Practice problems

RC time constant practice problems

A resistor and capacitor set a delay with one number, the time constant τ = R × C. These ten problems start with finding τ and work up to the voltage at any moment on the curve and the time a reset circuit takes to let a chip start. Type your answer to check it, and open the working whenever you want it.

The method

τ = R × C

With R in ohms and C in farads, τ is in seconds. Each τ, the voltage closes 63% of the gap to where it's heading:

After	Charging, reached	Discharging, left
1τ	63%	37%
2τ	86%	14%
3τ	95%	5%
5τ	99.3%	under 1%

Between the rows, use the curve itself:

Charging: V = V_supply × (1 − e^−t/τ)
Discharging: V = V_start × e^−t/τ

The problems

Type your answer and tap Check. Every one has the working, step by step.

Problem 1 of 10
What is the time constant of a 22 kΩ resistor with a 47 µF capacitor?
Show the working
1. Use τ = R × C in ohms and farads: 22 kΩ = 22,000 Ω and 47 µF = 0.000047 F.
2. τ = 22,000 Ω × 0.000047 F = 1.034 s ≈ 1.03 s
Answer: 1.03 s
Problem 2 of 10
A 3.3 kΩ resistor charges a 100 nF capacitor. What is the time constant, in microseconds?
Show the working
1. In ohms and farads: 3.3 kΩ = 3300 Ω and 100 nF = 10⁻⁷ F.
2. τ = 3300 Ω × 10⁻⁷ F = 0.00033 s
3. 0.00033 s is 330 µs.
Answer: 330 µs
Problem 3 of 10
An empty capacitor charges through a resistor from a 5 V supply. What voltage does it reach after one time constant?
Show the working
1. After one τ a charging capacitor has closed 63% of the gap to the supply.
2. V = 0.63 × 5 V ≈ 3.15 V. The exact share is 1 − e⁻¹ = 0.632, which gives 3.16 V.
Answer: 3.15 V
Problem 4 of 10
A capacitor charged to 12 V empties through a resistor. What voltage is left after one time constant?
Show the working
1. Discharging, it keeps 37% of its voltage in each τ.
2. V = 0.37 × 12 V ≈ 4.44 V. The exact share, e⁻¹ = 0.368, gives 4.41 V.
Answer: 4.44 V
Problem 5 of 10
A 33 µF capacitor charges through 68 kΩ. About how long until it counts as full?
Show the working
1. τ = 68,000 Ω × 0.000033 F = 2.244 s
2. It counts as full after about 5τ: 5 × 2.244 s ≈ 11.2 s.
Answer: 11.2 s
Problem 6 of 10
A 220 µF capacitor charges through 10 kΩ. How long until it reaches 95% of the supply?
Show the working
1. τ = 10,000 Ω × 0.00022 F = 2.2 s
2. 95% is 3τ: 3 × 2.2 s = 6.6 s
Answer: 6.6 s
Problem 7 of 10
15 kΩ charges an empty 100 µF capacitor from 12 V. What is the capacitor's voltage 3 s after switch-on?
Show the working
1. τ = 15,000 Ω × 0.0001 F = 1.5 s, so 3 s is exactly 2τ.
2. After 2τ it has reached 86%: 0.86 × 12 V ≈ 10.3 V
Answer: 10.3 V
Problem 8 of 10
You want a time constant of 0.5 s with a 10 µF capacitor. What resistor do you need, in kilohms?
Show the working
1. Turn τ = R × C round: R = τ over C, with 10 µF = 0.00001 F.
2. R = 0.5 s over 0.00001 F = 50,000 Ω = 50 kΩ. The nearest E12 value is 47 kΩ, which gives 0.47 s.
Answer: 50 kΩ
Problem 9 of 10
5 V charges an empty 47 µF capacitor through 10 kΩ. What is its voltage 0.2 s after switch-on?
Show the working
1. τ = 10,000 Ω × 0.000047 F = 0.47 s. 0.2 s isn't a whole number of τ, so use the curve itself: V = 5 V × (1 − e^(−t/τ)).
2. t over τ = 0.2 over 0.47 ≈ 0.426, and e^(−0.426) ≈ 0.653 (the eˣ key on a calculator).
3. V = 5 V × (1 − 0.653) = 5 V × 0.347 ≈ 1.73 V
Answer: 1.73 V
Problem 10 of 10
A reset circuit charges an empty 10 µF capacitor through 100 kΩ from 5 V. The chip starts once the capacitor reaches 3 V. How long after switch-on does it start?
Show the working
1. τ = 100,000 Ω × 0.00001 F = 1 s.
2. Charging: 5 V × (1 − e^(−t/τ)) = 3 V, so e^(−t/τ) = 1 − 3 over 5 = 0.4.
3. Take the natural log (the ln key) of both sides: t = τ × ln 1 over 0.4 = 1 s × ln 2.5.
4. t = 1 s × 0.916 ≈ 0.916 s: just under one τ, since 3 V is 60% of the supply and one τ reaches 63%.
Answer: 0.916 s

Build it in a free lesson

“RC charging” is lesson 2 of 5 in Capacitor Caves, Electronics World 3. Free with an account. Lessons unlock in order, so you’ll do the 16 before it first.

Free lesson

Now build it yourself

Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.

Try it yourself: free lesson

“RC charging” is lesson 2 of 5 in Capacitor Caves, Electronics World 3. Free with an account. Lessons unlock in order, so you’ll do the 16 before it first.

Or start from the very first lesson, no account needed

Check it with a calculator

RC time constant calculatorτ, charge time, cutoff frequency and the voltage at any moment.

The ideas behind it

Questions people ask

Is the time constant the time it takes to charge fully?

No. In one τ a charging capacitor only reaches 63% of the supply. It takes about 5τ to count as full, at 99.3%. So a 1 s time constant needs about 5 s to fill.

Where do 63% and 37% come from?

In each τ the capacitor's voltage closes 63% of the gap to where it's heading. Charging, it reaches 63% of the supply. Discharging, 37% is left. Both come from e⁻¹ ≈ 0.368, and after 2τ, 3τ and 5τ the figures are 86%, 95% and 99.3%.

Do I have to convert kilohms and microfarads?

τ = R × C gives seconds only in ohms and farads. There's a shortcut: kilohms times microfarads gives milliseconds, so 10 kΩ with 22 µF is 220 ms.