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HC Verma ii Solutions for Class 12-science Physics Chapter 19 - Electric Current Through Gases

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Question 1:

Cathode rays constitute a stream of
(a) electrons
(b) protons
(c) positive ions
(d) negative ions

(a) electrons

Cathode rays consist of a stream of fast moving electrons.

Question 2:

Cathode rays are passing through a discharge tube. In the tube, there is
(a) an electric field but no magnetic field
(b) a magnetic field but no electric field
(c) an electric as well as a magnetic field
(d) neither an electric nor a magnetic field

(c) an electric as well as a magnetic field

Cathode rays consist of beams of electrons that constitute current and, hence, magnetic field. We know electric field is produced by a charge, whether it is stationary or moving, So, electric field will also be present inside the tube.

Question 3:

How will the thermionic current vary if the filament current is increased?

If the filament current is increased, it will increase the temperature of the cathode (metal) and the cathode will emit more electrons. This will lead to an increase in the number of thermions emitted per unit time. As a result, the thermionic current will increase.

Question 4:

Would you prefer a material with a high melting point or a low melting point to be used as a cathode in a diode?

We will prefer a material with high melting point to be used as the cathode in a diode. The material of the cathode of a diode should be resistant to high temperature, have high melting point and be electrically conductive because thermionic emission occurs at high temperature.

Question 5:

Would you prefer a material with a high work-function or a low work-function to be used as a cathode in a diode?

We will prefer a material with low work-function to be used as a cathode in a diode, so that electron emission can occur using a small amount of energy.

Question 6:

An isolated metal sphere is heated to a high temperature. Will it become positively charged due to thermionic emission?

Yes, it will become positively charged due to thermionic emission. When the metal sphere is heated, average kinetic energy of the electrons will increases, due to which, the free electrons of the metal sphere will be able to escape. As a result, they will leave a positive charge on the isolated metal sphere.

Question 7:

A diode value is connected to a battery and a load resistance. The filament is heated, so that a constant current is obtained in the circuit. As the cathode continuously emits electrons, does it become more and more positively charged?

No, the cathode does not become more and more positively charged. In the diode circuit,  the cathode of the diode valve is always connected to the negative terminal of the battery. So, the battery will supply the electrons to the cathode and the cathode will continuously emit electrons without becoming more and more positively charged.

Question 8:

Why does thermionic emission not take place in non-conductors?

For thermionic emission, material should have low work function and large number of free electrons. But nonconductor does not have free electrons and they have higher work function. So, thermionic emission does not takes place in non-conductors.

Question 9:

The cathode of a diode valve is replaced by another cathode of double the surface area. Keeping the voltage and temperature conditions the same, will the place current decrease, increase or remain the same?

As the rate of emission of thermions is directly proportional to the surface area of the surface emitting it, the plate current doubles if the cathode of a diode valve is replaced by another cathode of double the surface area.Hence plate current increases.

Question 10:

Why is the linear portion of the triode characteristic chosen to operate the triode as an amplifier?

When the operating point lies on the linear portion of the characteristics curve, change in voltage across the load resistance follows the pattern of the input signal, but the amplitude is much larger. This can be done by choosing the linear portion of triode characteristic.

Question 7:

The plate current in a triode valve is zero. The temperature of the filament is high. It is possible that
(a) Vg > 0, Vp > 0
(b) Vg > 0, Vp < 0
(c) Vg < 0, Vp > 0
(d) Vg < 0, Vp < 0

(b) Vg > 0, Vp < 0
(c) Vg < 0, Vp > 0
(d) Vg < 0, Vp < 0

If the temperature of the filament is high, then it will emit electrons. Other conditions for the  plate current in the diode are:
(1) Positive grid voltage, Vg
(2) Positive plate voltage, VP
If any of the above conditions is not satisfied, then plate current in a triode will be zero.

Question 3:

Due to the space charge in a diode valve,
(a) the plate current decreases
(b) the plate voltage increases
(c) the rate of emission of thermions increases
(d) the saturation current increases

(a) the plate current decreases

A diode valve consists of a negatively-charged region between the cathode and anode, called the space charge region. This negatively-charged region repels the electrons coming from the cathode and, thus, it reduces the plate current. On the other hand, this region has no effect on plate voltage, rate of emission of thermions and the saturation current.

