**विद्युत सेल किसे कहते हैं ?**

- वह युक्ति जिसके द्वारा रासायनिक ऊर्जा को विद्युत ऊर्जा में रूपांतरित किया जाता है विद्युत सेल कहलाती है।
- प्रत्येक सेल में
व**धनाग्र**दो इलेक्ट्रोड होते हैं, जिन्हें क्रमशः**ऋणनाग्र**व**एनोड**कहते हैं।**कैथोड** - सेल में विद्युत अपघट्य होता है जिसमें आयन होते हैं, धनायन कैथोड की और ऋणायन एनोड की ओर गमन करते हैं।

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* Cube Resistance Problem | JEE Main | JEE Advanced :-* For a cube having equal resistance along its edges, let’s determine the equivalent resistance between :-

- Two vertices along the
**body diagonal** - Two vertices along the
**face diagonal** - Two
**adjacent vertices**

Let us consider a cube as shown in figure as shown in figure below :-

Equivalent resistance between two vertices along the body diagonal (AG) is = 5R/6. To prove this let us draw the equivalent representations of the above figure :-

The vertices BDE are at the same potential and also the vertices CFH are also at the same potential due to symmetry.

Again simplifying the circuit, we get :-

*Finally equivalent resistance between two vertices along the body diagonal (AG) = 5R/6.*

**Alternate Method**

Assume that the current flowing into vertex A is ** i**. Now because of the symmetry we can divide the current in different branches as shown in the figure below :-

Now to find potential difference difference, *V _{AG }*, i.e., potential difference between vertices A and G, let us follow the path A – B – C – G and write down the potential drop,

**⇒ R _{eq} = 5R/6**

Let us consider a cube as shown in figure as shown in figure below :-

Equivalent resistance between two vertices along the face diagonal (AC) is = 3R/4. To prove this let us draw the equivalent representations of the above figure :-

Note that the circuit is symmetric about B – F – H – D as indicated by red dotted line, it means all these vertices are at the same potential. This helps us to simplify the circuit again as shown below :-

On further simplifying :-

*Finally equivalent resistance between two vertices along the face diagonal (AC) = 3R/4*

Let us consider a cube as shown in figure as shown in figure below :-

Equivalent resistance between two adjacent vertices (AB) is = 7R/12. To prove this let us draw the equivalent representations of the above figure :-

The red dotted line indicates the horizontal symmetry. Further simplifying the above circuit, we get :-

This gives us :-

*Finally equivalent resistance between two adjacent vertices (AB) = 7R/12*

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The **Davisson and Germer experiment** was a 1923-27 experiment performed by * Clinton Davisson and Lester Germer *to establish the wave nature of slow moving electrons. This confirmed the de-Broglie hypothesis of wave-particle duality.

The Davisson and Germer experiment is contained in a vacuum chamber so that electron deflection and scattering by the medium are avoided. The major components of the experimental setup are following:

**Electron gun: **An electron gun is a Tungsten filament(F) coated with barium oxide(*barium oxide and other alkali oxides helps in decreasing the work function of the tungsten cathode. Work function of uncoated tungsten is 4.5 eV and of coated tungsten is 1.5 eV. Thus the cathode can emit electrons at a relatively low temperature which is highly economical*) that produces electrons by thermionic emission.

**Electrostatic particle accelerator:** To accelerate electrons at a known potential, two oppositely charged plates (positive and negative plate) are employed.

**Collimator:** The accelerator is housed within a cylinder C with a restricted path for electrons running along its axis. Its purpose is to prepare a narrow and straight beam of electrons.

**Target:** N is a nickel crystal cut along cubical diagonal. The nickel crystal is positioned in such a way that it may be rotated around a fixed axis.

**Detector:** D is an electron detector which is used to collect dispersed electrons from the Ni crystal. It is connected to a sensitive galvanometer. As illustrated in the picture below, the detector may be rotated on a semicircular arc.

- A fine beam of accelerated electrons obtained from the electron gun is made to fall normally on the surface of liquid crystal.
- The experiment was performed by varying the accelerating voltage from 44 V to 68 V.
- These accelerated electrons are scattered in different directions by the atoms of the Ni crystal.
- The intensity (
) of the scattered electron beam is recorded by the electron detector D by moving it on circular scale at different values scattering angle**I***ϕ*

- The intensity (
) of the electronic current received by the detector, as well as the scattering angle**I**, were investigated for different accelerating voltage.*ϕ* - The accelerated electrons intensity was not constant. It displayed a maximum and a lowest value.
- It was noticed that at an accelerating voltage of 54 V and at scattering angle
= 50*ϕ*^{º}the intensity I was maximum. - The appearance of peak in intensity in a particular direction is due to constructive interference of scattered electrons from different layers of regularly spaced atoms of the Ni crystal, i.e., the differentiation of electrons takes place.

In the above figure θ is called the glancing angle and ** ϕ** is scattering angle. For

θ + ** ϕ** + θ = 180

Or

θ = (180^{º} – ** ϕ**)/2 = 65

From X-ray scattering, the value of lattice spacing is *d* = 0.91 Å.

