Sure! Let's go through the detailed steps to find the focal length of the equi-convex lens given the distances:
1. Given Values:
- Distance of the image from the focus, [tex]\( v \)[/tex]: [tex]\( v = 30 \)[/tex] cm.
- Distance of the object from the first focus, [tex]\( u \)[/tex]: [tex]\( u = -30 \)[/tex] cm. (Note the negative sign, as per convention in lens formula, the object distance is taken as negative).
2. Lens Formula:
The lens formula is given by:
[tex]\[
\frac{1}{f} = \frac{1}{v} - \frac{1}{u}
\][/tex]
3. Substituting the Values:
Substitute [tex]\( v = 30 \)[/tex] cm and [tex]\( u = -30 \)[/tex] cm into the lens formula:
[tex]\[
\frac{1}{f} = \frac{1}{30} - \left( \frac{1}{-30} \right)
\][/tex]
4. Simplifying:
The expression becomes:
[tex]\[
\frac{1}{f} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30}
\][/tex]
5. Calculating Focal Length:
Invert the fraction to find [tex]\( f \)[/tex]:
[tex]\[
f = \frac{30}{2} = 15 \text{ cm}
\][/tex]
So, the focal length of the lens is [tex]\( \boxed{15 \text{ cm}} \)[/tex].
