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The photodiode is used to detect the optical signals. These diodes are preferably operated in reverse biased mode because :

2.

In the given figure, each diode has a forward bias resistance of $ 30 \, \Omega $ and infinite resistance in reverse bias. The current $ I_{1} $ will be:

3.

The truth table corresponding to the circuit given below is:

4.

Which of the following circuits is reverse - biased ?

5.

A zener diode with $ 5 \, \text{V} $ zener voltage is used to regulate an unregulated dc voltage input of $ 25 \, \text{V} $. For a $ 400 \, \Omega $ resistor connected in series, the zener current is found to be $ 4 $ times load current. The load current $ (I_{L}) $ and load resistance $ (R_{L}) $ are:

6.

A $ 8\,\text{V} $ zener diode along with a series resistance $ R $ is connected across a $ 20\,\text{V} $ supply (as shown in the figure). If the maximum Zener current is $ 25\,\text{mA} $, then the minimum value of $ R $ will be ______ $ \Omega $.

7.

The typical output characteristics curve for a transistor working in the common-emitter configuration is shown in the figure.

The estimated current gain from the figure is:

8.

In the given circuit if the power rating of Zener diode is $ 10\,\mathrm{mW} $, the value of series resistance $ R_s $ to regulate the input unregulated supply is:

9.

The given circuit works as:

10.

Which of the following circuits has the same output as that of the given circuit?

11.

A sinusoidal carrier voltage is amplitude modulated. The resultant amplitude modulated wave has maximum and minimum amplitude of $ 120 \, \text{V} $ and $ 80 \, \text{V} $ respectively. The amplitude of each sideband is:

12.

Find the correct combination of $ A $, $ B $, $ C $ and $ D $ inputs which can cause the LED to glow.

13.

A transistor is used in an amplifier circuit in common emitter mode. If the base current changes by $ 100 \, \mu \text{A} $, it brings a change of $ 10 \, \text{mA} $ in collector current. If the load resistance is $ 2 \, \text{k}\Omega $ and input resistance is $ 1 \, \text{k}\Omega $, the value of power gain is $ x \times 10^{4} $. The value of $ x $ is __________.

14.

Which one of the following statement is not correct in the case of light emitting diodes?
A. It is heavily doped p-n junction.
B. It emits light only when it is forward biased.
C. It emits light only when it is reverse biased.
D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.
Choose the correct answer from the options given below:

15.

In the given circuit the input voltage $ V_{in} $ is shown in figure. The cut-in voltage of p–n junction diode ($ D_1 $ or $ D_2 $) is $ 0.6 \, \text{V} $. Which of the following output voltage ($ V_0 $) waveform across the diode is correct?

16.

A zener diode of power rating $ 2 \, \text{W} $ is to be used as voltage regulator. If the zener diode has a breakdown of $ 10 \, \text{V} $ and it has to regular voltage fluctuated between $ 6 \, \text{V} $ and $ 14 \, \text{V} $, the value of $ R_{S} $ for safe operation should be _______ $ \Omega $.

17.

The following diagram shows a Zener diode as a voltage regulator. The Zener diode is rated at $ V_z = 5 \, \text{V} $ and the desired current in load is $ 5 \, \text{mA} $. The unregulated voltage source can supply upto $ 25 \, \text{V} $. Considering the Zener diode can withstand four times of the load current, the value of resistor $ R_s $ (shown in circuit) should be ______ $ \Omega $.

18.

The logic circuit shown above is equivalent to:

19.

What is the current through the battery in the circuit shown below?

20.

The logic operations performed by the given digital circuit is equivalent to:

21.

Zener breakdown occurs in a $ p $-$ n $ junction having $ p $ and $ n $ both:

22.

The output $ Y $ of following circuit for given inputs is:

23.

A diode has Zener voltage of $ 10 \, \text{V} $ and maximum power dissipation of $ 0.5 \, \text{W} $, then the minimum resistance to be used in series with this diode for safety when it is connected to a $ 25 \, \text{V} $ power supply is _____ $ \Omega $.

24.

