A capacitor with capacitance c=0.02 farads in series with a resistor with resistance r=20 ohms

General
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a capacitor with capacitance c=0.02 farads in series with a resistor with resistance r=20 ohms

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Understanding Capacitors and Resistors in Series

A capacitor with capacitance c = 0.02 farads and a resistor with resistance r = 20 ohms.

1. Circuit Basics

In a series circuit, components are connected end-to-end, so that the same current flows through each component. Here, we have a capacitor and a resistor in series:

  • Capacitor: Stores energy in the form of an electric field. Its capacitance, C, is given as 0.02 farads.
  • Resistor: Opposes the flow of electric current, dissipating energy as heat. Its resistance, R, is 20 ohms.

2. Charging and Discharging of Capacitors

When a capacitor is connected to a DC source, it goes through two main phases - charging and discharging.

Charging Phase

  1. Initial Condition: At the moment the circuit is closed, the capacitor starts with zero charge.
  2. Current Flow: Current, I(t), immediately begins flowing through the circuit, as there is an initial potential difference across the battery terminals.
  3. Charge on Capacitor: As time progresses, a charge builds up on the capacitor plates.

The voltage across a charging capacitor as a function of time can be described by:

V_c(t) = V_0 (1 - e^{-\frac{t}{RC}})

where:

  • V_c(t) = Voltage across the capacitor at time t.
  • V_0 = Supply voltage from the power source.
  • t = Time, in seconds.
  • RC = Time constant of the circuit, 0.4 seconds for this setup.

Discharging Phase

  1. Initial Charge: The capacitor begins with some charge when the circuit is disconnected from the battery.
  2. Decay of Charge: Over time, the current diminishes as the capacitor discharges through the resistor, eventually reaching zero.

The voltage across a discharging capacitor can be described by:

V_c(t) = V_0 e^{-\frac{t}{RC}}

3. Time Constant (\tau)

The time constant \tau is a crucial parameter for any RC circuit, defined as \tau = RC = 0.4 seconds in this case. It indicates the time it will take for the voltage across the capacitor to rise to about 63.2% of its maximum value when charging, or fall to 36.8% of its initial value when discharging.

4. Energy Stored in the Capacitor

The energy E stored in a capacitor is given by:

E = \frac{1}{2} C V^2

5. Effect of Resistance

In this setup, the resistor’s role is to limit the rate of charging and discharging:

  • Higher Resistance: Results in a slower rate of charge and discharge, increasing the time constant \tau.
  • Lower Resistance: Results in faster charging and discharging.

6. Practical Implications

Understanding the behavior of capacitors and resistors in series is important in designing electronic circuits that require timing elements, such as delay circuits, filtering applications, and others.

In summary, we’ve explored the behavior of a series circuit consisting of a capacitor and resistor, characterized by a time constant \tau, which determines charging and discharging rates. This type of setup is widely used in various electronic applications, making it an essential concept in both academic and practical contexts.