Lens Nature and Focal Length Determination

This solution determines the focal length and nature of a lens using the provided information about the object and its image. Key details are:

• Object distance: $u = 10\text{ cm}$. Using the standard sign convention where light travels left to right and the object is placed to the left of the lens, $u = -10\text{ cm}$.
• Image properties: Real, Inverted, and Same size as the object.

Magnification and Image Distance

The image being the 'same size' means the magnitude of magnification $|m|$ is 1. Since the image is 'real and inverted', the magnification $m$ is negative, so $m = -1$.

The magnification formula is $m = \frac{v}{u}$, where $v$ is the image distance.

Substituting the values: $-1 = \frac{v}{-10\text{ cm}}$ Solving for $v$: $v = (-1) \times (-10\text{ cm}) = 10\text{ cm}$ The positive value of $v$ confirms the image is real and formed on the opposite side of the lens from the object.

Focal Length Calculation

The lens formula relates focal length ($f$), object distance ($u$), and image distance ($v$): $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$

Substitute the values of $u$ and $v$: $\frac{1}{f} = \frac{1}{10\text{ cm}} - \frac{1}{-10\text{ cm}}$ $\frac{1}{f} = \frac{1}{10\text{ cm}} + \frac{1}{10\text{ cm}}$ $\frac{1}{f} = \frac{2}{10\text{ cm}} = \frac{1}{5\text{ cm}}$

Solving for $f$: $f = 5\text{ cm}$

Lens Nature Identification

The calculated focal length $f = +5\text{ cm}$ is positive. A positive focal length indicates that the lens is converging (also known as a convex lens). Converging lenses are capable of forming real, inverted images when the object is placed beyond the focal point.

Therefore, the lens has a focal length of $5\text{ cm}$ and is a converging type.