Question

Demonstrate that the minimum distance possible between a real object and its real image through a thin convex lens is \(4 f,\) where \(f\) is the focal length of the lens.

Short answer

Answer: The minimum distance possible between a real object and its real image through a thin convex lens is 4f, where f is the focal length of the lens.

Step-by-step solution

State the lens formula

Given the lens formula: \(\frac{1}{f} = \frac{1}{d_1} + \frac{1}{d_2}\). Here, \(f\) is the focal length of the lens, \(d_1\) is the distance between the object and the lens, and \(d_2\) is the distance between the image and the lens.

Minimize the distance between the object and the image

We want to minimize the distance \(s = d_1 + d_2\). So our goal is to find the critical point for \(s\). To do this, we will differentiate \(s\) with respect to \(d_1\), and then set the derivative equal to zero.

Differentiate and solve for the critical point

Differentiate the equation for \(s\) with respect to \(d_1\): \(\frac{ds}{dd_1} = 1 - \frac{f^2}{d_2^2}\). Now, set the equation to zero: \(0 = 1 - \frac{f^2}{d_2^2}\). We can solve this equation for \(d_2\): \(d_2^2 = f^2\) \(d_2 = f\)

Substitute the value for \(d_2\) into the lens formula

Substitute the value of \(d_2 = f\) into the lens formula: \(\frac{1}{f} = \frac{1}{d_1} + \frac{1}{f}\). Now, we will solve this equation for \(d_1\): \(d_1 = 2f\).

Calculate the minimum distance between the object and the image

We found that the minimum distance between the object and the lens is \(d_1 = 2f\) and the minimum distance between the image and the lens is \(d_2 = f\). So, the minimum distance between the object and the image is \(s = d_1 + d_2 = 2f + f = 4f\).

Conclusion

We have demonstrated that the minimum distance possible between a real object and its real image through a thin convex lens is \(4f\), where \(f\) is the focal length of the lens.

Key concepts

Lens Formula

In optics, the lens formula provides a relationship between an object's distance from a lens, the image's distance, and the lens's focal length. It is an essential tool for anyone working with lenses, either in a laboratory setting or for practical applications such as photography or vision correction.

The formula itself is expressed as \(\frac{1}{f} = \frac{1}{d_1} + \frac{1}{d_2}\), where \(f\) represents the focal length of the lens, \(d_1\) is the distance of the object from the lens, and \(d_2\) is the distance of the image from the lens. Through this formula, we can calculate any one of these three parameters when the other two are known, making it invaluable for predicting how a lens will form an image.

Focal Length

The focal length, represented by the symbol \(f\), is a measure of how strongly a lens converges or diverges light. In the case of a convex lens, focal length is the distance between the lens and the point where parallel rays of light converge to a single point, known as the focus.

The focal length determines the lens's power and how it will affect the resultant image; a shorter focal length indicates a stronger lens that brings rays of light to focus more quickly. This is particularly relevant when attempting to project real images onto screens, as well as in photography, where focal length affects field of view and magnification.

Real Image Distance

When working with a convex lens, understanding real image distance is critical for applications like imaging systems and cameras. Real image distance, \(d_2\), is defined as the distance from the lens to where a real image is formed. A real image is one that can be projected onto a screen: it is inverted and formed on the opposite side of the lens from the object.

To determine the shortest possible real object distance and its corresponding real image distance for a convex lens systematically, we apply differentiation to find the critical points where the sum of the object distance \(d_1\) and the real image distance \(d_2\) is at a minimum. This is relevant in identifying the limits of how close an object can be to the lens before no real image is formed.

Differentiation in Physics

Differentiation is a mathematical process used in physics to determine how a function changes as its input changes. It is particularly useful in finding the optimum values, such as minima or maxima, of certain physical quantities.

By differentiating the total distance \(s = d_1 + d_2\) with respect to one of the distances (in the given exercise, we differentiate with respect to \(d_1\)), we can set the derivative to zero to find the condition where the distance \(s\) is at a minimum. This concept of differentiation is not only pivotal in this particular problem involving the minimum distance between an object and its real image, but it is also widely applied in mechanics, thermodynamics, and other areas of physics to solve complex problems.