Learning objective: Connect the circuit differential equation to the exponential charging curve and calculate current and voltage at a given time.
Understand the idea
A capacitor voltage cannot change instantaneously in an ideal circuit unless an impulse of current is applied. When a DC source is connected through a resistor, the capacitor gradually approaches the source voltage. The product RC controls the speed of this response; it is called the time constant, τ.
Worked example: given data
- An initially uncharged 100 μF capacitor is connected to a 12 V DC supply through a 2 kΩ resistor at t = 0.
- Assume ideal components and an ideal step application of voltage.
- Calculate τ, the voltage at 0.2 s, the initial current and the approximate time to reach 95% of 12 V.
Detailed solution
Write the loop equation
KVL gives 12 = Ri + vC. Because i = C dvC/dt, we obtain RC dvC/dt + vC = 12. The initial condition is vC(0⁺) = 0 V.
Calculate the time constant
R = 2,000 Ω and C = 100 × 10⁻⁶ F. Therefore τ = RC = 0.2 s. Always convert microfarads to farads before multiplying.
Find the charging voltage
For a zero initial voltage, vC(t) = 12(1 − e−t/0.2) V. At t = 0.2 s, vC = 12(1 − e−1) ≈ 7.59 V, about 63.2% of its final value.
Determine current
The current is i(t) = (12/2000)e−t/0.2 A = 6e−t/0.2 mA. Immediately after switching, i(0⁺) = 6 mA; it then decays toward zero.
Reach 95% of the final voltage
Set 0.95 = 1 − e−t/τ, hence e−t/τ = 0.05 and t = −τ ln(0.05). The result is t ≈ 0.599 s, about three time constants.
What the answer means
One time constant does not mean the capacitor is fully charged: it means the voltage has completed approximately 63.2% of the total change. After five time constants the error to the final value is only about 0.67%. In GATE questions, check the capacitor's initial voltage; if it is not zero, the general solution is v(t) = Vfinal + (Vinitial − Vfinal)e−t/τ.
Three mistakes to avoid
- Treating C as 100 F instead of 100 μF.
- Using τ = R/C instead of τ = RC.
- Assuming the current starts at zero during charging from an uncharged capacitor.
Try a related question
For the same circuit, what are the capacitor voltage and current at t = 0.4 s?
Show the worked-answer check
Since t/τ = 2, vC = 12(1 − e⁻²) ≈ 10.38 V and i = 6e⁻² ≈ 0.812 mA.
How to revise this concept
Compare this response with an RL circuit, then solve switching problems with nonzero initial conditions. First attempt the example on paper without opening the answer, then compare the intermediate steps. Explain each assumption before memorising a formula.
Content note: This is a newly prepared practice example, not an official GATE previous-year question. It was drafted with AI assistance for this educational website. Students should verify calculations and exam requirements against standard textbooks and official sources; educator review is recommended before publication.