Space between the plates of a parallel plate capacitor of plate area 4 cm$^2$ and separation of $ d = 1.77 \, \text{mm} $, is filled with uniform dielectric materials with dielectric constants (3 and 5) as shown in figure. Another capacitor of capacitance 7.5 pF is connected in parallel with it. The effective capacitance of this combination is ____ pF.
Space between the plates of a parallel plate capacitor of plate area 4 cm$^2$ and separation of $ d = 1.77 \, \text{mm} $, is filled with uniform dielectric materials with dielectric constants (3 and 5) as shown in figure. Another capacitor of capacitance 7.5 pF is connected in parallel with it. The effective capacitance of this combination is ____ pF.
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Correct Answer: 15
Approach Solution - 1
The problem involves calculating the effective capacitance of a parallel plate capacitor arrangement with two dielectric materials and another capacitor connected in parallel.
First, the capacitance of the given capacitor with dielectric materials can be calculated. This capacitor is split into two capacitors in series due to the different dielectric constants.
Using the capacitance formula for a parallel plate capacitor: \( C = \frac{k\varepsilon_0A}{d} \), where \( k \) is the dielectric constant, \( \varepsilon_0 \) is the permittivity of free space \((8.85 \times 10^{-12} \, \text{F/m})\), \( A \) is the area, and \( d \) is the separation.
Calculate the capacitance for each layer:
- Top layer (\( k_1 = 5 \)): \( C_1 = \frac{k_1\varepsilon_0A}{d/2} = \frac{5 \times 8.85 \times 10^{-12} \times 4 \times 10^{-4}}{0.885 \times 10^{-3}} \)
- Bottom layer (\( k_2 = 3 \)): \( C_2 = \frac{k_2\varepsilon_0A}{d/2} = \frac{3 \times 8.85 \times 10^{-12} \times 4 \times 10^{-4}}{0.885 \times 10^{-3}} \)
Calculate \( C_1 \) and \( C_2 \):
- \( C_1 = 2 \times 10^{-12} \, \text{F} \)
- \( C_2 = 1.2 \times 10^{-12} \, \text{F} \)
The series combination \( C_s \) is given by:
\( \frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} \implies C_s = \frac{C_1 \times C_2}{C_1 + C_2} = \frac{(2 \times 10^{-12}) \times (1.2 \times 10^{-12})}{2 \times 10^{-12} + 1.2 \times 10^{-12}} = 0.75 \times 10^{-12} \, \text{F} \)
Convert to picofarads: \( C_s = 7.5 \, \text{pF} \).
Another 7.5 pF capacitor is connected in parallel, so the effective capacitance \( C_{\text{eff}} \) is:
\( C_{\text{eff}} = C_s + 7.5 = 7.5 + 7.5 = 15 \, \text{pF} \).
The computed capacitance falls within the given range of [15,15] pF.
Therefore, the effective capacitance is 15 pF.
Approach Solution -2
- Separation between the plates, \( d = 1.77 \, \text{mm} = 1.77 \times 10^{-3} \, \text{m} \),
- Dielectric constants for the two regions, \( k_1 = 5 \) and \( k_2 = 3 \), - Capacitance of an additional parallel capacitor, \( C_3 = 7.5 \, \text{pF} \),
- Permittivity of free space, \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \).
Step 1: Calculate the capacitance for each section with the dielectric constants. For the first section (with dielectric constant \( k_1 = 5 \)): The capacitance is given by: \[ C_1 = \frac{\varepsilon_0 A}{d/2} \cdot k_1 \] Substitute the values: \[ C_1 = \frac{(8.85 \times 10^{-12}) \times (4 \times 10^{-4})}{1.77 \times 10^{-3}/2} \cdot 5 \] First calculate the denominator \( d/2 = \frac{1.77 \times 10^{-3}}{2} = 8.85 \times 10^{-4} \, \text{m} \). Now substitute: \[ C_1 = \frac{8.85 \times 10^{-12} \times 4 \times 10^{-4}}{8.85 \times 10^{-4}} \cdot 5 = 4 \times 10^{-12} \times 5 = 20 \times 10^{-12} \, \text{F} = 20 \, \text{pF} \] For the second section (with dielectric constant \( k_2 = 3 \)): Using the same formula: \[ C_2 = \frac{\varepsilon_0 A}{d/2} \cdot k_2 \] Substitute the values: \[ C_2 = \frac{(8.85 \times 10^{-12}) \times (4 \times 10^{-4})}{1.77 \times 10^{-3}/2} \cdot 3 \] \[ C_2 = \frac{8.85 \times 10^{-12} \times 4 \times 10^{-4}}{8.85 \times 10^{-4}} \cdot 3 = 4 \times 10^{-12} \times 3 = 12 \times 10^{-12} \, \text{F} = 12 \, \text{pF} \]
Step 2: Find the total capacitance of the two dielectric regions in series. The two dielectric regions are in series, so the total capacitance \( C_{\text{total}} \) is given by: \[ \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} \] Substitute the values for \( C_1 \) and \( C_2 \): \[ \frac{1}{C_{\text{total}}} = \frac{1}{20 \, \text{pF}} + \frac{1}{12 \, \text{pF}} \] \[ \frac{1}{C_{\text{total}}} = \frac{3}{60 \, \text{pF}} + \frac{5}{60 \, \text{pF}} = \frac{8}{60 \, \text{pF}} = \frac{2}{15 \, \text{pF}} \] Thus, \[ C_{\text{total}} = \frac{15}{2} \, \text{pF} = 7.5 \, \text{pF} \]
Step 3: Add the capacitance of the additional parallel capacitor. The total capacitance of the system is the sum of the series capacitance and the parallel capacitance \( C_3 \): \[ C_{\text{effective}} = C_{\text{total}} + C_3 \] \[ C_{\text{effective}} = 7.5 \, \text{pF} + 7.5 \, \text{pF} = 15 \, \text{pF} \] Thus, the effective capacitance is \( \boxed{15 \, \text{pF}} \).
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