Question

The \(E_1\) dose of UV lamp is 1 minute at 100 cm. The treatment has to be given with an \(E_2\) dose at 50 cm distance. What would be the duration of the \(E_2\) dose?

• A. 60 seconds
• B. 50.5 seconds
• C. 30.5 seconds
• D. 37.5 seconds

Hint:

Inverse Square Law: \[ I \propto \frac{1}{d^2} \] If distance halves → intensity becomes 4 times → time reduces accordingly. Also remember: \(E_2 \approx 2.5 \times E_1\).

Answer 1

The Correct Option is D

Concept:
The intensity of ultraviolet (UV) radiation follows the Inverse Square Law: \[ I \propto \frac{1}{d^2} \] Where:
• \(I\) = Intensity
• \(d\) = Distance from the source Also, dose is given by: \[ \text{Dose} = \text{Intensity} \times \text{Time} \]

Step 1: Understanding the given data.

• \(E_1\) dose = 1 minute = 60 seconds at 100 cm
• Required: \(E_2\) dose at 50 cm

Step 2: Applying inverse square law.
When distance reduces from 100 cm to 50 cm: \[ \frac{I_2}{I_1} = \left(\frac{100}{50}\right)^2 = (2)^2 = 4 \] So, intensity becomes 4 times.

Step 3: Relation between dose and time.
Since: \[ \text{Dose} = I \times t \] For same dose: \[ I_1 \times t_1 = I_2 \times t_2 \] Substitute: \[ t_2 = \frac{I_1 \times t_1}{I_2} \]

Step 4: Substitute values.
\[ t_2 = \frac{60}{4} = 15 \text{ seconds (for same } E_1) \]

Step 5: Understanding \(E_2\) dose.
In UV therapy:
• \(E_1\) = minimal erythema dose
• \(E_2\) = approximately 2.5 times \(E_1\) Thus: \[ t_{E_2} = 15 \times 2.5 = 37.5 \text{ seconds} \]

Step 6: Final conclusion.
\[ \boxed{37.5 \text{ seconds}} \]