The eyepiece of a Galilean telescope is a diverging lens. The focal points Fo and Fe^' coincide.

step 1 This problem is about a Galilean telescope. This telescope uses a converging lens for the object

The eyepiece of a Galilean telescope is a diverging lens....

Key Concepts

Galilean Telescope

A Galilean telescope is an optical instrument that consists of a converging objective lens and a diverging eyepiece. In its design, the focal point of the objective coincides with the virtual focal point of the eyepiece, allowing the instrument to produce an upright, virtual final image. This configuration is historically significant and is used for low-power telescopes where image orientation is important.

Diverging Lens

A diverging lens (or concave lens) causes parallel incoming light rays to spread out as if they were emanating from a focal point on the same side of the lens as the object. In optical instruments like the Galilean telescope, the diverging lens is used for the eyepiece to create a virtual image that is easier for the human eye to view.

Focal Length and Focal Points

The focal length of a lens is the distance from its optical center to its focal point, where light rays either converge (in a converging lens) or appear to diverge from (in a diverging lens). Understanding the relationship between the focal lengths of the objective and eyepiece is crucial in telescope design, as it determines how and where images are formed.

Lens Equation

The thin lens equation, given by 1/f = 1/d_o + 1/d_i, relates a lens’s focal length (f) to the object distance (d_o) and the image distance (d_i). This equation is fundamental for predicting the position and nature (real or virtual) of the image produced by a lens, which is integral to solving problems in optics.

Angular Magnification

Angular magnification is the ratio of the angle subtended by the image when viewed through an optical instrument to the angle subtended by the object when viewed directly. It quantifies how much larger (or sometimes smaller) the object appears when seen through the instrument, making it a key parameter in evaluating the performance of telescopes and other imaging devices.

Virtual vs Real Images

Understanding the difference between virtual and real images is essential in optics. A real image is formed when light rays converge and can be projected onto a screen, whereas a virtual image is formed when light rays appear to diverge from a point behind the lens. Virtual images are not invertible on a screen and are typically observed directly by the eye, which is characteristic of the images produced by a Galilean telescope.

Transcript

00:03 This problem is about a galilean telescope.

00:07 This telescope uses a converging lens for the objective and a divergent lens for the ips.

00:18 The problem says that these two lenses are 32 ceres apart and that the focal length of the objective is 36 ceres.

00:29 Also, the problem says that the primary focal point of the objective is at the same place as the secondary focal point of the ips.

00:46 So it's this point here.

00:50 And with this information, we want to compute in the first part of the problem, what is the focal length of the ips.

00:59 So from the figure we see that this distance here is 36 ceres, is the distance between the primary focal point of the objective with the objective.

01:14 And also we have that this is 32 ceres, so from here we can say that this distance here is just four ceres and this represents a focal length of the ips.

01:34 This is just four centimeters, but we must remember that for a diverging lens, the focal length is negative, so this is minus four.

01:44 In the next part of the problem, we want to compute at what distance from the ips the final image is produced, so we can call it just qe.

01:56 Qe is the image distance from the ips, and we're gonna take just just the absolute value as we only want to get what this distance is.

02:07 So for the ips we can use the focal, sorry, the lens equation.

02:14 So the lens equation is this.

02:25 And this object distance for the ips is this distance here.

02:36 And this distance is for cere mirrors, but as this object is proposed, on the opposite side from the original object.

02:46 This is a negative object distance.

02:50 This is for the ips.

02:52 So if we replace this value here in this equation, this is minus 4 zero mirrors.

03:00 The focal length is also minus 4 3 meters.

03:03 So these two numbers are the same.

03:05 We get that 1 over qe is just 0.

03:11 So from here we can get two values.

03:12 From qe.

03:16 So this is infinity or this also can be minus infinity.

03:23 But in a telescope we want that the final image will be produced on the same side of the original object.

03:32 So the original object is in this region.

03:35 We also expect that the final image will be produced in this region.

03:40 This region is a negative image distance for the ips as it is on the same high of the original object.

03:51 So that's why we are going to take this value here.

03:57 But for the answer of the problem, of this part of the problem, we just want the absolute value of that.

04:04 So it's no problem, this is just infinity.

04:07 We just want to know what that distance is.

04:13 So the distance is just the absolute value.

04:15 This is different from the image distance.

04:18 Okay, in the next part of the problem, we want to know if this image is virtual or it's real, but as we already know that this number here is negative.

04:34 This is because the image is virtual.

04:39 So remember, when the final image distance is negative, it's virtual, and if it's positive, it is real.

04:47 In order to see if this image is inverted or upright, we're going to just compute the total transversal magnification.

04:58 This is not the same as the angular magnification.

05:03 The angular modification is other concept.

05:07 This is just the magnification from each lens.

05:12 And we multiply this value.

05:14 So for the first lens, for the objective, this is minus qo from objective over bo.

05:23 Minus qe for ibis over be...