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Step-by-Step Solution
Step 1: Understand the Physical Situation
In a single-slit diffraction experiment, the intensity of the diffraction pattern at an angle $ \theta $ is given by
$$I(\theta) = I_0 \left(\frac{\sin \theta}{\theta}\right)^2,$$
where $I_0$ is the maximum intensity at the principal maximum (central maximum).
Step 2: Identify the Principal Maximum Condition
The principal (central) maximum occurs at $y = 0$ on the screen, which corresponds to $ \theta = 0 $. Although the expression $\left(\frac{\sin \theta}{\theta}\right)$ seems indeterminate at $\theta = 0$, its limiting value as $ \theta \to 0 $ is $1$.
Step 3: Evaluate the Intensity at the Principal Maximum
Taking the limit, we have
$$ \lim_{\theta \to 0} \left(\frac{\sin \theta}{\theta}\right) = 1. $$
Therefore,
$$ I(\theta = 0) = I_0 \left(\frac{\sin 0}{0}\right)^2 \rightarrow I_0 \times 1^2 = I_0. $$
Thus, the intensity at the central maximum is $I_0$ when the slit width is $a$.
Step 4: Effect of Doubling the Slit Width on the Central Maximum
When the slit width is doubled, the overall diffraction pattern becomes narrower. However, the amplitude for the central maximum remains such that its value at $ \theta = 0 $ is still $I_0$. This is because the maximum value of the diffraction envelope (central maximum) depends on the total energy flow through the slit and the way it interferes constructively at $ \theta = 0 $. Hence, the intensity at the principal maximum does not change even if the slit width doubles.
Step 5: Final Answer
Therefore, the intensity of the principal maximum remains
$$ \boxed{I_0}. $$
