Question

It can be argued that a flat piece of glass, such as in a window, is like a lens with an infinite focal length. If so, where does it form an image? That is, how are${}^{{\mathit{d}}_{\mathbf{i}}}$and${}^{{\mathit{d}}_{\mathit{o}}}$related?

Short answer

The lens maker equation suggests:${}^{\frac{\mathbf{1}}{\mathit{f}}\mathbf{=}\frac{\mathbf{1}}{\mathit{u}}\mathbf{+}\frac{\mathbf{1}}{\mathit{v}}}$.

And as the focal length is infinite, then it is known that:${}^{\mathbf{v}\mathbf{=}\mathbf{-}\mathbf{u}}$.

Yes, the value of${}^{{\mathit{d}}_{\mathit{o}}}$and${}^{{\mathit{d}}_{\mathit{i}}}$are related as the distance between a viewer and an object is:${}^{{\mathit{d}}_{\mathit{o}}\mathbf{+}{\mathit{d}}_{\mathit{i}}}$.

Step-by-step solution

Concept Introduction

The focal length is the distance between a convex lens or a concave mirror and the focal point of a lens or mirror. It is the point at which two parallel light beams meet or converge. Depending on the nature of the lens and mirror (concave or convex), the focal length varies with the sign (positive or negative).

The equation for the lens

From the lens maker equation, it is known –

$\frac{\mathbf{1}}{\mathbf{f}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{u}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{v}}$

Where${}^{\mathit{f}}$is the focal length of the lens,role="math" localid="1653997519150" ${}^{\mathit{u}}$is the distance between the lens and the object, androle="math" localid="1653997555994" ${}^{\mathit{v}}$ is the distance between the image and the lens.

The focal length of the plane glass can be regarded as a lens with an infinite focal length. Hence, it can be represented as –

role="math" localid="1653997630021" $\frac{\mathbf{1}}{\mathit{u}}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathit{v}}$

or,

$\mathit{v}\mathbf{=}\mathbf{-}\mathit{u}$

Relation between the distance

And since the object is placed in front of the plane lens, hence the distance${}^{\mathit{u}}$must be positive. Therefore, the distance${}^{\mathit{v}}$is negative, which means that the image is located at the same side where the object is located and at a distance equal to the distance between the object and the plane lens. i.e., the image is formed at the same position where the object is located.

It forms an image in the relation –

${}^{{\mathit{d}}_{\mathbf{o}}\mathbf{+}{\mathit{d}}_{\mathbf{i}}}$

The relation between${}^{{\mathit{d}}_{\mathit{o}}}$and${}^{{\mathit{d}}_{\mathit{i}}}$is that both of them combined to give a fixed value, and that is the distance between a viewer and the object.

Therefore, the image is formed at a distance${}^{{\mathit{d}}_{\mathbf{o}}\mathbf{+}{\mathit{d}}_{\mathbf{i}}}$, in between the viewer and the object.