An object is placed at a distance of 20 cm from a convex lens of focal length 10 cm. The image is formed at a distance of:
20 cm
Calculating Image Distance for a Convex Lens
This question asks us to find the position where the image is formed by a convex lens when an object is placed at a specific distance. We are given the object distance and the focal length of the convex lens. To solve this, we need to use the lens formula, which relates object distance, image distance, and focal length.
Understanding the Lens Formula and Terms
The lens formula is a fundamental equation in optics that applies to both convex and concave lenses. It is given by:
$$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $$
Where:
- f is the focal length of the lens.
- v is the image distance (distance of the image from the optical centre of the lens).
- u is the object distance (distance of the object from the optical centre of the lens).
We also need to follow the sign conventions for lenses. According to the Cartesian sign convention:
- All distances are measured from the optical centre of the lens.
- Distances measured in the direction of incident light are taken as positive.
- Distances measured in the direction opposite to the incident light are taken as negative.
- For a real object, the object distance (u) is usually taken as negative because the object is placed on the left side of the lens (assuming light comes from the left).
- For a convex lens, the focal length (f) is taken as positive.
Applying the Lens Formula
Given in the question:
- Object distance, $u = -20$ cm (negative because the object is real and placed on the left).
- Focal length of the convex lens, $f = +10$ cm (positive for a convex lens).
We need to find the image distance, $v$.
Step-by-Step Calculation
Substitute the given values into the lens formula:
$$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $$
$$ \frac{1}{10} = \frac{1}{v} - \frac{1}{-20} $$
$$ \frac{1}{10} = \frac{1}{v} + \frac{1}{20} $$
Now, we need to solve for $1/v$. Subtract $1/20$ from both sides of the equation:
$$ \frac{1}{v} = \frac{1}{10} - \frac{1}{20} $$
To subtract the fractions, find a common denominator, which is 20:
$$ \frac{1}{v} = \frac{2}{20} - \frac{1}{20} $$
$$ \frac{1}{v} = \frac{2 - 1}{20} $$
$$ \frac{1}{v} = \frac{1}{20} $$
To find $v$, take the reciprocal of both sides:
$$ v = 20 \text{ cm} $$
Interpreting the Result
The image distance $v$ is calculated to be $+20$ cm. The positive sign for $v$ indicates that the image is formed on the right side of the lens (the side opposite to where the object is placed). For a real object and a convex lens, a positive image distance means the image is real and inverted.
Location and Nature of the Image
The image is formed at a distance of 20 cm from the convex lens, on the opposite side of the object. Since the object is placed at $u = 2f$ (where $2f = 2 \times 10 = 20$ cm), the image is formed at $v = 2f$. This is a specific case for a convex lens: when the object is at $2F_1$ (or $2f$ on the left), the image is formed at $2F_2$ (or $2f$ on the right) and is real, inverted, and of the same size as the object.
| Quantity | Symbol | Value | Sign Convention |
|---|---|---|---|
| Object Distance | u | 20 cm | -20 cm (real object) |
| Focal Length | f | 10 cm | +10 cm (convex lens) |
| Image Distance | v | ? | +v (calculated) |
Based on our calculation, the image is formed at a distance of 20 cm from the convex lens.
Revision Table: Convex Lens Concepts
| Concept | Description |
|---|---|
| Convex Lens | A converging lens, thicker at the center than at the edges. Has a positive focal length. |
| Focal Length (f) | Distance from the optical center to the principal focus. Positive for a convex lens. |
| Object Distance (u) | Distance from the optical center to the object. Negative for a real object. |
| Image Distance (v) | Distance from the optical center to the image. Positive for a real image, negative for a virtual image. |
| Lens Formula | $$ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $$ Relates f, v, and u. |
| Sign Conventions | Rules for assigning positive/negative signs to distances for calculations. |
Additional Information: Image Formation by Convex Lenses
The nature and position of the image formed by a convex lens depend on the position of the object. Here are some key cases:
- Object at infinity: Image is formed at the principal focus (F), real, inverted, and highly diminished (point size).
- Object beyond 2F: Image is formed between F and 2F, real, inverted, and diminished.
- Object at 2F: Image is formed at 2F on the other side, real, inverted, and of the same size as the object. This matches the scenario in the question.
- Object between F and 2F: Image is formed beyond 2F, real, inverted, and enlarged.
- Object at F: Image is formed at infinity, real, inverted, and highly enlarged.
- Object between the optical center and F: Image is formed on the same side as the object, virtual, erect, and enlarged.
Understanding these cases helps predict image characteristics without calculation, but the lens formula provides precise location.
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