Question

Galileo discovered the moons of Jupiter in the fall of \(1609 .\) He used a telescope of his own design that had an objective lens with a focal length of \(f_{o}=40.0\) inches and an eyepiece lens with a focal length of \(f_{e}=2.00\) inches. Calculate the magnifying power of Galileo's telescope.

Short answer

Answer: The magnifying power of Galileo's telescope is 20.

Step-by-step solution

Identify the focal lengths of the objective and eyepiece lenses

Galileo's telescope had an objective lens with a focal length of \(f_{o}=40.0\) inches and an eyepiece lens with a focal length of \(f_{e}=2.00\) inches.

Set up the formula for magnifying power of a telescope

To find the magnifying power (M) of a telescope, we use the formula: $$M = \frac{f_o}{f_e}$$ where \(f_o\) is the focal length of the objective lens and \(f_e\) is the focal length of the eyepiece lens.

Substitute the focal lengths of the lenses into the formula

Now, substitute the values from Step 1 into the formula: $$M = \frac{40.0}{2.00}$$

Calculate the magnifying power of the telescope

Divide the focal length of the objective lens by the focal length of the eyepiece lens to find the magnifying power: $$M = \frac{40.0}{2.00} = 20$$ So, the magnifying power of Galileo's telescope is \(20\).

Key concepts

Galileo's Telescope

When we talk about the magnifying power of a telescope, we must pay homage to Galileo Galilei, a pioneer who revolutionized astronomy with his innovative design. Galileo's telescope, which he called a 'perspicillum,' was notably simple yet groundbreaking. It consisted of two lenses: an objective lens that collects light from a distant object and an eyepiece lens through which the viewer observes the image.

In the case of Galileo's telescope, the objective lens had a relatively long focal length, which is the distance from the lens to the point where it forms a sharp image of an object on the optical axis. This focal length dramatically affected the telescope's ability to magnify distant objects, such as the moons of Jupiter he so famously observed. Despite its modest magnification factor, Galileo's telescope allowed him to make astronomical discoveries that had profound implications for our understanding of the universe.

Focal Length of Lens

Understanding the focal length of a lens is crucial when discussing the power of a telescope. The focal length is defined as the distance between the lens and the point where the parallel rays of light converge to a focus. It's a key factor in determining the magnifying ability of telescopic lenses. In essence, the longer the focal length of the objective lens, the larger the image that is projected for the eyepiece to magnify.

The focal length directly influences the telescope's field of view, and a longer focal length means a narrower field of view but more detail within it. It's a delicate balance that telescope makers, including Galileo, had to juggle. When calculating magnification power, as seen in Galileo's design, one would divide the focal length of the objective lens by that of the eyepiece lens. This fundamental optical principle hasn't changed and continues to guide current telescope design.

Astronomical Discovery

Astronomical discovery is the act of uncovering the unknowns of the cosmos, often greatly aided by technology such as telescopes. Each discovery, from identifying the moons of Jupiter to spotting distant galaxies, builds upon the work of astronomers who came before. Galileo's telescopic observations, for example, ushered in a new era by providing empirical evidence that supported the heliocentric model of the solar system.

The magnifying power of a telescope, while a technical tally, is much more than a number—it represents the potential for new discovery. With each increase in magnification, the universe reveals more of its secrets. When Galileo used his telescope to study Jupiter, he did not just see the planet; he found it accompanied by four moons, now known as the Galilean satellites. These findings challenged prevailing beliefs and expanded our understanding of the celestial mechanics governing the heavens. It's through tools like the telescope that we bridge the gap between mere observation and profound discovery.