| English | 意味 |
|---|---|
| time constant/taɪm ˈkɒnstənt/ | 時間定数 |
| exponential/ˌekspəˈnenʃl/ | 指数関数的 |
Why does a camera flash take a moment to recharge?
- Press the shutter, and the flash needs a second before it can fire again.
- A capacitor is refilling with charge through a resistor.
- The resistor limits the current, so the filling is gradual, not instant.
- A resistor and capacitor together make an RC circuit — the physics of timing.
Charging rises toward a limit
- Close the switch and charge flows onto the capacitor: $Q(t) = CV\left(1 - e^{-t/RC}\right)$.
- It climbs fast at first, then slows as the capacitor fills.
- It approaches, but never quite reaches, the final charge $CV$.
- The current does the opposite: large at first, fading to zero.
抵抗器を通る電荷
コンデンサが徐々に充電され、充電電流がゼロに向かう様子を見る。
The time constant τ = RC
- The product $\tau = RC$ is the time constant 时间常数 — the natural timescale.
- In one $\tau$, the charge reaches about $63\%$ of its final value.
- After about $5\tau$, it is essentially fully charged.
- Bigger $R$ or bigger $C$ means slower charging.
RC回路の時定数は：
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
$\tau = RC$ が充電および放電のスケーリングを設定する。
$5\ \Omega$ の抵抗が $4\ \text{F}$ のコンデンサを充電する。$\tau$ を求めよ（秒単位）。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
$\tau = RC = 5 \times 4 = 20\ \text{s}$.
時定数経過後、電荷は最終値の約63%に達する。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
$1 - e^{-1} \approx 0.63$ 経過後の $\tau$ 。
Discharging fades away
- Disconnect the source and let the capacitor drive the resistor.
- The charge decays: $Q(t) = Q_0\, e^{-t/RC}$ — an exponential 指数 fall.
- The current and voltage decay with the same time constant $\tau = RC$.
- Each $\tau$ the charge drops to about $37\%$ of what it was.
放電中のコンデンサは ____ 減衰として電荷を失い、$Q = Q_0 e^{-t/RC}$ 。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
放電は指数関数的である：$Q = Q_0 e^{-t/RC}$ 。
The capacitor's two extremes
- At the first instant ($t = 0$): the empty capacitor acts like a plain wire.
- After a long time ($t \to \infty$): the full capacitor acts like an open gap.
- So current starts high and ends at zero as the capacitor fills.
- Those two limits let you check any RC answer quickly.
充電の最初の瞬間（$t = 0$）において、未充電のコンデンサは次のように振る舞う：
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
空（未充電）の状態では逆電圧を示さず、導線のように振る舞う（最大電流）。
RC回路に関する正しい記述をすべて選べ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
指数関数的充電、τ = RC、満充電時は開いた隙間。充電は徐々に進行し、瞬時には達成されない。
A $2\ \Omega$ resistor charges a $3\ \text{F}$ capacitor. Find the time constant.
- $\tau = RC = 2 \times 3 = 6\ \text{s}$.
- After $6\ \text{s}$ the charge is about $63\%$ of its final value.
At $t = 0$ an uncharged capacitor behaves like a wire (max current), and at $t \to \infty$ a charged one behaves like an open gap (zero current). Swapping these two limits is the classic RC mistake — check which instant the question asks about.
In an RC circuit, charging follows $Q = CV(1 - e^{-t/RC})$ and discharging $Q = Q_0 e^{-t/RC}$, both with time constant $\tau = RC$ (about $63\%$ per $\tau$). A capacitor starts like a wire ($t=0$) and ends like an open gap ($t\to\infty$).