Combination Of Capacitors

For a parallel plate capacitor, the capacitance is given by: where:
- is the permittivity of free space,
- is the area of the plates,
- is the separation between the plates. When a dielectric is inserted between the plates, the capacitance increases by a factor of , the dielectric constant. The cases can be analyzed as follows: Case (a) In this case, the capacitor is filled with the dielectric constant in half of the space between the plates. We treat the capacitor as two capacitors in series, one filled with the dielectric and the other without it.
- The first capacitor has a dielectric material with constant and a distance of .
- The second capacitor has no dielectric material and also has a distance of . The capacitance of the first capacitor is: The capacitance of the second capacitor is: Since the capacitors are in series, the total capacitance is: Substituting the values of and : Simplifying: Thus, the total capacitance is: Case (b) In this case, the entire space between the plates is filled with the dielectric material with dielectric constant . The capacitance is simply:

Step 1: Voltage Limit Per Capacitor
Each capacitor can handle a maximum of 200 V. The supply voltage is 1200 V. To ensure no capacitor exceeds its voltage rating, the 1200 V must be distributed across capacitors in series. Let be the number of capacitors in series, then: So, we must use at least 6 capacitors in series to tolerate the full 1200 V. Step 2: Capacitance of Series Combination
Capacitance of capacitors in series (each of 1  ) is: Step 3: Achieving 2  Equivalent Capacitance
Let be the number of such series combinations connected in parallel. Parallel combination adds capacitances: Step 4: Total Capacitors Required
Each row (series) has 6 capacitors and there are 12 such rows in parallel. Total capacitors = % Final Answer Statement Answer: Minimum 72 capacitors are required, arranged as 12 rows of 6 capacitors in series, connected in parallel.