Laws of photo electric effect

Laws of photo electric effect : The laws of photo electric effect are----

• The number of electrons emitted per second will be directly proportional to the intensity of radiated light.

                  Number of photo electrons α intensity of incident light

                  Number of photo electrons α photo electric current

• There is a lower limit of frequency called threshold frequency below it no emission takes place however high the intensity of the incident radiation. So that ---

                                 E > W₀ ,    ν > ν₀   and       λ <  λ₀ 

• Above threshold frequency the maximum velocity with which electrons emerges is dependent on the frequency and not on the intensity of the radiated light.

                                    ½ mv2max  α  ν

• The emission of photo electron is an instantaneous process. It means time difference between incidence of light and emission of photo electron is very small may be less than even 10-9 second.{no time lag}

                                                                                 https://youtu.be/O3D2belxKBs

stopping potential or cut off potential

Cut-off potential : The minimum negative potential required to just make the photo electric current of the plate is zero, defined as stopping potential.

                 Kmax = eV₀           {eV₀ = stopping potential} 

Electron volt : One electron volt is the kinetic energy gained by an electron when it is accelerated through a potential difference of one volt.

                                     1eV = qV

                                     1eV = 1.602 x 10-19  x 1

                                     [1eV = 1.602 x 10-19 ]   

                                                                                             https://youtu.be/O3D2belxKBs      

Electric field due to uniformly charged thin wire

Electric field due to uniformly charged thin wire :

Consider a long  thin wire carrying charge Q distributed uniformly over the wire. Let the linear charge density of the wire is λ. Let a point p where we have to find the electric field making an angle α with the centre. Then the electric field will be -----

                   

 

       As we know,

                               λ = dQ/dy

                              dQ = λdy    .............................(1)

                In the above triangle,

                              tanα  = y/x

                              y= x tanα   

                             dy = x sec2αdα  ..................(2)

  

              In the above triangle,

                            cosα = x/r

                             r = x  cosα  .......................(3)

           Electric field at point due to dQ ----

                            dE= dQ/4πЄ₀r2

                            dE= λdy/4πЄ₀r2         {since, dQ=λdl from eq. 1}      .................(4)

  putting the value of dy and r from equation (2) and (3) in eq. (4), we get----

                            dE =  λ x sec2α dα/4πЄ₀ x2sec2α

                            dE = λdα/4πЄ₀x 

On x-axis the electric field will be-----

                            dEx  = λdα cosα/4πЄ₀x

                             ഽdEx  = ഽ-π/2π/2λ cosαdα/4πЄ₀x

                              ഽdEx  = λ /4πЄ₀xഽ-π/2π/2 cosαdα

                               Ex  = λ/4πЄ₀ x {[sinα]-π/2π/2}

                                Ex  = λ/4πЄ₀ x {Sinπ/2 + Sinπ/2} 

                                Ex  = 2λ/4πЄ₀ x

so the electric field in x-axis will be-

                                 Ex  = λ/2πЄ₀ x.

On the y-axis electric field will be -----

                               dEy  = λdα sinα/4πЄ₀x

                              ഽdEy  = ഽ-π/2π/2λ sinαdα/4πЄ₀x

                             ഽdEy  = λ /4πЄ₀xഽ-π/2π/2 sinαdα

                              Ey  = λ/4πЄ₀ x {[-cosα]-π/2π/2}

                              Ey  = λ/4πЄ₀ x {cosπ/2 + cosπ/2} 

                               Ey  = 0.

so the net electric field on the y-axis will be ---

                                  Ey  = 0.

  

                                                                                                                          https://youtu.be/O3D2belxKBs

Electric field on the axis of circular uniformly charged ring

Electric field on the axis of circular uniformly charged ring

Consider a circular ring of radius 'R' having charge 'Q' which is uniformly distributed. Let the linear charge density of the ring is λ.

Then electric field will be   ----

                                                    

         As we know,           λ = Q/l    
                                     λ = Q/2πR, 
                                     λ= dQ/dl,    so      dQ= λdl ..........(1)
                                      
                                      Q= λ(2πR) ..............(2)

                                       cosφ= x/r
                                        cosφ = x/√R2+x2

As we know,       

dE = dQ/4πЄ₀r2 

dE = λdl/4πЄ₀r2

              dE= λdl/4πЄ₀ (√R2+x2 )2

Net electric field at point  p will be----

ഽdE = ഽ02πRdE cosφ

ഽdE = λcosφ/4πЄ₀ (R2+x2 )ഽ02πRdl

   E= λx/4πЄ₀ (√R2+x2 )(R2+x2 )[l]02πR          {since, cosφ = x/√R2+x2 }   

E= λx2πR /4πЄ₀ (R2+x2 )3/2      
 E=QX/4πЄ₀ (R2+x2 )3/2     {since, Q=λ2πR}

       so net electric field at point p will be------------

 E = QX /4πЄ₀ (R2+x2 )3/2

Important conclusion : a) Electric field due to ring exist only in x-direction.

b)Net electric field due to ring in y-direction will be zero.