Question 4:

The saturation current in a triode valve can be changed by changing
(a) the grid voltage
(b) the plate voltage
(c) the separation between the grid and the cathode
(d) the temperature of the cathode

(a) the grid voltage

Since the triode value consists of a space charge region near the cathode, this space charge region is affected by the grid voltage. If the grid voltage is made negative, it repels the electrons coming from the cathode and reduces current. If this voltage is made positive, it helps the electrons move towards the anode. Hence, this voltage is more effective in changing the plate current. Since, the saturated current in the triode is the maximum value of plate current, the saturated current in the triode valve is effectively controlled by the grid voltage. More negative is the grid voltage, less is the value of saturated current.

Question 5:

Mark the correct options.
(a) A diode valve can be used as a rectifier.
(b) A triode valve can be used as a rectifier.
(c) A diode valve can be used as an amplifier.
(d) A triode valve can be used as an amplifier.

(a) A diode valve can be used as a rectifier.
(b) A triode valve can be used as a rectifier.
(d) A triode valve can be used as an amplifier.

A diode valve and a triode valve allow current to flow only in one direction. Since a rectifier is a device that converts alternating current (bi-directional) into direct current (uni-directional), a diode valve and a triode valve can be used as rectifiers.
A triode valve can control its output in proportion to the input signal. That is, it can act as an amplifier, whereas a diode valve cannot control its output in proportion to the input signal. So, it cannot be use as an amplifier.

Question 6:

The plate current in a diode is zero. It is possible that
(a) the plate voltage is zero
(b) the plate voltage is slightly negative
(c) the plate voltage is slightly positive
(d) the temperature of the filament is low

(a) the plate voltage is zero
(b) the plate voltage is slightly negative
(c) the plate voltage is slightly positive
(d) the temperature of the filament is low

The plate current varies directly with the plate voltage. Therefore, if the plate voltage is zero, the plate current is also zero. Due to the same reason, if the plate voltage is negative, the plate current will be zero. Now, if the plate is slightly positive, then it may be the reason that the plate voltage is not able to reduce the effect of space charge. Hence, the current may be zero. Now, as the temperature of the filament is low, it will not be able to emit electrons and the resulting plate current will be zero. Hence, all the options are possible.

Question 7:

A triode is operated in the linear region of its characteristics. If the plate voltage is slightly increased, the dynamic plate resistance will
(a) increase
(b) decrease
(c) remain almost the same
(d) become zero

(c) remain almost the same

Dynamic resistance (rP) is given by,
rP = $\frac{∆{V}_{P}}{∆{i}_{P}}$
The triode is operating in the linear region. Therefore,
Change in the value of voltage = Change in the value of current
So, (rP) will remain almost the same.

Question 8:

The plate current in a triode value is maximum when the potential of the grid is
(a) positive
(b) zero
(c) negative
(d) non-positive

(a) positive

If the grid voltage is made positive, it will help the electrons move towards the anode, which will help in increasing the current. Thus, the plate current in the triode value is maximum when the potential of the grid is positive.

Question 9:

The amplification factor of a triode operating in the linear region depends strongly on
(a) the temperature of the cathode
(b) the plate potential
(c) the grid potential
(d) the separations of the grid from the cathode and the anode

(d) the separations of the grid from the cathode and the anode

When the triode operating in the linear region, value of plate voltage, grid voltage and plate current is already specified (or fixed). If the grid is near the cathode, then we assume that grid lie in the region where the space charge density is more. There is a chance that grid will pick up electrons from the space charge region of cathode and effect the plate current and hence amplification factor (as we know that amplification factor will depend on the plate current). If the grid is far from the cathode, it will reduce the grid voltage and hence the amplification factor also get affected.

Question 1:

A discharge tube contains helium at low pressure. A large potential difference is applied across the tube. Consider a helium atom that has just been ionised due to the detachment of an atomic electron. Find the ratio of the distance travelled by the free electron to that by the positive ion in a short time dt after ionisation.

