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(D) For the $LED$ to just glow, the potential difference across the segment $PQ$ must be equal to the sum of the threshold voltage of the $LED$ and th

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$5$

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(D) For the $LED$ to just glow, the potential difference across the segment $PQ$ must be equal to the sum of the threshold voltage of the $LED$ and the breakdown voltage of the Zener diode.$V_{PQ} = V_{LED} + V_{Zener}$$V_{PQ} = 1.8 \,V + 3.2 \,V = 5.0 \,V$The potential divider circuit has a total voltage of $20 \,V$ across the total length $PR = 20 \,cm$.Using the principle of a potential divider, the voltage across a segment is proportional to its length:$\frac{V_{PQ}}{V_{PR}} = \frac{PQ}{PR}$$\frac{5 \,V}{20 \,V} = \frac{PQ}{20 \,cm}$$PQ = \left( \frac{5}{20} ight) 	imes 20 \,cm = 5 \,cm$Thus, the minimum length of $PQ$ required is $5 \,cm$.

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