If the height of transmitting and receiving antennas are $ 80 \, \text{m} $ each, the maximum line of sight distance will be: Given: Earth’s radius $ = 6.4 \times 10^{6} \, \text{m} $.

25.

Two $ 4 $ bits binary numbers, $ A = 1101 $ and $ B = 1010 $ are given in the inputs of a logic circuit shown in figure below. The output $ (Y) $ will be:

26.

For the forward biased diode characteristics shown in the figure, the dynamic resistance at $ I_D = 3 \, \text{mA} $ will be $ \_\_\_\_\_\_ \, \Omega $.

27.

The output of the given combination gates represents:

28.

For a transistor to act as a switch, it must be operated in

29.

The Boolean expression $Y = {A}\bar{B}{C} + \bar{A}\bar{C}$ can be realized with which of the following gate configurations.

A. One 3-input AND gate, 3 NOT gates and one 2-input OR gate, One 2-input AND gate,

B. One 3-input AND gate, 1 NOT gate, One 2-input NOR gate and one 2-input OR gate

C. 3-input OR gate, 3 NOT gates and one 2-input AND gate

Choose the correct answer from the options given below:

30.

The I-V characteristics of a p-n junction diode in forward bias is shown in the figure. The ratio of dynamic resistance, corresponding to forward bias voltage of $ 2 \, \text{V} $ and $ 4 \, \text{V} $ respectively, is:

31.

For using a multimeter to identify diode from electrical components, choose the correct statement out of the following about the diode:

32.

The correct truth table for the following logic circuit is:

33.

Identify the correct truth table of the given logic circuit.

34.

The output $ Y $ for the given inputs $ A $ and $ B $ to the circuit is:

35.

A zener diode having zener voltage $ 8 \, \text{V} $ and power dissipation rating of $ 0.5 \, \text{W} $ is connected across a potential divider arranged with maximum potential drop across zener diode is as shown in the diagram. The value of protective resistance $ R_P $ is ________ $ \Omega $.

36.

A zener of breakdown voltage $ V_Z = 8 \, \text{V} $ and maximum zener current, $ I_{ZM} = 10 \, \text{mA} $ is subjected to an input voltage $ V_i = 10 \, \text{V} $ with series resistance $ R = 100 \, \Omega $. In the given circuit $ R_L $ represents the variable load resistance. The ratio of maximum and minimum value of $ R_L $ is __________.

37.

The I–V characteristics of an electronic device shown in the figure. The device is:

38.

Refer to the circuit diagram given in the figure, which of the following observations are correct?
A. Total resistance of circuit is $ 6 \, \Omega $.
B. Current in Ammeter is $ 1 \, \text{A} $.
C. Potential across AB is $ 4 $ Volts.
D. Potential across CD is $ 4 $ Volts.
E. Total resistance of the circuit is $ 8 \, \Omega $.
Choose the correct answer from the options given below:

39.

In the circuit shown below, maximum zener diode current will be ________ $ \text{mA} $.

40.

A Zener diode of power ration $ 1.6 \, \text{W} $ is to be used as voltage regulator. If the Zener diode has a breakdown of $ 8 \, \text{V} $ and it has to regulate voltage fluctuating between $ 3 \, \text{V} $ and $ 10 \, \text{V} $. The value of resistance $ R_8 $ for safe operation of diode will be:

41.

If each diode has a forward bias resistance of $ 25 \, \Omega $ in the below circuit,

which of the following options is correct?

42.

In the following circuit, the correct relation between output ($ Y $) and inputs $ A $ and $ B $ will be:

43.

Assuming in forward bias condition there is a voltage drop of $ 0.7 \, \text{V} $ across a silicon diode, the current through diode $ D_1 $ in the circuit is ________ $ \text{ma} $. (Assume all diodes in the given circuit are identical)

44.

The value of net resistance of the network as shown in the given figure is:

45.

Statement I :
To get a steady dc output from the pulsating voltage received from a full wave rectifier we can connect a capacitor across the output parallel to the load $ R_L $.
Statement II :
To get a steady dc output from the pulsating voltage received from a full wave rectifier we can connect an inductor in series with $ R_L $.
In the light of the above statements, choose the most appropriate answer from the options given below:

46.

