Location of the images formed by a convex lens by the method of no-parallax and hence determination of the focal length of the lens

For a convex lens if $u$ is the object distance, $v$ is the image distance and $f$ the focal length,

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

If the graph is plotted using the new cartesian convention, the intercept of the graph of $1/v$ against $1/u$ will be $1/f$. The focal length of the lens can be calculated accordingly.

(For real images the value of $u$ is $+$ and the value of $v$ is $-$ and hence $1/u$ is $+$ and $1/v$ is $-$)

If the readings of the experiment are plotted using the sign convention, the graph obtained will be as follows.

Alternative methods of calculating focal length:

For real objects and real images,

(i) when the sign convention is applied to all $u$, $v$ and $f$,

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

When $\frac{1}{|v|}$ is plotted against $\frac{1}{|u|}$, the intercept of the graph $c=\frac{1}{|f|}$.

$$|f|=\frac{1}{c}$$

(ii) when the sign convention is applied to all $u$, $v$ and $f$,

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

Multiplying by $|uv|$

$$|u|+|v|=|\frac{u~v}{f}|$$ $$|u~v|=|f|(|u+v|)$$

When $|uv|$ is plotted against $|u+v|$, the gradient gives the value of $|f|$. From the readings it can be verified that for real objects and real images $|u|+|v|\ge4|f|$.

1. Point the given convex lens to a distant object and obtain its clear image on a screen. Measure the distance between the lens and the screen using the ruler and find the approximate focal length of the lens.
2. Draw a straight line in chalk on the table. Mid way on this line, place the lens mounted on the stand normal to the line.
3. On one side of the lens at a distance little further than the approximate focal length that was found, place the optical pin (O) fixed on the stand on the line drawn so that the tip of the optical pin is on the optical axis of the lens.
4. Place the background screen on the same side as this object pin but further away from the lens.
5. Place the eye on the opposite side of the lens and observe where an inverted clear image (I) is visible. If not push the object pin further from the lens until the inverted image is visible. (According to Figure 21.1.2, $x \ge$ least distance of distinct vision). <image>

<image> 6. Place the eye along the line and get confirmed that the inverted image of O is vertically situated along the line. If it is not, rotate the lens slightly with the stand and adjust it to be correctly normal to the line. Also if the tip of the pin is not seen at the centre of the lens adjust the place of the lens to be vertical.

7. Now place the other optical pin (P) as shown in Figure 21.1.2 along the axis and adjust its tip to be on the principal axis.

8. Adjust P either forwards or backwards until the tip of P coincides with the tip of the image I. In this situation of coincidence (or no-parallax), when the eye of the observer is moved sideways on the axis the tips of I and P should appear to move together without any relative motion.

9. Now measure the distance $u$ between the lens and the object and the distance $v$ between the lens and the image using the metre ruler.

10. By altering the object distance suitably obtain five more pairs of readings for $u$ and $v$ and record these readings (with the appropriate sign according to the new cartesian convention).

11. Plot $1/v$ against $1/u$. Calculate the focal length of the lens from the intercept of the graph as explained in the theory.