According to Bragg’s law, for first order diffraction maxima (n = 1), we get

2 d sinθ = 1 × λ

⇒ 2 × 0.91 sin 65^{º} = 1 × λ

⇒ **λ = 1.65 Å**

According to de-Broglie’s wave-particle duality, the wavelength of the wave associated with electron is given by

⇒ **λ = 1.66 Å**

**As a result, the experimental results correspond well with the theoretical values obtained from the de Broglie equation.**

** This proves the existence of the de-Broglie waves for the slow moving electrons.**

Next Topic :- Heisenberg Uncertainty Principle

Previous Topic :- Radiation Pressure

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* Radiation Pressure :-* Maxwell predicted that an electromagnetic wave carries momentum and when this electromagnetic wave is absorbed by an object it would experience a force in the direction of propagation of the wave. Maxwell’s prediction was confirmed in 1903 by Nichols and Hull by precisely measuring radiation pressures using a torsion balance.

From Einstein’s Mass-Energy Equivalence,

…..(1)

If the wave is incident perpendicular to a surface and it is completely absorbed by the surface then equation (1) tells us the linear momentum imparted to the surface.

*If the surface is perfectly reflecting then change in momentum of the wave is doubled consequently the momentum imparted to the surface is also doubled.*

According to Newton’s second law the force exerted by an electromagnetic wave on a surface is the rate of change of linear momentum, i.e.,

From equation (1)

…..(2)

Intensity of a wave is given by,

From equation (2)

Hence radiation pressure,

(I/c) represent energy density (energy per unit volume) *u.*

Hence,

If the surface is perfectly reflecting than the pressure on the surface is doubled. Thus we can write,

and

**Next Topic :-** Davisson and Germer Experiment

**Previous Topic :- **de Broglie wavelength associated with an electron

**Solution.**

Energy of electron after first collision

E_{1} = (90 % of E) = 2.79 *eV*

Energy of electron after second collision

E_{2} = (90 % of E_{1}) = 2.51 *eV*

Hence, kinetic energy of the electron as it comes out of out of the metal surface

*KE = (2.51 – 2.2) eV = 0.31 eV*

Now

Energy of electron after third collision

E_{3} = (90 % of E_{2}) = 2.26 *eV*

Energy of electron after forth collision

E_{4} = (90 % of E_{3}) = 2.03 *eV < Φ _{0}*

After forth collision, the energy of electron is less than the work function of the metal, so the electron will not be able to come out of the metal.

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**Solution :-**

Using Einstein photoelectric equation,

Now, as

**⇒ v _{max} = 8.4 × 10^{5} ms^{-1}**

**Solution.**

Pitch of helical path,

(as θ = 60º)

Here

Hence Pitch,

Now

E = Φ_{0} + K_{max}

⇒ Φ_{0} = E – K_{max}

⇒ Φ_{0} = (4.9 – 0.4) eV

⇒ Φ_{0} = 4.5 eV

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**Solution.**

Intensity is given by

Where *N* is number of photons incident in *t* time.

As 53% of the incident photon eject photoelectrons, so

Number of photoelectrons emitted per sec

Minimum kinetic energy of photoelectrons, **K _{min} = 0** and

Maximum kinetic energy of photoelectrons,

**K _{max}** = E – Φ

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* State Heisenberg Uncertainty Principle :-* According to Heisenberg

If Δx and Δp are the uncertainties in determining the position and momentum of the particle simultaneously, then

…..(1)

here **h = 6.63 × 10 ^{-34}Js** is Plank’s constant

From equation (1) if we are able to measure the exact position of the particle(Δx = 0) then the uncertainty in the measurement of its linear momentum is infinite(Δp→∞). In the same way if we are able to measure the exact linear momentum of the particle(Δp = 0) then we cannot measure the exact position of the particle at that time.

∴ **Δx = 0, then Δp→∞** and if **Δp = 0 then Δx→∞**

If we have to observe an electron then it cannot be done without light and when light(photons) falls on electron, then it’s momentum changes as shown in figure below :-

Uncertainty Principle is a fundamental part of nature and it holds for all microscopic and macroscopic particles.

*(a) In terms of position and velocity :-*

As momentum p = mv, Δp = mΔv, so from equation (1), we get

…..(2)

*(b) In terms of energy and time :-*

Now by dividing and multiplying equation (1) by Δt, we get

As Δp = FΔt, so

As FΔx = ΔE, hence

…..(3)

*(c) In terms of angular momentum and angular displacement :-*

If the trajectory of the particle is circular with radius r, then

where Δ*l* is the uncertainty in the measurement of position along the trajectory.

Thus

…..(4)

* Previous Topic :-* Davisson and Germer Experiment

**प्रतिरोधों के समान्तर क्रम संयोजन में :-**

- प्रत्येक प्रतिरोध के सिरों पर विभवान्तर समान होता है।
- प्रत्येक प्रतिरोध में से प्रवाहित धारा प्रतिरोध के मान के व्युत्क्रमानुपाती होती है।

- परिपथ में प्रवाहित कुल धारा प्रत्येक प्रतिरोध में से अलग-अलग प्रवाहित धाराओं के योग के बराबर होती है।

अतः

**उदाहरण 1.**

प्रतिरोध R, 2R, 4R, 8R … ∞ समान्तर क्रम में जुडे हुए हैं। उनका तुल्य प्रतिरोध कितना होगा?

**हल :-**

**उदाहरण 2.**

दिए गए परिपथ का तुल्य प्रतिरोध ज्ञात कीजिये।

**हल :-**

यहां सभी प्रतिरोध बिंदु A व B के मध्य जुड़े हुए हैं। अतः तुल्य परिपथ है।

अथवा

अतः

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