(I) A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle θ (Fig. 10–50) of 1.4 x 10⁻⁵ rad. What diameter spot will it make on the Moon?


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Verified step by step guidance

1. Identify the given values: the distance to the Moon (D) is 380,000 km, and the divergence angle of the laser beam (θ) is 1.4 x 10⁻⁵ radians.
2. Understand that the diameter of the spot created by the laser on the Moon can be modeled as the arc length of a circle segment, where the circle's radius is the distance from the Earth to the Moon, and the angle subtended by the arc is the divergence angle of the laser.
3. Convert the distance from Earth to the Moon from kilometers to meters by multiplying by 1,000, as the standard unit for distance in physics calculations is meters.
4. Use the formula for the arc length, which is given by the equation: Arc length = radius × angle. Here, the radius is the distance to the Moon and the angle is the divergence angle of the laser.
5. Substitute the values into the formula to find the diameter of the spot on the Moon. This is calculated by multiplying the distance to the Moon (in meters) by the divergence angle (in radians).

Key Concepts

Divergence of a Laser Beam

Divergence refers to the spreading of a laser beam as it travels through space. It is typically measured in radians and indicates how much the beam expands over a distance. A smaller divergence angle results in a more focused beam, while a larger angle leads to a wider spread. In this question, the divergence angle of 1.4 x 10⁻⁵ rad is crucial for calculating the diameter of the spot on the Moon.

Distance and Geometry

The distance from Earth to the Moon is approximately 380,000 km, which is essential for determining how far the laser beam travels. The geometry of the situation can be analyzed using simple trigonometric relationships, where the diameter of the spot on the Moon can be calculated by considering the angle of divergence and the distance. This relationship allows us to find the size of the spot created by the laser beam on the lunar surface.

Trigonometric Relationships

Trigonometric relationships, particularly involving right triangles, are fundamental in physics for relating angles to distances. In this scenario, the diameter of the spot can be calculated using the formula: diameter = 2 * (distance to the Moon) * tan(θ/2). This formula utilizes the tangent function to relate the angle of divergence to the linear dimensions of the spot, providing a clear method to find the answer.