Charge on the electron, $-$q = −1.6 × 10−19 C
Charge on the helium ion = +q = +1.6 × 10−19 C
Let the electric field in the tube be E.
Mass of electron, me = 9.1 × 10−31 kg
Mass of proton, mp = 1.67 × 10−27 kg
Mass of helium ion, = 4mp = 4×(1.67 × 10−27 ) kg
Magnitude of the force experienced by the electron,
F = qE
Magnitude of acceleration of the electron,
a = $\frac{qE}{{m}_{\mathrm{e}}}$
Distance travelled by the electron in time t,

Acceleration of the positive ion $=\frac{qE}{4×{m}_{p}}$
Distance travelled by the helium ion,

The ratio of distance travelled by the given particles is given by
$\frac{{S}_{\mathrm{e}}}{{S}_{\mathrm{He}}}=\frac{4{m}_{\mathrm{p}}}{{m}_{\mathrm{e}}}\phantom{\rule{0ex}{0ex}}\frac{{S}_{\mathrm{e}}}{{S}_{\mathrm{He}}}=\frac{4×1.67×{10}^{-27}}{9.1×{10}^{-31}}\phantom{\rule{0ex}{0ex}}\frac{{S}_{\mathrm{e}}}{{S}_{\mathrm{He}}}=0.73406×{10}^{4}=7340.6\phantom{\rule{0ex}{0ex}}⇒\frac{{S}_{\mathrm{e}}}{{S}_{\mathrm{He}}}=7340$

Question 2:

A molecule of a gas, filled in a discharge tube, gets ionised when an electron is detached from it. An electric field of 5.0 kV m−1 exists in the vicinity of the event. (a) Find the distance travelled by the free electron in 1 µs, assuming there's no collision. (b) If the mean free path of the electron is 1.0 mm, estimate the time of transit of the free electron between successive collisions.

Given:
Electric field inside discharge tube, E = 5.0 kV/m

(a)
Let S be distance travelled by the free electron in 1 µs.

(b)
Let the time of transit of the free electron between successive collisions be t.

Question 3:

The mean free path of electrons in a gas in a discharge tube is inversely proportional to the pressure inside it. The Crookes dark space occupies half the length of the discharge tube when the pressure is 0.02 mm of mercury. Estimate the pressure at which the dark space will fill the whole tube.

Let the mean free path of the electrons be L and pressure inside the tube be P.
It is given that the mean free path of electrons and the pressure inside the tube are related as:
$L\propto \frac{1}{P}$
Here, L = half of tube's length
Pressure inside the tube, P = 0.02 mm of Hg

As pressure 'P' becomes half, mean free path of electrons, L, doubles. So, the whole tube is filled with Crookes dark space.
Thus, the pressure required for filling the whole tube with Crookes dark space,

Question 4:

Two discharge tubes have identical material structures and the same gas is filled in them. The length of one tube is 10 cm and that of the other tube is 20 cm. Sparking starts in both the tubes when the potential difference between the cathode and the anode is 100 V. If the pressure in the shorter tube is 1.0 mm of mercury, what is the pressure in the longer tube?

According to Paschen's equation, sparking potential (V) is a function of the product of pressure (P) inside the tube and the length (d) of the tube.
$V=f\left(Pd\right)$
As the sparking potential is same in both the tubes, the product of P and d must be equal for both the tubes.
Length of the smaller tube, ds = 10 cm
Pressure inside the smaller tube, Ps = 1.0 mm of mercury
Let the pressure inside the longer  tube, Pl,, be x mm of mercury.
Length of the longer tube, dl = 20 cm

Hence, the pressure in the longer tube is 0.5 mm of Hg.

Question 5:

Calculate n(T)/n(1000 K) for tungsten emitter at T = 300 K, 2000 K and 3000 K, where n(T) represents the number of thermions emitted per second by the surface at temperature T. Work function of tungsten is 4.52 eV.

According to Richardson-Dushman equation, the number of thermions (n) emitted by a surface, in a given time (t), is given by
$i=ne=AS{T}^{2}{e}^{-\varphi /kT}\phantom{\rule{0ex}{0ex}}A\text{'}=\frac{A}{e}\phantom{\rule{0ex}{0ex}}⇒n=A\text{'}S{T}^{2}{e}^{-\varphi /kT}$
Here,

$\frac{n\left(3000K\right)}{n\left(1000\mathrm{K}\right)}=\frac{A\text{'}S×\left(3000{\right)}^{2}×{e}^{\left(-4.52×1.6×{10}^{-19}\right)/\left(1.38×{10}^{-23}×3000\right)}}{A\text{'}S×1.7396×{10}^{-17}}\phantom{\rule{0ex}{0ex}}\frac{n\left(3000K\right)}{n\left(1000\mathrm{K}\right)}=\frac{\left(9×{10}^{6}\right)×\left(2.5913×{10}^{-8}\right)}{\left(1.7396×{10}^{-17}\right)}\phantom{\rule{0ex}{0ex}}\frac{n\left(3000K\right)}{n\left(1000\mathrm{K}\right)}=1.34×{10}^{16}$

