Answer:
Step-by-step explanation:
To determine how long it will take for the voltage Vo(t) to decay, we need to first determine the time constant of the circuit. The time constant is given by the product of the resistance and capacitance of the circuit:
τ = RC
In this circuit, the resistance R is 5 kΩ and the capacitance C is 100 nF. Converting capacitance to farads, we have:
C = 100 nF = 100 × 10^-9 F
Substituting these values into the equation for the time constant, we get:
τ = RC = (5 × 10^3 Ω)(100 × 10^-9 F) = 0.5 ms
The voltage Vo(t) will decay to approximately 36.8% of its initial value after one time constant, which is given by the equation:
V(t) = Vo(0) e^(-t/τ)
where Vo(0) is the initial voltage and t is the time elapsed since the voltage was applied. Substituting the values we have, we get:
0.368Vo(0) = Vo(0) e^(-t/τ)
Dividing both sides by Vo(0), we get:
0.368 = e^(-t/τ)
Taking the natural logarithm of both sides, we get:
ln(0.368) = -t/τ
Solving for t, we get:
t = -τ ln(0.368)
Substituting the values we have, we get:
t = -(0.5 ms) ln(0.368) ≈ 0.276 ms
Therefore, it will take approximately 0.276 ms for the voltage Vo(t) to decay to approximately 36.8% of its initial value.