Given below are two statements:
Statement I: In a typical transistor, all three regions emitter, base and collector have same doping level.
Statement II: In a transistor, collector is the thickest and base is the thinnest segment.
In the light of the above statements, choose the most appropriate answer from the options given below.

47.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Photodiodes are used in forward bias usually for measuring the light intensity.
Reason R: For a $ p - n $ junction diode, at applied voltage $ V $ the current in the forward bias is more than the current in the reverse bias for $ |V_z| \ge \pm V \ge |V_0| $ where $ V_0 $ is the threshold voltage and $ V_z $ is the breakdown voltage.
In the light of the above statements, choose the correct answer from the options given below.

48.

Name the logic gate equivalent to the diagram attached:

49.

Consider a circuit consisting of a capacitor $ (20 \, \mu \text{F}) $, resistor $ (100 \, \Omega) $ and two identical diodes as shown in figure. The resistance of diode under forward biasing condition is $ 10 \, \Omega $. The time constant of the circuit is $ \alpha \times 10^{-3} \, \text{s} $. The value of $ \alpha $ is ______.

50.

To measure the temperature coefficient of resistivity $ \alpha $ of a semiconductor, an electrical arrangement shown in the figure is prepared. The arm BC is made up of the semiconductor.The experiment is being conducted at $ 25^\circ $ and resistance of the semiconductor arm is $ 3\,\text{m}\Omega $. Arm BC is cooled at a constant rate of $ 2^\circ\text{C}/\text{s} $. If the galvanometer G shows no deflection after $ 10\,\text{s} $, then $ \alpha $ is:

51.

Four NOR gates are connected as shown in figure. The truth table for the given figure is:

52.

Identify the solar cell characteristics from the following options:

53.

Conductivity of a photodiode starts changing only if the wavelength of incident light is less than $ 660 \, \text{nm} $. The band gap of photodiode is found to be $ \left(\dfrac{X}{8}\right) \, \text{eV} $. The value of $ X $ is:
(Given, $ h = 6.6 \times 10^{-34} \, \text{J s}, \, e = 1.6 \times 10^{-19} \, \text{C} $ )

54.

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Photodiodes are preferably operated in reverse bias condition for light intensity measurement.
Reason (R): The current in the forward bias is more than the current in the reverse bias for a p – n junction diode.
In the light of the above statement, choose the correct answer from the options given below:

55.

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: $ n\text{-}p\text{-}n $ transistor permits more current than a $ p\text{-}n\text{-}p $ transistor.
Reason R: Electrons have greater mobility as a charge carrier.
Choose the correct answer from the options given below:

56.

The effect of increase in temperature on the number of electrons in conduction band $ (n_e) $ and resistance of a semiconductor will be is:

57.

Choose the correct statement about Zener diode:

58.

The following logic gate is equivalent to:

59.

The resistivity $ (\rho) $ of semiconductor varies with temperature. Which of the following curve represents the correct behaviour:

60.

Identify the logic operation carried out.

61.

For a transistor in CE mode to be used as an amplifier, it must be operated in:

62.

A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is $ 1.42 \, \text{eV} $. The wavelength of light emitted from the LED is:

63.

Match List I with List II:

[Choose the correct answer]

64.

Choose the correct logic circuit for the given truth table having inputs $ A $ and $ B $.

65.

The output of the circuit is low (zero) for:

(A) $ X = 0, \, Y = 0 $
(B) $ X = 0, \, Y = 1 $
(C) $ X = 1, \, Y = 0 $
(D) $ X = 1, \, Y = 1 $
Choose the correct answer from the options given below:

66.

In the circuit shown here, assuming threshold voltage of diode is negligibly small, then voltage $ V_{AB} $ is correctly represented by:

67.

If the potential barrier across a p-n junction is $ 0.6 \, \text{V} $. Then the electric field intensity, in the depletion region having the width of $ 6 \times 10^{-6} \, \text{m} $, will be __________ $ \times 10^{5} \, \text{N/C} $

68.

In the following circuit, the reading of the ammeter will be: (Take Zener breakdown voltage $ = 4 \, \text{V} $)

69.