Question 6:

The saturation current from a thoriated-tungsten cathode at 2000 K is 100 mA. What will be the saturation current for a pure-tungsten cathode of the same surface area operating at the same temperature? The constant A in the Richardson−Dushman equation is 60 × 104 Am −2 K−2 for pure tungsten and 3.0 × 104 Am −2 k−2 for thoriated tungsten. The work function of pure tungsten is 4.5 eV and that of thoriated tungsten is 2.6 eV.

According to Richardson-Dushman equation, the current of thermions is given by
$i=AS{T}^{2}{e}^{-\varphi /KT}$
For the thoriated tungsten cathode:
Saturation current, i = 100 mA
Temperature, T = 2000 K
A = 3.0 × 104 Am−2K−2
ϕ = 2.6 eV.

For the pure tungsten cathode:
Let the saturation current be i.
Temperature, T = 2000 K
A = 60 × 104 Am −2 K−2
ϕ = 2.6 eV

k = 1.38× 10−23 J/K

Substituting the values of the quantities in the Richard-Dushman equation, we get:

Dividing equation (1) by (2)

Question 7:

A tungsten cathode and a thoriated-tungsten cathode have the same geometric dimensions and are operated at the same temperature. The thoriated-tungsten cathode gives 5000 times more current than the other cathode. Find the operating temperature. Take relevant data from the previous problem.

For the pure tungsten cathode,
work function, ϕ = 4.5 eV
A = 60×104 A/m2-K2
For the thoriated-tungsten cathode
work function, ϕ = 2.6 eV
A = 3×104 A/m2-K2

Saturation current,
$i=AS{T}^{2}{e}^{-\varphi /KT}$
For thoriated tungsten,

Taking 'ln' of both sides, we get:

Question 8:

If the temperature of a tungsten filament is raised from 2000 K to 2010 K, by what factor does the emission current change? Work function of tungsten is 4.5 eV.

Given:
Work function of tungsten = 4.5 eV
Initial temperature of tungsten filament, T = 2000 K
Final temperature of tungsten filament, T' = 2010 K
Let the emission current at (T = 2000 K) be i.
Let the emission current at (T' = 2010 K) be i'.
The emission currents are given by
$i=AS{T}^{2}{e}^{-\varphi /kT}$
$i\text{'}=AST{\text{'}}^{2}{e}^{-e/kT\text{'}}$
Dividing i by i', we get:
$\frac{i}{i\text{'}}=\frac{{T}^{2}}{T{\text{'}}^{2}}\frac{{e}^{-\varphi /kT\text{'}}}{{e}^{-\varphi /kT\text{'}}}\phantom{\rule{0ex}{0ex}}\frac{i}{i\text{'}}={\left(\frac{T}{T\text{'}}\right)}^{2}{e}^{-\varphi /kT+\varphi /kT\text{'}}\phantom{\rule{0ex}{0ex}}={\left(\frac{T}{T\text{'}}\right)}^{2}{e}^{\frac{\varphi }{k}\left(\frac{1}{T\text{'}}-\frac{1}{T}\right)}\phantom{\rule{0ex}{0ex}}\frac{i}{i\text{'}}={\left(\frac{2000}{2010}\right)}^{2}{e}^{\frac{4.5×1.6×{10}^{-19}}{1.38×{10}^{-23}}\left(\frac{1}{2010}-\frac{1}{2000}\right).}\phantom{\rule{0ex}{0ex}}=\frac{40000}{\left(201{\right)}^{2}}{e}^{\frac{4.5×1.6}{1.38}\left(-0.0248\right)}\phantom{\rule{0ex}{0ex}}=\frac{0.8786}{\left(201{\right)}^{2}}×40000=0.8699\phantom{\rule{0ex}{0ex}}\frac{i}{i\text{'}}=\frac{1}{0.8699}=1.1495=1.14$

Question 9:

The constant A in the Richardson−Dushman equation for tungsten is 60 × 104 A m−2 K−2. The work function of tungsten is 4.5 eV. A tungsten cathode with a surface area 2.0 × 10−5 m2 is heated by a 24 W electric heater. In steady state, the heat radiated by the heater and the cathode equals the energy input by the heater and the temperature becomes constant. Assuming that the cathode radiates like a blackbody, calculate the saturation current due to thermions. Take Stefan's Constant = 6 × 10−8 W m−2 K−1. Assume that the thermions take only a small fraction of the heat supplied.

