To determine the focal length of a convex lens by plotting graphs between:

1. Object distance (u) and image distance (v)
2. Reciprocal of object distance (1/u) and reciprocal of image distance (1/v)

• Optical bench with upright stands
• Convex lens with holder
• Object needle (illuminated pin or cross-wire)
• Image screen or observation pin
• Half-meter scale
• Plumb line
• Graph paper
• White sheet of paper (for screen)
• Laboratory stands and clamps

A convex lens has the property of converging parallel rays of light to a point called the focus. The distance between the optical center of the lens and the focus is called the focal length.

According to the lens formula:

Where:

• $f$ = focal length of the lens
• $u$ = object distance (distance between object and optical center)
• $v$ = image distance (distance between image and optical center)

Rearranging the lens formula:

This is a linear equation in the form of $y = mx + c$ where:

• $y = \frac{1}{v}$
• $x = \frac{1}{u}$
• $m = -1$ (slope)
• $c = \frac{1}{f}$ (y-intercept)

Therefore, if we plot a graph between $\frac{1}{u}$ and $\frac{1}{v}$, we get a straight line with:

• The y-intercept equals $\frac{1}{f}$
• The x-intercept equals $\frac{1}{f}$
• The slope equals -1

Alternatively, from the lens formula, we can derive:

This gives us a relationship between $u$ and $v$. When we plot a graph between $u$ and $v$, we get a hyperbola. This graph will meet the coordinate axes at points that help us determine the focal length.

The key formulas used in this experiment are:

For the graphical method using $\frac{1}{u}$ vs $\frac{1}{v}$:

1. Set up the optical bench with the illuminated object pin, convex lens, and screen in a straight line.
2. Adjust the position of the lens and screen to obtain a clear, magnified image of the object on the screen.
3. Measure and record the distance between the object and the lens ($u$) and the distance between the lens and the screen ($v$).
4. Repeat the experiment by changing the position of the object and obtaining a clear image on the screen each time.
5. For each set of readings, calculate the values of $\frac{1}{u}$ and $\frac{1}{v}$.
6. Plot a graph between $u$ and $v$, marking appropriate scales on both axes.
7. Plot another graph between $\frac{1}{u}$ and $\frac{1}{v}$, marking appropriate scales on both axes.
8. Determine the focal length from the graph using the appropriate formula.

Method 1: Using the $u$ vs $v$ Graph

Plot a graph with object distance $u$ on the x-axis and image distance $v$ on the y-axis.

The graph will be a hyperbola. Draw asymptotes parallel to the x and y axes.

The distances of these asymptotes from the origin give the focal length.

Method 2: Using the $\frac{1}{u}$ vs $\frac{1}{v}$ Graph

Plot a graph with $\frac{1}{u}$ on the x-axis and $\frac{1}{v}$ on the y-axis.

The graph will be a straight line. Extend this line to meet the x and y axes.

If the x-intercept is $a$ and the y-intercept is $b$, then:

Calculate the mean value of the focal length from both intercepts:

The focal length of the given convex lens as determined by:

1. The $u$ vs $v$ graph method: $f = _____ $ cm
2. The $\frac{1}{u}$ vs $\frac{1}{v}$ graph method: $f = _____ $ cm

Mean value of focal length: $f = _____ $ cm

1. The optical bench should be placed on a horizontal surface.
2. All upright stands must be perpendicular to the optical bench.
3. The optical center of the lens should be aligned with the axis of the optical bench.
4. The object pin, the optical center of the lens, and the center of the screen should be at the same height.
5. Parallax error should be avoided when taking measurements.
6. The lens should be clean and free from dust.
7. The room should be sufficiently dark to obtain a clear image.
8. For each observation, ensure that the image formed on the screen is sharp and clear.
9. Take at least six observations with different values of $u$ for accurate results.
10. Ensure that you measure distances from the optical center of the lens, not from the lens holder.

1. Incorrect identification of the optical center of the lens.
2. Parallax error in measuring distances.
3. The object, lens, and screen may not be perfectly aligned.
4. The upright stands may not be exactly perpendicular to the optical bench.
5. Difficulty in obtaining a perfectly sharp image due to spherical or chromatic aberration of the lens.
6. Measurement errors in reading the scale.
7. The lens may have manufacturing defects.
8. Improper illumination of the object.
9. Ambient light interference causing difficulty in observing the image clearly.
10. Human errors in judging the sharpness of the image.