Question

A small surface of area \(A=3 \mathrm{~cm}^{2}\) emits radiation with an intensity of radiation that can be expressed as \(I_{e}(\theta, \phi)=100 \phi \cos \theta\), where \(I_{e}\) has the units of \(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{sr}\). Determine the emissive power from the surface into the hemisphere surrounding it, and the rate of radiation emission from the surface.

Short answer

Answer: The emissive power from the surface into the hemisphere surrounding it is approximately \(100\pi^2 \frac{W}{m^2}\), and the rate of radiation emission from the surface is approximately \(2.976 \mathrm{W}\).

Step-by-step solution

Integrate over the solid angle

First, integrate \(I_{e}(\theta, \phi)\) over the solid angle represented by the hemisphere surrounding the surface. The solid angle \(d\Omega\) in polar coordinates can be expressed as \(\sin(\theta)d\theta d\phi\). The integration of \(I_{e}\) over the solid angle will be: $$\int_{0}^{\pi/2}\int_{0}^{2\pi} I_{e}(\theta, \phi) \sin(\theta) d\theta d\phi$$

Integrate \(\phi\)

Now we integrate over \(\phi\): $$\int_{0}^{2\pi} 100 \phi d\phi = \left[50\phi^{2}\right]_{0}^{2\pi} = 200\pi^2$$

Integrate \(\theta\)

Now integrate over \(\theta\): $$\int_{0}^{\pi/2} 200\pi^2 \cos(\theta) \sin(\theta) d\theta = 200\pi^2 \left[-\frac{1}{2}\cos^{2}(\theta)\right]_{0}^{\pi/2} = 100\pi^2$$

Calculate the emissive power

Now the integration result gives us the emissive power into the hemisphere surrounding it: $$P_{e} = 100\pi^2 ~ \frac{W}{m^2}$$

Calculate the rate of radiation emission

Finally, we can find the rate of radiation emission from the surface by multiplying the area A and the emissive power \(P_{e}\): $$P = A \cdot P_{e} = 3 \mathrm{cm}^2 \cdot 100\pi^2 ~ \frac{W}{m^2}$$ $$P \text{ (in watts)} = 3 \times 10^{-4} m^2 \cdot 100\pi^2 ~ \frac{W}{m^2} \approx 2. 976 \mathrm{W}$$ In conclusion, the emissive power from the surface into the hemisphere surrounding it is approximately \(100\pi^2 \frac{W}{m^2}\), and the rate of radiation emission from the surface is approximately \(2.976 \mathrm{W}\).

Key concepts

Radiative Heat Transfer

Radiative heat transfer is a fundamental mode of thermal energy transfer, distinct from conduction and convection. It doesn’t require any medium and occurs through electromagnetic waves emitted by bodies due to their temperature. It’s described by the Stefan-Boltzmann law, which states that the power emitted by a black body is proportional to the fourth power of its absolute temperature.

The rate of energy emission also depends on the surface properties such as emissivity, which describes how well a surface emits infrared energy compared to an ideal black body. A perfect black body, which absorbs all incident radiation, would have an emissivity of 1. Real-world materials have emissivities less than 1, which affects the calculation of radiative heat transfer. For instance, the presence of an intensity function in the exercise demonstrates how directional emission is considered, where intensity varies with angles \theta (theta) and \( \phi \) (phi).

Solid Angle Integration

Solid angle integration is crucial in calculating the total radiation emitted or received by an object. A solid angle, denoted by \( \Omega \), is a three-dimensional analog of the two-dimensional angle, measuring the portion of space 'covered' by an object as seen from a given point. It is measured in steradians (sr).

When discussing radiation emission, each infinitesimal area on the surface emits radiation in all directions within a hemisphere. Integrating over this hemisphere accounts for the variation in intensity with direction. In our exercise, the integration of emission intensity \( I_e(\theta, \phi) \) over the hemisphere's solid angle is performed to determine the total emitted power.

To perform such an integration, one typically uses spherical coordinates, involving the angles \( \theta \) and \( \phi \) and integrating them over their respective ranges. The integration boundaries for \( \theta \) (0 to \( \pi/2 \) for a hemisphere) and \( \phi \) (0 to \( 2\pi \) for a full circle) reflect the geometry of the problem—ensuring that all directions over which radiation is emitted are included.

Emissive Power

Emissive power represents the thermal radiation energy emitted by a surface per unit area and is a fundamental concept in thermal physics. It is typically denoted by \( E \) or \( P_e \) and measured in watts per square meter \( \frac{W}{m^2} \).

Emissive power is not constant and can change depending on the temperature and the material's emissivity. In our exercise, the emissive power of the surface is obtained by integrating the given intensity function over the hemisphere surrounding the surface. It illustrates how different angles contribute differently to the total emissive power, highlighting the anisotropic nature of radiation in many real-world scenarios.

Once the emissive power for a particular orientation is known, it can be scaled by the surface area to find the total rate of radiation emission in watts. This value is crucial for a wide range of applications, including thermal management in electronic devices, energy efficiency in building design, and even astrophysical observations.