SME
University Physics II: Electricity & Magnetism · Problem type
Charging a capacitor through a resistor
Use this for “A 2 μF capacitor charges through 50 kΩ from a 12 V battery. How much charge does it hold after 0.1 s, and what current flows then?”
Chart
q(t) = 900(1 − exp(−t/τ)), τ = 1 s
After τ the response has made 63.2% of its change; after 5τ, 99.3%.
At t = 2 s: q = 778.2 μC.
This is an example. Type your own values to start a new problem.
τ (
)
=
R (
)
×
C (
)
q (
)
=
C (
)
×
ε (V)
×
(1
−
e
−
s
÷
s
)
Qf
Full charge Cε
calculated
I
Current at t
calculated
Formulas
τ
=
R
×
C
×
10
-3
1
=
10
×
100 ×
10
-3
Qf
=
C
×
ε
900 = 100 × 9
q
=
Qf
×
(1
−
e
−
t
÷
τ
)
778.2
=
900
×
(1
−
e
−2 ÷ 1
)
I
=
(
ε
R
)
×
e
−
t
÷
τ
0.1218
=
(
9
10
)
×
e
−2 ÷ 1
Assumptions
•
The capacitor starts uncharged and the switch closes at t = 0; the battery and wires are ideal.
•
Loop rule: ε = IR + q ÷ C with I = dq/dt, solved by q = Cε(1 − e−t/τ) with the time constant τ = RC.
•
The current I = (ε ÷ R)e−t/τ starts at ε ÷ R and dies away as the capacitor fills toward Cε.
•
kΩ × μF gives ms, so × 10⁻³ gives s; μF × V gives μC, and V ÷ kΩ gives mA.
Step-by-step
WE KNOW
R = 10 kΩ, C = 100 μF, ε = 9 V, t = 2 s
FIND
time constant (τ), full charge Cε (Qf), charge at t (q), current at t (I)
Step 1 · Find time constant (τ)
Use τ = R × C × 10⁻³
The
time
constant
is
RC:
kΩ
×
μF
gives
ms,
and
×
10
-3
gives
seconds.
τ
=
10
×
100 ×
10
-3
τ
=
1
s
Step 2 · Find full charge Cε (Qf)
Use Qf = C × ε
Fully charged, no current flows, so the capacitor holds the whole emf: μF × V gives μC.
Qf = 100 × 9
Qf = 900 μC
Step 3 · Find charge at t (q)
Use q = Qf × (1 − e−t ÷ τ)
The
share
of
the
full
charge
reached
so
far:
63.2%
after
one
τ
,
86.5%
after
2
τ
.
q
=
900
×
(1
−
e
−2 ÷ 1
)
q
=
900
×
(1
−
0.1353)
q
=
900
×
0.8647
q
=
778.2
μC
Step 4 · Find current at t (I)
Use I = (ε ÷ R) × e−t ÷ τ
I
=
dq/dt:
at
t
=
0
the
empty
capacitor
drops
no
voltage,
so
I
starts
at
ε
R
(
V
kΩ
gives
mA),
then
dies
away.
I
=
(
9
10
)
×
e
−2 ÷ 1
I
=
9
10
×
0.13534
I
=
0.9
×
0.13534
I
=
0.1218
mA
Check
Put the numbers back in. Both sides should match.
1
=
10
×
100 ×
10
-3
✓
900 = 100 × 9 ✓
778.2
=
900
×
(1
−
e
−2 ÷ 1
)
✓
0.1218
=
(
9
10
)
×
e
−2 ÷ 1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.