A logic gate circuit has two inputs $ A $ and $ B $ and output $ Y $. The voltage waveforms of $ A $, $ B $ and $ Y $ are shown below:

The logic gate circuit is:

70.

The energy band gap of semiconducting material to produce violet (wavelength $ = 4000\text{ }\mathring{\text{A}} $) LED is ________ $ \text{eV} $. (Round off to the nearest integer).

71.

If $ V_{A} $ and $ V_{B} $ are the input voltages (either $ 5 \, \text{V} $ or $ 0 \, \text{V} $) and $ V_{0} $ is the output voltage then the two gates represented in the following circuits (A) and (B) are:

72.

Following gates section is connected in a complete suitable circuit.

For which of the following combination, bulb will glow (ON):

73.

A Zener diode of breakdown voltage $ 10 \, \text{V} $ is used as a voltage regulator as shown in the figure. The current through the Zener diode is:

74.

The truth table for this given circuit is:

75.

A voltage regulating circuit consisting of Zener diode, having break-down voltage of $ 10 \, \text{V} $ and maximum power dissipation of $ 0.4 \, \text{W} $, is operated at $ 15 \, \text{V} $. The approximate value of protective resistance in this circuit is ______ $ \Omega $.

76.

From the given transfer characteristic of a transistor in CE configuration, the value of power gain of this configuration is $ 10^{x} $, for $ R_{B} = 10 \, \text{k}\Omega $, and $ R_{C} = 1 \, \text{k}\Omega $. Then value of $ x $ is _____.

77.

Consider a $ n $-type semiconductor in which $ n_{e} $ and $ n_{h} $ are number of electrons and holes, respectively.
(A) Holes are minority carriers
(B) The dopant is a pentavalent atom
(C) $ n_{e} n_{h} \neq n_{i}^{2} $
(where $ n_{i} $ is number of electrons or holes in semiconductor when it is intrinsic form)
(D) $ n_{e} n_{h} \ge n_{i}^{2} $
(E) The holes are not generated due to the donors
Choose the correct answer from the options given below:

78.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: A diode under reverse-biased condition provides very small current which is nearly independent of voltage until a critical limit at which the current increases drastically.
Reason R: Below the critical voltage limit, only majority charge carriers flow which increases drastically above critical voltage.
Choose the correct answer from the options given below:

79.

For a constant collector-emitter voltage of $ 8 \, \text{V} $, the collector current of a transistor reached to the value of $ 6 \, \text{mA} $ from $ 4 \, \text{mA} $, whereas base current changed from $ 20 \, \mu\text{A} $ to $ 25 \, \mu\text{A} $ value. If transistor is in active state, small signal current gain (current amplification factor) will be:

80.

In an $ n\!\!-\!p\!\!-\!n $ common emitter $ (CE) $ transistor, the collector current changes from $ 5 \,\text{mA} $ to $ 16 \,\text{mA} $ for the change in base current from $ 100 \,\mu\text{A} $ and $ 200 \,\mu\text{A} $ respectively. The current gain of transistor is _______.

81.

Statement I: When a Si sample is doped with Boron, it becomes P type and when doped by Arsenic it becomes N-type semi-conductor such that P-type has excess holes and N-type has excess electrons.
Statement II: When such P-type and N-type semi-conductors, are fused to make a junction, a current will automatically flow which can be detected with an externally connected ammeter.
In the light of above statements, choose the most appropriate answer from the options given below:

82.

The cut-off voltage of the diodes (shown in figure) in forward bias is $ 0.6 \,\text{V} $. The current through the resister of $ 40 \,\Omega $ is __________ $ \text{mA} $.

83.

In the given circuit, the value of current $ I_{L} $ will be ________ $ \text{mA} $. (When $ R_{L} = 1 \, \text{k}\Omega $)

84.

The truth table for the circuit given below is:

85.

Given below are two statements:
Statement : I PN junction diodes can be used to function as transistor, simply by connecting two diodes, back to back, which acts as the base terminal
Statement : II In the study of transistor, the amplification factor $\beta$ indicates ratio of collector current to the base current
In the light of the above statements, choose the correct answer from the options given below:

86.

