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Q2.9:
The photograph of a house occupies an area of 1.75 cm$^2$ on a 35 mm slide. The slide is projected on to a screen, and the area of the house on the screen is 1.55 m$^2$. What is the linear magnification of the projector-screen arrangement.

Solution :

Step 1: Identification of Given Physical Quantities

We are provided with the area of the object on the slide (the initial image area) and the area of the projected image on the screen. Let the given variables be:

  • Area on the slide ($A_{slide}$) = $1.75 \text{ cm}^2$
  • Area on the screen ($A_{screen}$) = $1.55 \text{ m}^2$

Step 2: Conversion to SI Base Units

To perform calculations, all units must be consistent. We convert $A_{slide}$ from $\text{cm}^2$ to $\text{m}^2$ [By the definition of unit prefix conversion, $1 \text{ cm} = 10^{-2} \text{ m}$, therefore $1 \text{ cm}^2 = (10^{-2} \text{ m})^2 = 10^{-4} \text{ m}^2$]:

$A_{slide} = 1.75 \times 10^{-4} \text{ m}^2$

Step 3: Determination of Areal Magnification

Areal magnification ($M_A$) is defined as the ratio of the area of the image to the area of the object [Per the definition of magnification in optics]:

$M_A = \frac{A_{screen}}{A_{slide}}$

$M_A = \frac{1.55 \text{ m}^2}{1.75 \times 10^{-4} \text{ m}^2}$

$M_A = \frac{1.55}{1.75} \times 10^{4}$

$M_A \approx 0.8857 \times 10^{4} = 8857.14$

Step 4: Derivation of Linear Magnification

Linear magnification ($m$) is related to areal magnification by the square relationship: $M_A = m^2$ [Derived from the fact that area is the square of a linear dimension]. Thus, the linear magnification is the square root of the areal magnification:

$m = \sqrt{M_A}$

$m = \sqrt{8857.14}$

$m \approx 94.11$

Visual Representation of Magnification

Slide ($A_1$) Screen ($A_2$) Magnification m

Step 5: Significant Figures Analysis

The given values $1.75$ and $1.55$ both contain 3 significant figures. Therefore, the final result should be rounded to 3 significant figures [Per the rules of propagation of uncertainty in multiplication and division].

$m = 94.1$

Final Solution: The linear magnification of the projector-screen arrangement is 94.1 (dimensionless).


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