_{23}$) is:
$C_{23} = \frac{\epsilon_0 A}{d}$ - Since terminal 'a' is at P1 and the bottom terminal is at P3, the combined top section ($C_{12}$) and the bottom section ($C_{23}$) are in series.
3. Calculate the numerical values:
Given $\frac{\epsilon_0 A}{d} = 4\ \mu\text{F}$:
- $C_{12} = 3 \times 4\ \mu\text{F} = 12\ \mu\text{F}$
- $C_{23} = 4\ \mu\text{F}$
4. Calculate the equivalent capacitance ($C_{eq}$):
For two capacitors in series:
$C_{eq} = \frac{C_{12} \times C_{23}}{C_{12} + C_{23}}$
$C_{eq} = \frac{12 \times 4}{12 + 4}$
$C_{eq} = \frac{48}{16} = 3\ \mu\text{F}$
The equivalent capacitance of the configuration is 3 $\mu$F.
Similar Questions
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Important Questions from Electricity and Magnetism
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$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - An inductor stores $16\text{ J}$ of magnetic field energy and dissipates $32\text{ W}$ of thermal energy due to its resistance when an a.c. current of $2\text{ A}$ (rms) and frequency $50\text{ Hz}$ flows through it. The ratio of inductive reactance to its resistance is _________. ($\pi = 3.14$)
- A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^\circ$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ___________ mV. (Take $g = 10\text{ m/s}^2$)
Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is ___________ $\mu\text{F}$.