If $ X $ and $ Y $ are the inputs, the given circuit works as ________.

87.

To obtain the given truth table, following logic gate should be placed at G:

88.

For a given transistor amplifier circuit in CE configuration $ V_{CC} = 1 \, \text{V} $, $ R_{C} = 1 \, \text{k}\Omega $, $ R_{b} = 100 \, \text{k}\Omega $ and $ \beta = 100 $. Value of base current $ I_{b} $ is:

89.

Identify the correct Logic Gate for the following output $ (Y) $ of two inputs $ A $ and $ B $.

90.

For a transistor $ \alpha $ and $ \beta $ are given as $ \alpha = \dfrac{I_C}{I_E} $ and $ \beta = \dfrac{I_C}{I_B} $. Then the correct relation between $ \alpha $ and $ \beta $ will be:

91.

In a semiconductor p-n diode, the doping concentrations on p-side and n-side are $ 10^{15} $ $ \text{atoms}/\text{cm}^3 $ and $ 10^{18} $ $ \text{atoms}/\text{cm}^3 $, respectively. Which one of the following statements is true?

92.

As per the given circuit, the value of current through the battery will be __________ $ \mathrm{A} $.

93.

The typical transfer characteristics of a transistor in CE configuration is shown in figure. A load resistor of $ 2 \, \text{k}\Omega $ is connected in the collector branch of the circuit used. The input resistance of the transistor is $ 0.50 \, \text{k}\Omega $. The voltage gain of the transistor is __________.

94.

A circuit is arranged as shown in figure. The output voltage $ V_0 $ is equal to ______ V.

95.

The output voltage in the following circuit is: (Consider ideal diode case)

96.

An n.p.n. transistor with current gain $ \beta = 100 $ in common emitter configuration is shown in figure. The output voltage of the amplifier will be:

97.

In the circuit, the logical value of $ A = 1 $ or $ B = 1 $ when potential at $ A $ or $ B $ is $ 5\,\text{V} $ and the logical value of $ A = 0 $ or $ B = 0 $ when potential at $ A $ or $ V $ is $ 0\,\text{V} $.

The truth table of the given circuit will be :

98.

In the following logic circuit the sequence of the inputs A, B are $ (0, 0) $, $ (0, 1) $, $ (1, 0) $ and $ (1, 1) $. The output $ y $ for this sequence will be:

99.

The output of the given circuit diagram is:

100.

The logic gate equivalent to the given circuit diagram is:

101.

The correct relation between $ \alpha $ (ratio of collector current to emitter current) and $ \beta $ (ratio of collector current to base current) of a transistor is:

102.

In the given circuit, the voltage across load resistance $ (R_L) $ is:

103.

The maximum rated power of the LED is $ 2 \, \text{mW} $ and it is used in the circuit with input voltage of $ 5 \, \text{V} $ as shown in the figure below. The current through resistance $ R_S $ is $ 0.5 \, \text{mA} $. The minimum value of the resistance of $ R_S $, to ensure that the LED is not damaged is _____ $ \text{k}\Omega $.

104.

In the given circuit, the current $ (I) $ through the battery will be

105.

Identify the logic operation performed by the given circuit:

106.

For the given circuit (shown in part (A)) the time dependent input voltage $ v_{in}(t) $ and corresponding output $ v_{0}(t) $ are shown in the part (B) and part (C), respectively. Identify the components that the used in the circuit between points $ X $ and $ Y $.

107.

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

108.

Which of the following circuits represents a forward biases diode?

Choose the correct answer from the options given below:

109.

The output $ (Y) $ of logic circuit given below is $ 0 $ only when:

110.

A potential barrier of $ 0.4 \, \text{V} $ exists across a $ p-\!n $ junction. An electron enters the junction from the $ n $ - side with a speed of $ 6.0 \times 10^{5} \, \text{m s}^{-1} $. The speed with which electron enters the $ p $ side will be $ \frac{x}{3} \times 10^{5} \, \text{m s}^{-1} $ the value of $ x $ is ______.
(Given mass of electron $ = 9 \times 10^{-31} \, \text{kg} $, charge on electron $ = 1.6 \times 10^{-19} \, \text{C} $.)

