| A parallel plate capacitor is made of two dielectric blocks in series. One of the blocks has thickness \[{{d}_{1}}\] and dielectric constant \[{{k}_{1}}\] and the other has thickness \[{{d}_{2}}\] and dielectric constant \[{{k}_{2}}\]as shown in Figure. This arrangement can be thought as a dielectric slab of thickness d (=\[{{d}_{1}}\]+\[{{d}_{2}}\]) and effective dielectric constant k. The k is: |
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A) \[\frac{{{k}_{1}}{{d}_{1}}+{{k}_{2}}{{d}_{2}}}{{{d}_{1}}+{{d}_{2}}}\]
B) \[\frac{{{k}_{1}}{{d}_{1}}+{{k}_{2}}{{d}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]
C) \[\frac{{{k}_{1}}{{k}_{2}}({{d}_{1}}+{{d}_{2}})}{({{k}_{1}}{{d}_{1}}+{{k}_{2}}{{d}_{2}})}\]
D) \[\frac{2{{k}_{1}}2{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}\]
Correct Answer: C
Solution :
| Option [c] is correct |
| Explanation: Capacitance of a parallel plate capacitor filled with dielectric of constant \[{{k}_{1}}\] and thickness \[{{d}_{1}}\] is, |
| \[{{\operatorname{C}}_{1}}=\frac{{{k}_{1}}{{\varepsilon }_{0}}\operatorname{A}}{{{d}_{1}}}\] |
| Similarly, for other capacitance of a parallel plate capacitor filled with dielectric of constant \[{{k}_{2}}\] and thickness \[{{d}_{2}}\]is, |
| \[{{\operatorname{C}}_{2}}=\frac{{{k}_{2}}{{\varepsilon }_{0}}\operatorname{A}}{{{d}_{2}}}\] |
| Both capacitors are in series so equivalent capacitance C is related as : |
| \[\frac{1}{C}=\frac{1}{{{C}_{1}}}+\frac{1}{{{C}_{2}}}=\frac{{{d}_{1}}}{{{k}_{1}}{{\varepsilon }_{0}}\operatorname{A}}+\frac{{{d}_{2}}}{{{k}_{2}}{{\varepsilon }_{0}}\operatorname{A}}\] |
| \[=\frac{1}{{{\varepsilon }_{0}}\operatorname{A}}\left[ \frac{{{k}_{2}}{{d}_{1}}+{{k}_{1}}{{d}_{2}}}{{{k}_{1}}{{k}_{2}}} \right]\] |
| So, \[\operatorname{C}=\frac{{{k}_{1}}{{k}_{2}}{{\varepsilon }_{0}}A}{({{k}_{1}}{{d}_{2}}+{{k}_{2}}{{d}_{1}})}\] .(i) |
| \[\operatorname{C}'=\frac{k{{\varepsilon }_{0}}A}{d}=\frac{k{{\varepsilon }_{0}}A}{({{d}_{1}}+{{d}_{2}})}\] .(ii) |
| where, d = \[({{d}_{1}}+{{d}_{2}})\] |
| Comparing eqns. (i) and (ii), the dielectric constant of new capacitor is : |
| \[k=\frac{{{k}_{1}}{{k}_{2}}({{d}_{1}}+{{d}_{2}})}{{{k}_{1}}{{d}_{2}}+{{k}_{2}}{{d}_{1}}}\] |
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