                                            Ey  =0

c) If point p is very far away from centre, i.e., x>>R,

                                       x2 >>>>R2  

so,       R2 can be neglected, then the electric field will be-

                                       E=Q/4πЄ₀x2

it means the ring will behave as a point charge.

                                                            https://youtu.be/O3D2belxKBs

EAN

Effective Atomic Number (EAN) : It can be defined as the resultant number of electrons with the metal atom or ion after gaining electron from the donor atom of the ligands.

In order to explain the stability of a complex, Sidgwick proposed effective atomic number.The EAN generally coincides with the atomic number of next noble gas in some case.

EAN can be calculated by following relation-

   EAN = atomic number of the metal - number of electron lost in the ion formation + number of electron gained by the donor atom of the ligands.

                                                                           https://youtu.be/O3D2belxKBs

Valence bond theory

Valence bond theory : The salient feature of this theory are-

• The central metal ion has a number of empty orbitals for accommodating electrons donated by the ligands. The number of empty orbitals is equal to the co-ordination number of the metal ion for the particular complex.
• The atomic orbitals (s, p or d) of a metal ion hybridize to form hybrid orbitals with definite directional properties. These hybrid orbitals now overlaps with the ligands orbitals to form strong chemical bonds.
• The d-orbitals involved in the hybridisation may be either inner (n-1)d or outer n-d orbitals.The complexes formed in the two ways are referred to as low  spin or high spin complexes, respectively.
• Each ligand contains a lone pair of electrons.
• A covalent bond is formed by the overlapping of a vacant hybridised metal orbital and a filled orbital of a ligand. The bond is also sometime called as co-ordinate bond.
• If a complex contains an unpaired electrons, it is paramagnetic in nature, while if it does not contains unpaired electron, it is diamagnetic in nature.
• The number of unpaired electrons in the complex, points out the geometry of the complex as well as hybridisation of the central metal ion and vice-versa. In practice, the number of unpaired electrons in a complex as formed from magnetic moment measured from-

                                       μ={n(n+2)}1/2

                   { where n= no. of lone pair of electrons.}

                                                                                           https://youtu.be/O3D2belxKBs

Werner's co-ordination theory

Werner's Co-ordination Theory : The postulates of this theory are -

• Each metal in co-ordination compound possesses two type of valencies :

i)Principal valency or Primary valency or ionisable valency.

ii)Secondary valency or non-ionisable valency.

• Primary valencies are satisfied by anion only. The number of primary valencies depends upon the oxidation state of the central metal.These are represented by dotted lines between central metal atom and anion.
• Secondary valencies are satisfied only by electron pair donor, the ions or the neutral species. These are represented by thick lines.
• Every central ions tends to satisfy its primary as well as secondary valencies.
• The secondary valencies are directional and are directed in space about the central metal ions. The primary valencies are non-directional.
• The presence of secondary valencies gives rise to stereoisomerism in complexes.
• The ions attached to the primary valency possesses ionizing nature whereas, the ions attached to secondary valencies do not ionizes when the complex is dissolved in a solvent.
•  Initially, Werner had pointed out co-ordination number of a metal to be four or six.

Note: Now it has been proposed that coordination number of a metal may be any whole number between 2 and 9.

Sidgwick theory and EAN rule : 

Sidgwick suggested that the metal ion will continue accepting electron pair till the total number of electron in the metal ion and those donated by ligands is equal to that to the next noble gas. The total number of electron is called EAN of the metal.

                                                                                 https://youtu.be/O3D2belxKBs

Superposition theorem

             Superposition Theorem

 According to this theorem,"In any linear network containing linear impedances and two or more than two potential sources, the current flowing in any element is the algebraic sum of current that would flow in that element by each potential source, all other sources being replaced at that time by their internal impedances".

Let the current due to E1 and E2 acting together are I1 and I2 (Fig. a) and let the current due to e.m.f. E1 acting alone is I1’ and I2’ (Fig. b) and due to E2 acting alone is I1” and I2”.

Applying Kirchoff’s second law to the mesh of fig. (a) we have

E1 = I1 (Z1 + Z3) + I2Z3    ...............(1)

  And              E2 = I2 (Z2 + Z3) + I1Z3    ..............(2)  

When E1 is considered to act alone, mesh of fig. (b) gives

                        E1 = I1’ (Z1 + Z3) + I2’Z3    .............(3)

                        0 = I2’ (Z2 + Z3) + I1’ Z3      ..............(4)

And, when E2  considered alone mesh if fig. (c ) gives

                        0 = I1” (Z1 + Z3) + I2” Z3     ...............(5)

                         E2 = I2” (Z2 + Z3) + I1” Z3   ...............(6)

 adding equation (3) and (5), we get—

          E1 + 0 =  I1’ (Z1 + Z3) + I2’Z3  +  I1” (Z1 + Z3) + I2” Z3

          E1 = (I1’ + I1”) (Z1 + Z3) + (I2’ + I2”)Z3     .......…..…(7)

Adding equation (4) and (6), we get ------

          0 + E2 = I2’ (Z2 + Z3) + I1’ Z3  + I2” (Z2 + Z3) + I1” Z3

          E2 = (I2’ + I2”) (Z2 + Z3) + ( I1’ + I1”)Z3   ..….....…    (8)

equation (7) and (8) are identical with equation (1) and (2), respectively

  If                          

                     I1 = (I1’ + I1”)          and I2 = (I2’ + I2”)

 This proves the superposition theorem.

