Series Capacitors with Voltage Rating Limit

← Back to Problems
Electrostatics Intermediate Capacitor

Source: High school physics (Chinese)

Problem

Two capacitors are rated as $C_1: 200\ \mathrm{pF}, 500\ \mathrm{V}$ and $C_2: 300\ \mathrm{pF}, 900\ \mathrm{V}$.

Can the two capacitors safely be connected in series and have a voltage of $1000$ V applied across the combination?

No --- $C_1$ would see $600$ V, exceeding its $500$ V rating. The maximum safe series voltage is only about $833$ V.

In series, both capacitors carry the same charge $Q$, so the voltages split inversely with capacitance: $V_1 : V_2 = 1/C_1 : 1/C_2 = C_2 : C_1 = 300 : 200 = 3 : 2$. For a total of $1000$ V:

$$V_1 = \dfrac{3}{5} \times 1000 = 600\ \mathrm{V}, \qquad V_2 = \dfrac{2}{5} \times 1000 = 400\ \mathrm{V}.$$

The voltage across $C_1$ is $600$ V, which exceeds its $500$ V rating; $C_1$ would break down. The combination is not safe at $1000$ V.

(For reference, the maximum safe series voltage is limited by $C_1$: when $V_1 = 500$ V (max for $C_1$), $V_2 = (2/3) \times 500 \approx 333$ V, giving a total of about $833$ V.)