Given:
A = 60 × 104 A m−2 K−2
Work function of tungsten, ϕ = 4.5 eV
Stefan's Constant, σ = 6 × 10−8 W m−2 K−1
Surface area of tungsten cathode, S = 2.0 × 10−5 m2
Boltzmann's Constant, k = 1.38× 10−23 J/K
Heat supplied by the heater, H = 24 W

The cathode acts as a black body; thus, its emissivity is 1.
According Stefan's Law:
The power (P) radiated by a blackbody with surface area (A) and temperature (T),

According to the Richard-Dushmann equation, emission current,

Question 10:

A plate current of 10 mA is obtained when 60 volts are applied across a diode tube. Assuming the Langmuir-Child relation ${i}_{p}\infty {V}_{p}^{3/2}$ to hold, find the dynamic resistance rp in this operating condition.

According to Lamgmuir-Child Law,
the relation between plate current (ip) and the plate voltage (Vp) is given by

Differentiating equation (1) with respect Vp, we get:

Dividing (2) and (1), we get:
$\frac{1}{{i}_{p}}\frac{d{i}_{p}}{d{v}_{p}}=\frac{3/2C{{V}_{p}}^{1/2}}{C{{V}_{p}}^{3/2}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{{i}_{\mathrm{p}}}.\frac{d{i}_{\mathrm{p}}}{d{v}_{\mathrm{p}}}=\frac{3}{2{V}_{\mathrm{p}}}\phantom{\rule{0ex}{0ex}}$

The dynamic resistance is given by:

Question 11:

The plate current in a diode is 20 mA when the plate voltage is 50 V or 60 V. What will be the current if the plate voltage is 70 V?

For 50 V or 60 V, the plate current is 20 mA. That means 20 mA is the saturation current.
At the given temperature, the plate current is 20 mA for all other values of voltages.

Thus, the current at 70 V will be 20 mA.

Question 12:

The power delivered in the plate circular of a diode is 1.0 W when the plate voltage is 36 V. Find the power delivered if the plate voltage is increased to 49 V. Assume Langmuir-Child equation to hold.

Given:

When plate voltage, Vp, is 36 V, power delivered in the plate circular of a diode, P, is 1.0 W.
Let the plate current be Ip.

Let the power delivered be P' and plate current be Ip' when plate voltage, Vp, is increased to 49 V

According to Langmuir-Child equation,

Thus, power delivered when the plate voltage is increased to 49 V,

Question 13:

A triode value operates at Vp = 225 V and Vg = −0.5 V.
The plate current remains unchanged if the plate voltage is increased to 250 V and the grid voltage is decreased to −2.5 V. Calculate the amplification factor.

Amplification factor for a triode valve,

At constant plate current,

This implies that the amplification factor of the triode is 12.5 .

Question 14:

Calculate the amplification factor of a triode valve that has plate resistance of 2 kΩ and transconductance of 2 millimho.

Given:
Plate resistance,

Transconductance of plate,

Therefore, the amplification factor is given as:
$\mu ={r}_{\mathrm{P}}×{g}_{\mathrm{m}}\phantom{\rule{0ex}{0ex}}\mu =2×{10}^{3}×2×{10}^{-3}=4\phantom{\rule{0ex}{0ex}}$
That is, amplification factor is 4.

Question 15:

The dynamic plate resistance of a triode value is 10 kΩ. Find the change in the plate current if the plate voltage is changed from 200 V to 220 V.

Given:
Plate resistance,
Change in plate voltage,
Plate resistance at constant grid voltage is given as:

Question 16:

Find the values of rp, µ and gm of a triode operating at plate voltage 200 V and grid voltage −6. The plate characteristics are shown in the figure.
Figure

Dynamic plate resistance,
${r}_{p}=\left(\frac{{\mathrm{\delta V}}_{p}}{{\mathrm{\delta I}}_{p}}\right)$ , at constant grid voltage
We need to find the slope of the graph for a particular value of grid voltage, i.e. Vg = −6 V.

