Question

Minimum number of capacitors each of \[8\,\mu F\] and 250 V used to make a composite capacitor of \[16\,\mu F\] and 1000 V are
A. 8
B. 32
C. 16
D. 24

Answer 1

Hint: To make the voltage tolerance 1000 V, the four capacitors each of voltage 250 V should be connected in series. But the series combination of four \[8\,\mu F\] capacitors will give an equivalent capacitance of \[2\,\mu F\]. Thus, we have to form n set of parallel arrangements of series of 4 capacitors of \[8\,\mu F\].

Formula used:
Equivalent capacitance in series arrangement of capacitors,
\[{C_{eq}} = {C_1} + {C_2} + ...... + {C_n}\]
where, C is the capacitance of each capacitor.

Complete step by step answer:
To answer this question, we first need to know that the voltage supplies connected in a series adds up. Also, the voltage is the parallel arrangement of any electrical components that remains constant.We have the formula for the equivalent capacitance of the number of capacitors connected in a series,
\[\dfrac{1}{{{C_{eq}}}} = \dfrac{1}{{{C_1}}} + \dfrac{1}{{{C_1}}} + ..... + \dfrac{1}{{{C_n}}}\]
Also, the formula for the equivalent capacitance of the number of capacitors connected in a parallel combination,
\[{C_{eq}} = {C_1} + {C_2} + ...... + {C_n}\]
Thus, we can see, the capacitance of the parallel combination increases with connecting more capacitors in the parallel arrangement.Now, to have 1000 V tolerance, we need to stack up 4 capacitors of \[8\,\mu F\]in series so that we can get,
\[250\,V + 250\,V + 250\,V + 250\,V = 1000\,{\text{V}}\]
But, connecting 4 capacitors of \[8\,\mu F\] in series would make an equivalent capacitance of,
\[\dfrac{1}{{{C_{eq}}}} = \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} + \dfrac{1}{8} = \dfrac{1}{2}\]
\[ \Rightarrow {C_{eq}} = 2\,\mu F\]
But, we need the equivalent capacitance of \[16\,\mu F\]. Thus, to make the total of \[16\,\mu F\] that can tolerate 1000 V, we need 8 sets of such series combination of 4 capacitors arranged in parallel combination. Therefore, the total number of capacitors is,\[8 \times 4 = 32\] capacitors.In this way, the voltage across the capacitors will remain 1000 V since the voltage across the parallel combination does not change.

So, the correct answer is option B.

Note:The crucial step in the solution is to know that the voltage across the parallel combination remains the same and the voltage increases in the series of emf supplies. Note that the equivalent resistance and equivalent capacitance have quite opposite formulae since the capacitance decreases in the series and the resistance increases in the series combination.