Understanding the Concave Lens Problem

This problem asks us to find the distance of an object from a concave lens, given its focal length and the image distance. We will use the lens formula to solve this.

Given Information

We are given the following properties of the concave lens and the image formed:

• Type of lens: Concave lens
• Focal length ($\text{f}$) of the concave lens: 20 cm
• Image distance ($\text{v}$) from the lens: 10 cm

Understanding Sign Conventions for Lenses

For calculations involving lenses, we use the Cartesian sign convention. For a concave lens:

• The focal length ($\text{f}$) is always taken as negative. So, $\text{f} = -20 \text{ cm}$.
• A concave lens always forms a virtual, erect, and diminished image.
• This image is always formed on the same side of the lens as the object. Therefore, the image distance ($\text{v}$) is also taken as negative. So, $\text{v} = -10 \text{ cm}$.
• The object distance ($\text{u}$) is also usually taken as negative as the object is placed to the left of the lens.

Applying the Lens Formula

The lens formula relates the object distance ($\text{u}$), the image distance ($\text{v}$), and the focal length ($\text{f}$) of a lens:

\(\frac{1}{\text{f}} = \frac{1}{\text{v}} - \frac{1}{\text{u}}\)

We need to find the object distance ($\text{u}$). We can rearrange the formula to solve for \(\frac{1}{\text{u}}\):

\(\frac{1}{\text{u}} = \frac{1}{\text{v}} - \frac{1}{\text{f}}\)

Step-by-Step Calculation

Now, substitute the given values (with their correct signs) into the rearranged lens formula:

\(\frac{1}{\text{u}} = \frac{1}{-10 \text{ cm}} - \frac{1}{-20 \text{ cm}}\)

Simplify the expression:

\(\frac{1}{\text{u}} = \frac{1}{-10} + \frac{1}{20}\)

To add the fractions, find a common denominator, which is 20:

\(\frac{1}{\text{u}} = \frac{-2}{20} + \frac{1}{20}\)

\(\frac{1}{\text{u}} = \frac{-2 + 1}{20}\)

\(\frac{1}{\text{u}} = \frac{-1}{20 \text{ cm}}\)

Now, take the reciprocal of both sides to find $\text{u}$:

\(\text{u} = -20 \text{ cm}\)

Interpreting the Result

The object distance ($\text{u}$) is -20 cm. The negative sign confirms that the object is placed on the left side of the lens, which is the standard convention for placing objects. The question asks for the distance, which is the magnitude of $\text{u}$.

Therefore, the distance of the object from the lens is 20 cm.

Conclusion

Using the lens formula and applying the correct sign conventions for a concave lens, we found the object distance.

Revision Table: Concave Lens Formulas and Terms

Additional Information: Concave Lens Properties

A concave lens is also known as a diverging lens because it diverges parallel rays of light that fall on it. Key properties:

• It is thinner at the center and thicker at the edges.
• The principal focus is on the same side as the object (where the diverging rays appear to meet).
• It always forms a virtual, erect, and diminished image, regardless of the object's position.
• The image is always located between the optical center and the principal focus on the same side as the object.

Understanding these properties helps in predicting the nature and position of the image formed by a concave lens and in applying the lens formula correctly.