Steradian

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steradian
Solid Angle, 1 Steradian.svg
A graphical representation of two different steradians.
The sphere has radius r, and in this case the area A of the highlighted spherical cap is r2. The solid angle Ω equals [A/r2] sr which is 1 sr in this example. The entire sphere has a solid angle of 4π sr.
General information
Unit system SI
Unit of solid angle
Symbolsr
Conversions
1 sr in ...... is equal to ...
   SI base units   1 m2/m2

The steradian and radian on physics steradian (symbol: sr) or square radian [1] [2] is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radians, projected onto a circle, gives a length on the circumference, a solid angle in steradians, projected onto a sphere, gives an area on the surface. The name is derived from the Greek στερεόςstereos 'solid' + radian.

Contents

The steradian, like the radian, is a dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. Both the numerator and denominator of this ratio have dimension length squared (i.e. L2/L2 = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol "sr" is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian (W⋅sr−1). The steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit.

Solid angle of countries and other entities relative to the Earth. BlankMap-World6 steradian.svg
Solid angle of countries and other entities relative to the Earth.

Definition

A steradian can be defined as the solid angle subtended at the centre of a unit sphere by a circular unit area on its surface. For a general sphere of radius r, any portion of its surface with area A = r2 subtends one steradian at its centre. [3]

The solid angle is related to the area it cuts out of a sphere:

where

Because the surface area A of a sphere is 4πr2, the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4π (≈ 0.07958) of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.

Other properties

Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere Steradian cone and cap.svg
Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere

If A = r2, it corresponds to the area of a spherical cap (A = 2πrh) (where h stands for the "height" of the cap) and the relationship h/r = 1/2π holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by:

This angle corresponds to the plane aperture angle of 2θ 1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere, or to (180°/π)2
≈ 3282.80635 square degrees.

The solid angle of a cone whose cross-section subtends the angle 2θ is:

SI multiples

Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams. [4] [5] Other multiples are rarely used.

See also

Related Research Articles

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<span class="mw-page-title-main">Radian</span> SI derived unit of angle

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<span class="mw-page-title-main">Central angle</span> Measure of two radii meeting

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<span class="mw-page-title-main">Spherical cap</span> Section of a sphere

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<span class="mw-page-title-main">Directivity</span> Measure of how much of an antennas signal is transmitted in one direction

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<span class="mw-page-title-main">Golden triangle (mathematics)</span>

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<span class="mw-page-title-main">Circular arc</span> Part of a circle between two points

A circular arc is the arc of a circle between a pair of distinct points. If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the centre of the circle that is less than π radians ; and the other arc, the major arc, subtends an angle greater than π radians. The arc of a circle is defined as the part or segment of the circumference of a circle. A straight line that connects the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.

The goat problem is either of two related problems in recreational mathematics involving at least figuratively a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing the exterior of a circular area.

In the field of heat transfer, intensity of radiation is a measure of the distribution of radiant heat flux per unit area and solid angle, in a particular direction, defined according to

<span class="mw-page-title-main">Spherical sector</span> Intersection of a sphere and cone emanating from its center

In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed by the center of the sphere and the base of the cap. It is the three-dimensional analogue of the sector of a circle.

References

  1. Stutzman, Warren L; Thiele, Gary A (2012-05-22). Antenna Theory and Design. ISBN   978-0-470-57664-9.
  2. Woolard, Edgar (2012-12-02). Spherical Astronomy. ISBN   978-0-323-14912-9.
  3. "Steradian", McGraw-Hill Dictionary of Scientific and Technical Terms, fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997. ISBN   0-07-052433-5.
  4. Stephen M. Shafroth, James Christopher Austin, Accelerator-based Atomic Physics: Techniques and Applications, 1997, ISBN   1563964848, p. 333
  5. R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer" IRE Transactions on Antennas and Propagation9:1:22-30 (1961)