Question Details

Five capacitors of capacitances C1 = C2 = C3 = C4 = 10µF and C5 = 2.5µF are connected as shown, along with a battery of 50 V. The equivalent capacitance and the charges on each capacitor respectively are: ____.


Options

A

5 µF, 125 µC on C1  to C4 and 25µC on C5

B

4µF, 250 µC on C1 to C4 and 125µC on C5

C

5µF, 250 µC on all capacitors

D

5 µF, 125 µC on all capacitors

Show Answer

Correct Answer :

Option D

5 µF, 125 µC on all capacitors

5 µF, 125 µC on all capacitors

Solution :

The diagram supplied in the image shows the five capacitors connected to a 50 V battery as follows:

‑ Capacitors C1, C2, C3 and C4 (each 10 µF) are arranged **in series**.

‑ The series string is placed **in parallel** with capacitor C5 (2.5 µF).

Thus the overall circuit consists of a series group of four equal capacitors and a single capacitor that shares the same terminals as that group.

**Step 1: Find the equivalent capacitance of the series group.**

C_{\text{series}} = \frac{1}{\displaystyle\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\frac{1}{C_4}}

Since C1=C2=C3=C4=10 µF,

C_{\text{series}} = \frac{1}{\displaystyle 4\left(\frac{1}{10\,\mu\text{F}}\right)} = \frac{1}{0.4\,\mu\text{F}^{-1}} = 2.5\,\mu\text{F}

**Step 2: Combine with C5 in parallel.**

C_{\text{eq}} = C_{\text{series}} + C_5 = 2.5\,\mu\text{F} + 2.5\,\mu\text{F} = 5\,\mu\text{F}

**Step 3: Determine the voltage across each branch.**

The series group and C5 are in parallel, so they share the full battery voltage:

V_{\text{branch}} = V_{\text{battery}} = 50\ \text{V}

**Step 4: Find the charge on each capacitor.**

For the parallel capacitor C5:

Q_5 = C_5 \, V_{\text{branch}} = 2.5\,\mu\text{F} \times 50\ \text{V} = 125\ \mu\text{C}

For the series capacitors (C1–C4):

In a series connection the charge on each capacitor is the same. The voltage across each of the four equal‑capacitance capacitors is one‑fourth of the total branch voltage:

V_{C_i} = \frac{V_{\text{branch}}}{4} = \frac{50\ \text{V}}{4} = 12.5\ \text{V}

Hence the charge on any Ci (i = 1…4) is

Q_i = C_i \, V_{C_i} = 10\,\mu\text{F} \times 12.5\ \text{V} = 125\ \mu\text{C}

All five capacitors therefore store the same charge of 125 µC.

**Result:** The equivalent capacitance of the network is 5 µF, and each capacitor (C1 through C5) carries a charge of 125 µC, exactly matching the given correct option.

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