The object distances from lens are recorded by student as (u, v) –P₁(−30, 60), P₂(30, 12), P₃(20, 60), P₄(−25, 100)  (values are magnitudes with sign). If power of lens is 5D, which readings are correct?

Correct Answer: P₁, P₄

Step-by-Step Explanation:

Step 1: Determine the focal length of the lens
The power ($P$) of the lens is given as +5 D (dioptres). The relationship between the power and the focal length ($f$) of a lens in meters is given by:
$f=\frac{1}{P}$
Substituting the given power:
$f=\frac{1}{5\text{D}}=0.2\text{m}=20\text{cm}$
Since the power is positive, the focal length is also positive ($f=+20\text{cm}$), which confirms that it is a convex (converging) lens.

Step 2: Use the lens formula to verify the readings
The lens formula relates the focal length ($f$), object distance ($u$), and image distance ($v$):
$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$
For a reading to be correct, the values of $u$ and $v$ must satisfy this equation with $f=20\text{cm}$ (so $\frac{1}{f}=\frac{1}{20}=0.05{\text{cm}}^{-1}$).

Step 3: Check each recorded reading:

1. For reading P₁(−30, 60):
Here, $u=-30\text{cm}$ and $v=60\text{cm}$.
Let us calculate the focal length:
$\frac{1}{f}=\frac{1}{60}-\frac{1}{-30}$
$\frac{1}{f}=\frac{1}{60}+\frac{1}{30}=\frac{1+2}{60}=\frac{3}{60}=\frac{1}{20}$
This gives $f=20\text{cm}$. Thus, reading P₁ is correct.

2. For reading P₂(30, 12):
Here, $u=30\text{cm}$ and $v=12\text{cm}$.
Calculating the focal length:
$\frac{1}{f}=\frac{1}{12}-\frac{1}{30}=\frac{5-2}{60}=\frac{3}{60}=\frac{1}{20}$
Although this mathematically gives $f=20\text{cm}$, the object distance is positive ($u=+30\text{cm}$), which represents a virtual object. In standard practical setups with real objects, the object distance $u$ must be negative.

3. For reading P₃(20, 60):
Here, $u=20\text{cm}$ and $v=60\text{cm}$.
Calculating the focal length:
$\frac{1}{f}=\frac{1}{60}-\frac{1}{20}=\frac{1-3}{60}=-\frac{2}{60}=-\frac{1}{30}$
This gives $f=-30\text{cm}$, which is incorrect.

4. For reading P₄(−25, 100):
Here, $u=-25\text{cm}$ and $v=100\text{cm}$.
Calculating the focal length:
$\frac{1}{f}=\frac{1}{100}-\frac{1}{-25}$

This gives $f=20\text{cm}$. Thus, reading P₄ is correct.

Therefore, the correct readings correspond to P₁, P₄.