Convex Lens Focal Length Calculation

This problem involves calculating the focal length of a convex lens using information about the object and its image formed during an experiment. We will use the concepts of linear magnification and the lens formula.

Given Data Analysis

The following information is provided:

• Image length ($L_i$): 1 cm
• Object length ($L_o$): 5 cm
• Object distance ($u$): 40 cm

Magnification Calculation Explained

Linear magnification ($m$) is defined as the ratio of the size of the image to the size of the object. It also relates the image distance ($v$) to the object distance ($u$).

The formula for magnification based on lengths is:

$ m = \frac{L_i}{L_o} $

Substituting the given values:

$ m = \frac{1 \text{ cm}}{5 \text{ cm}} = \frac{1}{5} $ $ m = 0.2 $

Since the image length is smaller than the object length, the magnification is less than 1, indicating that the image is diminished.

Lens Formula and Sign Convention

To find the focal length, we use the lens formula:

$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $

We need to apply the standard sign conventions:

• Object distance ($u$) is measured from the optical center. Since the object is placed in front of the lens, it is taken as negative: $u = -40$ cm.
• For a convex lens forming a diminished image, it's usually a real image formed on the opposite side, meaning the image distance ($v$) is positive.
• Magnification ($m$) is negative for real, inverted images. Thus, $m = -0.2$.

Image Distance Determination

We use the magnification formula relating distances:

$ m = \frac{v}{u} $

Substitute the known values of $m$ and $u$:

$ -0.2 = \frac{v}{-40 \text{ cm}} $

Solve for $v$:

$ v = (-0.2) \times (-40 \text{ cm}) $ $ v = 8 \text{ cm} $

The positive value for $v$ confirms that the image formed is real.

Focal Length Calculation

Now, substitute the values of $u$ and $v$ into the lens formula:

$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $ $ \frac{1}{f} = \frac{1}{8 \text{ cm}} - \frac{1}{-40 \text{ cm}} $ $ \frac{1}{f} = \frac{1}{8 \text{ cm}} + \frac{1}{40 \text{ cm}} $

To add the fractions, find a common denominator, which is 40:

$ \frac{1}{f} = \frac{5}{40 \text{ cm}} + \frac{1}{40 \text{ cm}} $ $ \frac{1}{f} = \frac{6}{40 \text{ cm}} $

Simplify the fraction:

$ \frac{1}{f} = \frac{3}{20 \text{ cm}} $

Calculate the focal length ($f$) by taking the reciprocal:

$ f = \frac{20}{3} \text{ cm} $ $ f \approx 6.67 \text{ cm} $

The calculated focal length is positive, which is characteristic of a convex lens.

Conclusion

The focal length of the convex lens is calculated to be approximately 6.67 cm.