An object is located 16.0 cm in front of a converging lens with focal length 12.0 cm . To the ri

An object is located 16.0 cm in front of a converging lens...

An object is located $16.0 \mathrm{cm}$ in front of a converging lens with focal length $12.0 \mathrm{cm} .$ To the right of the converging lens, separated by a distance of $20.0 \mathrm{cm},$ is a diverging lens of focal length $-10.0 \mathrm{cm} .$ Find the location of the final image by ray tracing and verify using the lens equations.

Key Concepts

Focal Length

Focal length is a measure of how strongly a lens converges or diverges light. For converging lenses, the focal length is positive, meaning that parallel rays converge to a point after passing through the lens. Conversely, diverging lenses have a negative focal length and cause parallel rays to spread out as if they were emanating from a virtual focal point.

Multiple Lens Systems

In systems with multiple lenses, the overall image formation is analyzed sequentially by treating the image formed by the first lens as the object for the next lens. This approach requires careful application of the lens equations and sign conventions at each stage, as well as consideration of the distances between lenses, to accurately predict the final image's location and characteristics.

Sign Conventions

Sign conventions are critical in optics to ensure consistency in calculations involving lens equations. They determine the positive and negative directions for measurements of object distances, image distances, and focal lengths. For example, in the Cartesian sign convention, distances measured in the direction of the incident light are typically considered positive, while those measured against it are negative, thereby affecting the interpretation of the lens equation.

Lens Equation

The lens equation, 1/f = 1/d_o + 1/d_i, is fundamental in geometric optics and relates the object distance (d_o), the image distance (d_i), and the focal length (f) of a lens. It is used to compute where an object’s image will form given the lens’s properties and is applicable to both converging and diverging lenses when proper sign conventions are followed.

Ray Tracing

Ray tracing is a graphical method used in optics to determine the path of light through a system of lenses. By drawing a few principal rays from the object through the lens(es), one can locate the point where these rays converge. This visual technique helps illustrate concepts such as image location, size, and orientation and provides an intuitive check against analytical methods such as the lens equation.

Transcript

00:01 All right, we've got a situation here where we've got one lens that's converging here.

00:06 Converging has a focal length of 12 centimeters.

00:12 And then 16 centimeters over here, we have our object, v -o -1, it's 16 centimeters away.

00:20 And then over here, 20 centimeters away, we have a diverging lens, near lens, right? 20 centimeters over here.

00:29 And so we're going to try to figure out where does the object end up? and what size is it, and et cetera, et cetera.

00:36 All right, so the deal is that we're going to start off by taking this object right here, and we're going to find out where its image is using the focal equation.

00:46 So 1 over 12, because of this converging lens, equals 1 over 16 plus 1 over d .i.

00:56 So that means our image, we're going to convert this into 48s, 4 over 48 equals 3 over 48.

01:04 Plus 1 for 48.

01:07 So that means our di equals 48 centimeters.

01:12 And then also, we can look at this by making a ray tracing, make our center ray come through here.

01:20 And way over here, we're going to get a pretty good -sized inverted image up here.

01:25 And di over do minus di over do.

01:29 It's going to give us our magnification.

01:31 It's going to be a minus 4.

01:32 That's why it's inverted.

01:34 It's negative.

01:35 And it's large.

01:37 All right, so now we have, i'm going to scroll down, and we're going to start looking at when we have this for our object, for our second lens.

01:49 All right, so here is our diverging lens right here...

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