SME
Thermal & Statistical Physics · Problem type
A blackbody: Wien’s and the Stefan–Boltzmann laws
Use this for “A star’s surface is at 3200 K and its area is 1.5 × 10¹⁷ m². Where does its spectrum peak, and what total power does it radiate?”
Chart
5,800
K:
λₘₐₓ
=
b
T
=
2,898
μm·
K
5,800
K
=
0.4997
μm
=
499.7
nm
Tinted: the visible band, 380 to 750 nm
This is an example. Type your own values to start a new problem.
λₘₐₓ
=
b
÷
T (K)
=
λₘₐₓ (nm)
P
=
σ
(
T (K)
)
4
×
A (m²)
=
P (W)
σT⁴
Power per area
calculated · W/m²
R
Radius of a ball or star
calculated · m
Formulas
λₘₐₓ
=
2.898 ×
10
6
T
499.7
=
2.898 ×
10
6
5,800
σT⁴
=
5.670 ×
10
-8
×
T
4
6.4165 ×
10
7
=
5.670 ×
10
-8
×
5,
800
4
P
=
σT⁴
×
A
3.9076 ×
10
26
=
6.4165 ×
10
7
×
(6.09
×
10
18
)
A
=
4
×
π
×
R
2
6.09 ×
10
18
=
4
×
π
×
(6.9615 ×
10
8
)
2
Assumptions
•
A blackbody absorbs everything; its glow is a photon gas in equilibrium at T, with Planck’s spectrum.
•
Wien’s law: λₘₐₓ T = b = 2.898 × 10⁻³ m·K, so λₘₐₓ = 2.898 × 10⁶ ÷ T in nm.
•
Each square meter gives off σT⁴, σ = 5.670 × 10⁻⁸ W/(m²·K⁴); a real surface gives εσT⁴, ε ≤ 1.
Step-by-step
WE KNOW
T = 5,800 K, A = 6.09 × 10¹⁸ m²
FIND
peak wavelength (λₘₐₓ), power per area (σT⁴), total power (P), radius of a ball or star (R)
Step 1 · Find peak wavelength (λₘₐₓ)
Use λₘₐₓ = 2.898 × 10⁶ ÷ T
Wien’s
law:
the
peak
moves
to
shorter
waves
as
T
rises;
b
=
2.898 ×
10
-3
m·K
is
2.898 ×
10
6
nm·K.
λₘₐₓ
=
2.898 ×
10
6
5,800
λₘₐₓ = 499.7 nm
Step 2 · Find power per area (σT⁴)
Use σT⁴ = 5.670 × 10⁻⁸ × T⁴
Stefan–Boltzmann: summing Planck’s spectrum over every wavelength gives σT⁴ per square meter.
σT⁴
=
5.670 ×
10
-8
×
5,
800
4
σT⁴
=
(5.670
×
10
-8
)
×
(1.1316
×
10
15
)
σT⁴
=
6.4165 ×
10
7
W/
m
2
Step 3 · Find total power (P)
Use P = σT⁴ × A
Every square meter of the surface gives off σT⁴, so multiply by the area.
P
=
6.4165 ×
10
7
×
(6.09
×
10
18
)
P
=
3.9076 ×
10
26
W
Step 4 · Find radius of a ball or star (R)
Use A = 4 × π × R²
Undo
A
=
4πR²:
divide
by
4π
and
take
the
root.
R
=
√(
A
4 × π
)
R
=
√(
6.09 ×
10
18
4 × π
)
R
=
√
4.846 ×
10
17
R
=
6.9615 ×
10
8
m
Check
Put the numbers back in. Both sides should match.
499.7
=
2.898 ×
10
6
5,800
✓
6.4165 ×
10
7
=
5.670 ×
10
-8
×
5,
800
4
✓
3.9076 ×
10
26
=
6.4165 ×
10
7
×
(6.09
×
10
18
)
✓
6.09 ×
10
18
=
4
×
π
×
(6.9615 ×
10
8
)
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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