23.03 seconds
The time constant (τ) of an RC circuit is given by the product of the resistance (R) and the capacitance (C):
τ = RC
In this case, R = 1 MΩ = \(10^6\) Ω and C = 10 µF = \(10 \times 10^{-6}\) F. Therefore,
τ = \(10^6 \times 10 \times 10^{-6}\) = 10 seconds
The charging equation for a capacitor in an RC circuit is:
\(V(t) = V_f (1 - e^{-t/τ})\)
where:
- \(V(t)\) is the voltage across the capacitor at time t
- \(V_f\) is the final voltage across the capacitor
- \(τ\) is the time constant
We want to find the time it takes for the capacitor to reach 90% of its final charge, meaning \(V(t) = 0.9 V_f\). Substituting this into the equation:
\(0.9 V_f = V_f (1 - e^{-t/τ})\)
Dividing both sides by \(V_f\):
\(0.9 = 1 - e^{-t/τ}\)
\(e^{-t/τ} = 1 - 0.9 = 0.1\)
Taking the natural logarithm of both sides:
\(-t/τ = \ln(0.1)\)
Solving for t:
\(t = -τ \ln(0.1)\)
Substituting τ = 10 seconds:
\(t = -10 \times \ln(0.1) \approx -10 \times (-2.303) \approx 23.03\) seconds
Therefore, it will take approximately 23.03 seconds for the capacitor to reach 90% of its final charge.
Important Questions from Time Constant
The time constant of an inductive circuit is defined as the time required for the circuit current to rise to _____ % (approximately) of its final steady state value.
Which of the following groups have similar units?
Symbols have their own common meanings.
- What will be the transient time (in seconds) of a series RC circuit, when the capacitance of the circuit is 8 mF and the resistance of the circuit is 8 kilo-Ohms?
Which of the following is/are not discharging equation/equations for an R-C circuit?
1. Capacitor voltage: \[ v_C = V_0 e^{-\tfrac{t}{RC}} \] 2. Resistor voltage: \[ v_R = E e^{-\tfrac{t}{RC}} \] 3. Discharging current: \[ i = \frac{V_0}{R} e^{-\tfrac{t}{RC}} \]
Select the correct answer using the code given below.
- Match the following Lists regarding time constant with capacitor discharging voltage:Select the correct answer using the code given below.
List-I (Time constant) List-II (Capacitor discharging voltage) P. $1\tau$ 1. 0.67% of E Q. $2\tau$ 2. 36.8% of E R. $3\tau$ 3. 13.5% of E S. $4\tau$ 4. 5% of E T. $5\tau$ 5. 1.87% of E
