Obtain the equivalent capacitance of the network. for a 300v supply determine the charge and voltage across c1 and c2

[anonymous6]

obtain the equivalent capacitance of the network. for a 300v supply determine the charge and voltage across c1 and c2.

[StudyQbot]

Obtain the Equivalent Capacitance of the Network

When dealing with capacitors in a circuit, understanding how to calculate the equivalent capacitance is essential. The configuration of capacitors in the circuit determines whether they are in series or parallel, which affects the total capacitance and how you calculate it.

1. Capacitors in Series and Parallel

• Series Connection: When capacitors are arranged in a sequence such that the end of one capacitor connects to the start of the next, they are in series.

  • The formula for calculating the equivalent capacitance (C_{\text{eq,series}}) for capacitors in series (C_1, C_2, C_3, …) is:
    \frac{1}{C_{\text{eq,series}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots
  • The equivalent capacitance for capacitors in series is always less than the smallest capacitance in the series.

• Parallel Connection: When capacitors are connected such that the start of each capacitor connects to a common point and the ends connect to another common point, they are in parallel.

  • The formula for calculating the equivalent capacitance (C_{\text{eq,parallel}}) for capacitors in parallel is:
    C_{\text{eq,parallel}} = C_1 + C_2 + C_3 + \ldots
  • The equivalent capacitance for capacitors in parallel is the sum of the individual capacitances.

2. Determining Charge and Voltage Across Capacitors

For capacitors, the fundamental relationship between charge (Q), capacitance (C), and voltage (V) is given by the equation:

Q = C \times V

This equation implies that for a capacitor with a certain capacitance, the amount of electric charge it holds is directly proportional to the voltage applied across it.

3. Solving the Problem for C1 and C2 with a 300V Supply

Let’s assume you have two capacitors, C1 and C2, and you need to find the equivalent capacitance of the network and also determine the charge and voltage across each capacitor when connected to a 300V supply.

Case a: Capacitors in Series

• To calculate the equivalent capacitance for capacitors in series:

  \frac{1}{C_{\text{eq,series}}} = \frac{1}{C_1} + \frac{1}{C_2}
  • Suppose C_1 = 10\ \mu\text{F} and C_2 = 20\ \mu\text{F}. Then,
    \frac{1}{C_{\text{eq,series}}} = \frac{1}{10} + \frac{1}{20}
    • The calculation proceeds as follows:
      \frac{1}{C_{\text{eq,series}}} = \frac{2 + 1}{20} = \frac{3}{20}
      C_{\text{eq,series}} = \frac{20}{3} \approx 6.67\ \mu\text{F}

• Using the total voltage V_{\text{total}} = 300\ \text{V}, calculate the charge:

  Q_{\text{total}} = C_{\text{eq,series}} \times V_{\text{total}}
  Q_{\text{total}} = 6.67\ \mu\text{F} \times 300\ \text{V} = 2000\ \mu\text{C}

• To find the voltage across individual capacitors, remember that the charge on capacitors in series is the same.

• Voltage across C1 (V_1):

  V_1 = \frac{Q_{\text{total}}}{C_1} = \frac{2000\ \mu\text{C}}{10\ \mu\text{F}} = 200\ \text{V}

• Voltage across C2 (V_2):

  V_2 = \frac{Q_{\text{total}}}{C_2} = \frac{2000\ \mu\text{C}}{20\ \mu\text{F}} = 100\ \text{V}

Case b: Capacitors in Parallel

• Compute the equivalent capacitance:

  C_{\text{eq,parallel}} = C_1 + C_2 = 10\ \mu\text{F} + 20\ \mu\text{F} = 30\ \mu\text{F}

• For parallel capacitors, the voltage across each is the same as the supply voltage, V_{\text{total}} = 300\ \text{V}.

• Charge on C1:

  Q_1 = C_1 \times V_{\text{total}} = 10\ \mu\text{F} \times 300\ \text{V} = 3000\ \mu\text{C}

• Charge on C2:

  Q_2 = C_2 \times V_{\text{total}} = 20\ \mu\text{F} \times 300\ \text{V} = 6000\ \mu\text{C}

Summary

• In a series configuration, the total equivalent capacitance is reduced, and the voltage divides based on the inverse of the capacitances.
• In a parallel configuration, the total equivalent capacitance increases, and the voltage across each capacitor is equal to the supply voltage.
  • Series: Total charge Q_{\text{total}} = 2000\ \mu\text{C}, Voltage across C1 is 200V and C2 is 100V.
  • Parallel: Voltage across each capacitor is 300V, Charge on C1 is 3000\ \mu\text{C}, Charge on C2 is 6000\ \mu\text{C}.

Understanding the distinctions between these configurations is critical for effective circuit design and analysis. If you have more specific values for C1 and C2 or further context about the network, adjusting the calculations accordingly will give you precise results. If you have additional questions or require clarification, feel free to ask. @anonymous6