                                                                          http://youtu.be/O3D2belxKBs

Network theorem

                   NETWORK THEOREM

 NETWORKS: An electrical circuit containing impedances (or elements like resistance, inductance, capacitance etc.) and generators (sources of e.m.f. or power) is known as electrical network.

 Types of Networks :

Linear network : A network is said to be linear when the current in all the branches is directly proportional to the driving voltage.

Non-linear network :  If the relation between current and driving voltage in any branch of the network is non-linear, the network is said to be non-linear network.

Active Networks :  If a network contains energy sources as well as other circuit elements(resistor, inductor,capacitor etc), it is called as an active network.

 Passive Network : The network containing circuit elements without any energy source is called as Passive network.

Distributed Network : If the inductors, resistors and capacitors are not electrically separated in a network (e.g., transmission line), then it is called as a distributed network.

Lumped Network : In a lumped network, the inductors, resistors and capacitors are electrically separated.

                                                 https://youtu.be/OfFfMFTEFlw

Thevenin’s theorem

                            Thevenin’s theorem

"The current in a load impedance connected between two terminal of a network of generators and linear impedance is same as if this load impedance is connected  to a single voltage generator whose e.m.f. is equal to open circuit (when there is no load) voltage between the terminals". 

The impedance will be equal to the impedance of the network into the terminal when all the generators are replaced by their internal impedance.

 Note : It is a reduction technique by which we can reduce a complicated network containing several generators and impedance into a single equivalent voltage generator.

   In fig. (a) N is the network containing a number of generators and impedance with output terminals A and B. ZL  is the load impedance. Let E’ be the open circuit voltage across A and B when all the generators are replaced by their internal impedance. The network N will produce the same current in external load impedance ZL as a single voltage generator of e.m.f. E’ and internal impedance Z’ would do (Fig (b)).

  

          To established the theorem reduce the network to an equivalent T-section arrangement of impedance Z1 ,Z2 and Z3 as shown in fig.(c) . Here E is the source of e.m.f., Current I1 is supplied by the source and IL be the current flowing in the load impedance ZL . According to Thevenin’s theorem, the circuit with e.m.f., E’ and impedance Z’  in fig.(b) will be equivalent to the circuit with impedance ZL and e.m.f.  E in fig.(c).

To find the expression of the load current IL , apply Kirchoff’s second law in the mesh of fig.(c), we have –

                    E = I1 (Z1 + Z3) – IL Z3  ………………….(1)

 and             0 = IL (Z2 + Z3 + ZL) – I1 Z3  ……………(2)

 from equation (2), we get ----

                   I1 =  IL (Z2 + Z3 + ZL) / Z3

Substituting this value of I1 in equation (1), we get -----

                  E = IL (Z1 + Z3) (Z2 + Z3 + ZL) / Z3   - IL Z3

                  E =  IL {(Z1 + Z3) (Z2 + Z3 + ZL) / Z3  - Z3 }

                  IL =  E / {(Z1 + Z3) (Z2 + Z3 + ZL) / Z3  - Z3 }

  I_L  = E / {(Z₂(Z₁ + Z₃) + Z₃² +Z₁ Z₃ + Z_L (Z₁ + Z₃) – Z₃² ) / Z₃ }

  I_L = E Z₃ / {Z₂(Z₁ + Z₃) + Z₁ Z₃ + Z_L (Z₁ + Z₃)}

         Multiply and divide by (Z₁ + Z₃), we get ---

  I_L  = E(Z₃ /(Z₁ + Z₃)) / { Z₂ + (Z₁Z₃/(Z₁ + Z₃)) + Z_L }  ……(3)

From fig.(c) , the open circuit voltage at terminal A and B when load  is disconnected isgiven by---

  E^’ = E Z₃/ (Z₁ + Z₃) = E { Z₃/ (Z₁ + Z₃)}  …………(4)

The impedance between the terminal A and B after disconnected the load Z_L in fig.(c) is given by ---

   Z^’ = Z₂ + Z₁Z₃/(Z₁ + Z₃)                ………………..(5)

Substituting the values of E^’ and Z^’ from equation (4), (5) in equation (3) …..

                      I_L = E^’ / (Z^’ + Z_L)

This is the expression of current for network in fig.(b).

                                                         https://youtu.be/UCHIvvAvekA