111.

In the digital circuit shown in figure, for the given inputs the $ P $ and $ Q $ values are:

112.

The truth table of the given circuit diagram is:

113.

For the given circuit, the power across zener diode is __________ mW.

114.

Identify the logic operation performed by the given circuit.

115.

For the logic circuit shown, the output waveform at $ Y $ is:

116.

Consider the following statements:
(A) The junction area of solar cell is made very narrow compared to a photo diode.
(B) Solar cells are not connected with any external bias.
(C) LED is made of lightly doped p–n junction.
(D) Increase of forward current results in continuous increase of LED light intensity.
(E) LEDs have to be connected in forward bias for emission of light.
Choose the correct answer from the options given below:

117.

For the given logic gates combination, the correct truth table will be:

118.

For the given logic circuit, which of the following inputs combination will make both LED-1 and LED-2 to glow?

119.

The logic performed by the circuit shown in figure is equivalent to:

120.

A transistor is used in common-emitter mode in an amplifier circuit. When a signal of $ 10 \, \text{mV} $ is added to the base-emitter voltage, the base current changes by $ 10 \, \mu\text{A} $ and the collector current changes by $ 1.5 \, \text{mA} $. The load resistance is $ 5 \, \text{k}\Omega $. The voltage gain of the transistor will be ________.

121.

In the given circuit, the breakdown voltage of the Zener diode is $ 3.0 \, \text{V} $. What is the value of $ I_z $?

122.

For the following circuit and given inputs $A$ and $B$, choose the correct option for output ‘$Y$’.

123.

A transistor is connected in common emitter circuit configuration, the collector supply voltage is $ 10 \, \text{V} $ and the voltage drop across a resistor of $ 1000 \, \Omega $ in the collector circuit is $ 0.6 \, \text{V} $. If the current gain factor $( \beta) $ is $ 24 $, then the base current is _______ $ \mu \text{A} $. (Round off to the Nearest Integer)

124.

The output waveform of the given logical circuit for the following inputs $ A $ and $ B $ as shown below, is:

125.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Diffusion current in a $ p\text{-}n $ junction is greater than the drift current in magnitude if the junction is forward biased.
Reason R: Diffusion current in a $ p\text{-}n $ junction is from the $ n $-side to the $ p $-side if the junction is forward biased.
In the light of the above statements, choose the most appropriate answer from the options given below

126.

In an experiment of CE configuration of n-p-n transistor, the transfer characteristics are observed as given in figure.

If the input resistance is $ 200 \, \Omega $ and output resistance is $ 60 \, \Omega $, the voltage gain in this experiment will be _______.

127.

In a semiconductor, the number density of intrinsic charge carriers at $ 27^\circ \text{C} $ is $ 1.5 \times 10^{16} \, /\text{m}^{3} $. If the semiconductor is doped with impurity atom, the hole density increases to $ 4.5 \times 10^{22} \, /\text{m}^{3} $. The electron density in the doped semiconductor is ______ $ \times 10^{9} \, /\text{m}^{3} $.

128.

For the circuit shown above, equivalent GATE is:

129.

The output $ Y $ for the inputs $ A $ and $ B $ of circuit is given by:

Truth table of the shown circuit is:

130.

Two ideal diodes are connected in the network as shown in figure. The equivalent resistance between $ A $ and $ B $ is __________ $ \Omega $.

131.

Find the truth table for the function $ Y $ of $ A $ and $ B $ represented in the following figure.

132.

An npn transistor operates as a common emitter amplifier with a power gain of $ 10^{6} $. The input circuit resistance is $ 100 \, \Omega $ and the output load resistance is $ 10 \, \text{k}\Omega $. The common emitter current gain $ \beta $ will be ______________. (Round off to the Nearest integer)

133.

For the circuit shown below, calculate the value of $ I_z $:

134.

Which one of the following will be the output of the given circuit?

135.

In the truth table of the above circuit the value of $X$ and $Y$ are:

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JEE Main 2024 8th April Evening Shift

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

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