Consider two points for the plot of Vg = −6 V:

Consider two points on the 200 V line:

Amplification factor,

Question 17:

The plate resistance of a triode is 8 kΩ and the transconductance is 2.5 millimho. (a) If the plate voltage is increased by 48 V and the grid voltage is kept constant, what will be the increase in the plate current? (b) With plate voltage kept constant at this increased value, by how much should the grid voltage be decreased in order to bring the plate current back to its initial value?

Given:
Plate resistance,
Change in plate voltage,
Formula for plate resistance:

(b) Now, Vp is kept constant.
Change in plate current,

Transconductance,

Question 18:

The plate resistance and amplification factor of a triode are 10kΩ and 20. The tube is operated at plate voltage 250 V and grid voltage −7.5 V. The plate current is 10 mA. (a) To what value should the grid voltage be changed so as to increase the plate current to 15 mA? (b) To what value should the plate voltage be changed to take the plate current back to 10 mA?

Given:
Plate resistance,
Amplification factor, $\mu =20$
Plate voltage,
Grid voltage,

(a) Transconductance,

Question 19:

The plate current, plate voltage and grid voltage of a 6F6 triode tube are related as
ip = 41 (Vp + 7 Vg)1.41
Here, Vp and Vg are in volts and ip in microamperes.
The tube is operated at Vp = 250 V, Vg = −20 V. Calculate (a) the tube current, (b) the plate resistance, (c) the mutual conductance and (d) the amplification factor.

Given:
Plate voltage, VP = 250 V
Grid voltage, VG = $-$20 V

(a) As given in the question, plate current varies as,

(b) ${i}_{\mathrm{p}}=41\left({V}_{\mathrm{p}}+7{V}_{\mathrm{G}}{\right)}^{1.41}$
Differentiating this equation, we get:

(c) From above:

(d) Amplification factor,

Question 20:

The plate current in a triode can be written as
${i}_{p}=k{\left({v}_{g}+\frac{{V}_{p}}{\mathrm{\mu }}\right)}^{3/2}$
Show that the mutual conductance is proportional to the cube root of the plate current.

Given: The plate current varies with plate and grid voltage as

Differentiating the equation w.r.t ${V}_{\mathrm{G}}$ ,we get:

From (1), plate current can be written in terms of transconductance as:

Question 21:

A triode has mutual conductance of 2.0 millimho and plate resistance of 20 kΩ. It is desired to amplify a signal by a factor of 30. What load resistance should be added in the circuit?

Plate resistance, rp = 20 kΩ
Mutual Conductance, gm = 2.0 milli mho
gm = 2 × 10−3 mho

Amplification factor, µ = 30
Load Resistance = RL = ?
We know: $A\mathit{=}\frac{\mu }{\mathit{1}\mathit{+}\frac{{r}_{\mathrm{p}}}{{R}_{\mathrm{L}}}}$,
where A is the voltage amplification factor.
Also, amplification factor,

Question 22:

The gain factor of an amplifier in increased from 10 to 12 as the load resistance is changed from 4 kΩ to 8 kΩ. Calculate (a) the amplification factor and (b) the plate resistance.

We know:
Voltage gain = $\frac{\mu }{1+\frac{{r}_{\mathrm{p}}}{{R}_{\mathrm{L}}}}$                   ...(1)
When voltage amplification factor, A = 10,
RL = 4 kΩ

Now, increased gain,  A = 12
Substituting this value in (1) ,we get:

On solving equations (2) and (3), we get:

Question 23:

The figure shows two identical triode tubes connected in parallel. The anodes are connected together and the cathodes are connected together. Show that the equivalent plate resistance is half the individual plate resistance, the equivalent mutual conductance is double the individual mutual conductance and the equivalent amplification factor is the same as the individual amplification factor.
Figure

From the circuit,
Cathode of both the triodes are connected to a common point and their anodes are also connected together. Thus, the 2 triodes are connected in parallel.

As plate resistance is obtained by observing the variation of plate voltage with plate current, keeping grid voltage as constant.

Hence equivalent plate resistance

To find the equivalent transconductance
We need to look at the variation of current with respect to grid voltage, keeping plate voltage as constant.

From the circuit
Now the 2 triodes are in series w.r.t grid voltage.
Hence equivalent transconductance will be equal to

Now the equivalent amplification factor is,

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