A converging lens with a focal length of 70.0 cm forms an image of a 3.20-cm-tall real object th

A converging lens with a focal length of 70.0 cm forms an...

Key Concepts

Converging Lens

A converging lens (also known as a convex lens) is a lens that bends incoming parallel light rays so that they converge to a point called the focus. This type of lens is used to form images by refracting light through its structure and, depending on the object's position relative to the focus, can produce real and inverted images or virtual and upright ones.

Focal Length

The focal length of a lens is the distance from the center of the lens to the focal point, where parallel rays of light converge (or appear to diverge from in the case of a virtual focus). For converging lenses, the focal length is positive, and its value helps determine the image formation characteristics with respect to the object’s placement.

Thin Lens Equation

The thin lens equation, given by 1/f = 1/o + 1/i, relates the focal length (f) of a lens to the object distance (o) and the image distance (i). This equation is essential in geometric optics for predicting where an image will form when an object is placed at a given distance from a lens.

Magnification

Magnification in lens systems quantifies how much larger or smaller the image is compared to the object. It is calculated as m = -i/o, where the negative sign indicates that an inverted image is formed. This concept allows us to understand the scaling of dimensions from the object to the image and its orientation.

Real vs. Virtual Images

Images formed by lenses are classified as real or virtual based on how light rays converge or diverge after passing through the lens. A real image is produced when light rays actually converge at a point and can be projected onto a screen, often being inverted. In contrast, virtual images occur when the light rays only appear to diverge from a common point and cannot be projected onto a screen; these images are typically upright.

Sign Conventions

Sign conventions are fundamental in applying the lens equation correctly. They dictate that distances measured in the direction of incoming light (usually to the left of the lens) are considered negative for objects, and real images have a positive image distance (to the right of the lens). These conventions ensure consistent and accurate determination of object and image characteristics.

Transcript

00:01 In this question, we are given some information about the object and image for a converging lens of known focal length, and we are asked to find the object and image distances.

00:14 So the first thing to note here, so we have a real object to the left of the lens, so we'll just like sketch in a thin lens, and we're going to sketch in a focal length on each side, and we're given that that focal length is 70 .0 centimeters, and we are told that the object height is 3 .20 centimeters tall, and then we're told that the image is 4 .5 centimeters tall and inverted.

01:01 So converging lenses form real images that are inverted.

01:18 Virtual images are always upright regardless of the kind of lens.

01:22 So the fact that this is inverted tells us that it is real, and that helps us because we now know that this real image is on the right side of the lens and will therefore have a positive image distance.

01:58 That lets us check if we get the, you know, a reasonable answer here at the end.

02:13 All right, and the fact that it is inverted tells us that we need to assign a value of negative 4 .50 centimeters.

02:25 So we're going to come back to our magnification equation, which has negative image height over object height equals image distance over object distance.

02:43 So we can find our magnification, and i get that ratio to be 1 .40625 equals di over do, which means that we're going to be able to use our lens equation and make a substitution that our image distance is 1 .40625 times our object distance.

03:27 So now let's come to our lens equation.

03:32 One over the focal length equals one over object distance plus one over image distance.

03:42 So one over our, we can go ahead and plug in actually, we can go ahead and plug in that 70 centimeters for our focal length, equals one over the object distance plus one over 1 .40625 times our object distance, and that gives me two, well our next step will be to find a common denominator.

04:13 So we'll multiply, and let's sketch this in...

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