Question

A potential divider circuit is connected with a dc source of 20 V, a light emitting diode of glow in voltage 1.8 V and a zener diode of breakdown voltage of 3.2 V. The length (PR) of the resistive wire is 20 cm. The minimum length of PQ to just glow the LED is …………. cm.

Answer 1

Correct Answer: 5

To determine the minimum length PQ required to just glow the LED, we must ensure the voltage across the LED is 1.8 V. The circuit involves a potential divider comprised of a resistive wire PR, with a total voltage of 20 V from the DC source. The zener diode, with a breakdown voltage of 3.2 V, ensures a stable voltage across PR. The LED needs 1.8 V to glow. 

First, calculate the voltage drop across the zener and LED once the LED starts glowing: 
Total Voltage across PQ + QR = 20 V
1.8 V is across LED and the zener does not conduct below 3.2 V.
To make LED glow, 1.8 V (LED) + 3.2 V (Zener) = 5 V across PQ is needed.

Now that we have determined a 5 V drop is needed to just start the LED glowing, PQ must drop 5 V of the total 20 V.

Given, PR = 20 cm, we determine the position of Q such that voltage drop across PQ is 5 V.

Voltage across PQ/Total Voltage = Length of PQ/Total Length of PR

\( \frac{5 \text{ V}}{20 \text{ V}} = \frac{\text{Length of PQ}}{20 \text{ cm}} \)

Cross-multiplying gives:
\(\text{Length of PQ} = \frac{5}{20} \times 20 \text{ cm} = 5 \text{ cm}\)

The minimum length PQ is 5 cm. This solution falls within the specified range. Therefore, the correct answer is 5 cm.

Answer 2

The total voltage across $PR$ is:
\[V_{PR} = 20 \, \text{V}.\]
The resistive wire $PR$ has a total length of:
\[\ell_{PR} = 20 \, \text{cm}.\]
The voltage across $QR$ is determined by the zener diode:
\[V_{QR} = 3.2 \, \text{V}.\]
The voltage across $PQ$ is:
\[V_{PQ} = V_{PR} - V_{QR} = 20 - 3.2 = 16.8 \, \text{V}.\]
The fraction of voltage across $PQ$ relative to $PR$ is:
\[\frac{V_{PQ}}{V_{PR}} = \frac{16.8}{20} = \frac{1}{4}.\]
Using the proportionality of voltage and length:
\[\ell_{PQ} = \frac{1}{4} \times \ell_{PR}.\]
Substitute $\ell_{PR} = 20 \, \text{cm}$:
\[\ell_{PQ} = \frac{1}{4} \times 20 = 5 \, \text{cm}.\]
Thus, the minimum length of $PQ$ to just glow the LED is:
\[\ell_{PQ} = 5 \, \text{cm}.\]