Question

By integrating the element of solid angle \((14.15), d \Omega=\sin \theta d \theta d \phi\), over all directions, verify that the solid angle corresponding to all directions is 4 \(\pi\) steradians.

Short answer

The solid angle encompassing all directions is 4\(\pi\) steradians.

Step-by-step solution

Understand the Element of Solid Angle

The element of solid angle in spherical coordinates is given by \( d\Omega = \sin \theta \, d\theta \, d\phi \). This describes a small patch of area on the unit sphere, which is determined by angles \( \theta \) (polar angle) and \( \phi \) (azimuthal angle). We are tasked to integrate over all possible directions to find the total solid angle.

Set Integration Limits

To find the total solid angle, we need to integrate over all possible values of \( \theta \) and \( \phi \). For \( \theta \), which is the angle from the positive z-axis, it ranges from \( 0 \) to \( \pi \). For \( \phi \), which is the angle in the xy-plane from the positive x-axis, it ranges from \( 0 \) to \( 2\pi \).

Integrate Over \( \phi \)

The integral over \( \phi \) from 0 to \( 2\pi \) is straightforward as \( \phi \) does not appear in the expression for \( d\Omega \). Evaluating the integral we have: \[ \int_0^{2\pi} d\phi = 2\pi\]

Integrate Over \( \theta \)

Next, integrate the remaining part over \( \theta \): \[ \int_0^{\pi} \sin\theta \, d\theta \] The antiderivative of \( \sin\theta \) is \( -\cos\theta \). Therefore, substituting the limits, we get: \[ \left[ -\cos\theta \right]_0^{\pi} = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2\]

Multiply Results from Step 3 and Step 4

Now, multiply the results of the two integrals to find the total solid angle: \[ \text{Total solid angle} = (2\pi) \times (2) = 4\pi \, \text{steradians}\]

Conclusion: Verification Complete

The calculated total solid angle of \( 4\pi \) steradians confirms the known fact that the solid angle encompassing all directions in space is indeed \( 4\pi \) steradians.

Key concepts

Spherical Coordinates

Spherical coordinates are an essential tool in dealing with 3D problems, especially when spherical symmetry is involved. In contrast to Cartesian coordinates which use \(x, y, z\) representations, spherical coordinates describe a point in space using three values: the radial distance \(r\), the polar angle \(\theta\), and the azimuthal angle \(\phi\). This is particularly useful in physics and engineering when analyzing phenomena involving spheres, cylinders, cones, etc.

• The radial distance \(r\) measures how far a point is from the origin, similar to the radius of a sphere.
• The polar angle \(\theta\) measures the angle from the positive z-axis down towards the xy-plane.
• The azimuthal angle \(\phi\) represents the angle in the xy-plane from the positive x-axis.

Spherical coordinates simplify the mathematics of certain integrals, like the one you're exploring for the solid angle integration.

Integration Limits

When integrating to find total solid angles, it's crucial to set the correct integration limits. In spherical coordinates, these limits correspond to full sweeps of the angles \(\theta\) and \(\phi\).

• The polar angle \(\theta\) varies from 0 to \(\pi\). This range covers from the top of the sphere (the positive z-axis) to the bottom (negative z-axis).
• The azimuthal angle \(\phi\) changes from 0 to \(2\pi\), completing a full circle in the xy-plane.

These specific limits ensure that all possible directions on a unit sphere are considered, thereby covering an entire solid angle as required.

Unit Sphere

The concept of a unit sphere is pivotal in understanding the representation of solid angles in spherical coordinates. A unit sphere is simply a sphere with a radius of one unit. This makes calculations simpler and results more generalizable.
Here's why it's important:

• All points on a unit sphere are equidistant from the origin, simplifying distance-related calculations.
• On a unit sphere, any area element can directly be described with the angular measurements only, thus the element of solid angle is linked to the angles alone.
• This concept is utilized frequently in physics to normalize equations and quantify directional quantities like radiation or field intensities using steradians.

Utilizing a unit sphere helps in creating standard solutions that are easily scalable and applicable across various scenarios.

Polar Angle

The polar angle \(\theta\) is a crucial component of spherical coordinates, as it describes the angle from the positive z-axis of the sphere. Understanding \(\theta\) is necessary for integrating areas on a sphere accurately.

• It varies between \(0\) (pointing directly at the positive z-axis) to \(\pi\) (directly down the negative z-axis).
• As \(\theta\) increases, points are angled progressively downwards away from the z-axis towards the xy-plane.
• This measurement is critical when determining areas and volumes involved in 3D space revolving around a center point.

In the context of integrating the solid angle, the correct limits of \(\theta\) are essential for encompassing the entire sphere.

Azimuthal Angle

The azimuthal angle \(\phi\) is another essential angle in spherical coordinates, representing the direction in the xy-plane. This angle facilitates a full rotational perspective around the z-axis.

• This angle ranges from 0 to \(2\pi\), allowing for a complete 360-degree rotation around the z-axis.
• At \(\phi = 0\), the direction is aligned with the positive x-axis, and progressive increases in \(\phi\) rotate the point counterclockwise in the xy-plane.
• The azimuthal angle helps determine positions and directions in a plane, making it vital for tasks involving full spatial orientation.
  The limits for \(\phi\) ensure that integration covers all horizontal directions, completing the calculation for the total solid angle.