the total resistance is 13.0 kΩ, and the battery's emf is 28.0 V. If the time constant is measured to be 34.0 µs . Calculate the total capacitance of the circuit. Calculate the time it takes for the voltage across the resistor to reach 11.0 V after the switch is closed.
R
^^^
=E C=
S
To calculate the total capacitance of the circuit, we need to use the formula for the time constant (τ) of an RC circuit:
τ = RC
Given that the time constant is measured to be 34.0 µs and the total resistance is 13.0 kΩ, we can rearrange the formula to solve for the total capacitance (C):
C = τ / R
Substituting the values:
C = (34.0 × 10^-6 s) / (13.0 × 10^3 Ω)
= 2.615 × 10^-9 F (F stands for Farads)
Therefore, the total capacitance of the circuit is 2.615 nF (nanofarads).
To calculate the time it takes for the voltage across the resistor to reach 11.0 V after the switch is closed, we need to use the formula for the transient response of an RC circuit:
V = V0 × (1 - e^(-t/τ))
Where:
- V is the voltage across the resistor at time "t"
- V0 is the initial voltage across the resistor (0 V in this case since the switch is closed at t = 0)
- τ is the time constant of the circuit (34.0 µs in this case)
We need to solve this equation for time (t), given that V = 11.0 V and V0 = 0 V:
11.0 = 0 × (1 - e^(-t/(34.0 × 10^-6 s)))
Simplifying the equation:
1 - e^(-t/(34.0 × 10^-6 s)) = 0
Since the exponential function will never be zero, we can conclude that the voltage across the resistor will not reach 11.0 V in this circuit configuration.
Therefore, the time it takes for the voltage across the resistor to reach 11.0 V is infinite.