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# Blackbody radiation for OCR A-level Physics

This page covers the following topics:

1. Stefan's law

2. The Hertzsprung-Russel diagram

3. Wien's displacement

4. Black holes

5. Electromagnetic radiation from stars

The power of a black body is directly proportional to the surface area of the object and its absolute temperature. This is given by Stefan's Law: P = σAT⁴, where σ is Stefan's constant (5.76 × 10⁻⁸ m⁻²K⁻⁴), A is the surface area and T is the absolute temperature.

The Hertzsprung-Russel diagram is a diagram which classifies stars. It has luminosity as compared to the Sun on the y-axis and temperature in Kelvin on the x axis. The dimmer stars are found at the bottom, whereas the brighter ones are found at the top. The hotter planets are found on the left, whereas the cooler ones are on the right.

Wien's displacement Law can be used to find the temperature of a black body being observed using its peak wavelength. Wien's displacement Law states that the temperature of the black body is inversely proportional to the wavelength which gives peak intensity. This is given by the formula: λmaxT = 2.9 × 10⁻³ mK. The peak wavelength of a black body decreases when its temperature is increased. The intensity of a black body increases at any wavelength when its temperature is increased.

When the core of a giant star collapses, gravity forces the neutrons together to create a singularity called a black hole. Black holes have immense gravitational pull, to which even light is susceptible. Escape velocity is the the velocity an object would need to be at to move out of the gravitational pull of a black hole. The point at which the escape velocity becomes greater than the speed of light is called the event horizon. The radius of the event horizon is given by the Schwarzchild radius.

An electron will leave its excited state by moving down energy levels. By doing this, it releases energy in the form of a photon. The energy of this photon is given by the following equation: E = hf, where h is the Planck constant and f is the frequency of the photon. Since different energy levels have different values, a different photon with different energy will be released for different electron transitions. These different photons will have different colours and will produce line emission spectra. The line emission spectrum is not continuous, since the energy being released will only have specific colours. Since different elements have different energy levels, their line spectra will look different and therefore they can be used to identify elements. When electrons are shared between many atoms, continuous spectra are formed.

# 1

Use the Hertzsprung-Russel diagram to compare the brightness main sequence stars and supergiants.

Most supergiants are significiantly brighter than main sequence stars, with only a few of them having about the same luminosity.

# 2

Describe Stefan's Law.

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# 3

Calculate the wavelength of the photon released in the given electron transition. Use the fact that 1 eV = 1.6 × 10⁻¹⁹ J and h = 6.63 × 10⁻³⁴ m²kg/s.

E = 22 − 8.4 = 13.6 eV.

13.6 × 1.6 × 10⁻¹⁹ = 2.176 × 10⁻¹⁸ J.

2.176 × 10⁻¹⁸ J = (6.63 × 10⁻³⁴ m²kg/s)(3 × 10⁸)/λ, so λ = 9.14 × 10⁻⁸ m (to 3 significant figures).

# 4

Calculate the absolute temperature of a black body which has a peak wavelength of 8 × 10⁻⁶ m.

T = 2.9 × 10⁻³ mK/(8 × 10⁻⁶ m) =362.5 K.

# 5

Explain what happens to the peak wavelength of a black body when its temperature is increased.

The peak wavelength of a black body decreases when its temperature is increased.

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