All Derivations - Physics Class 12 (1).pdf

chapter wise derivations of grade 12 CBSE board - physics
electric charges, electrostatics, current electricity, moving charges, magnetism, alt current, EM induction, waves, ray optics, wave optics, dual nature, atoms, nuclei, semi conductors chapters 2025

1. 3% Loulombis Law of Electrostatics? Lin Yector form) |
lonsides Mo charges +9 and -, separated by distance "Ss"

where, ta aah ‘exerted on Y, by qa 1 a E
2 forte exerted on ga by Qe 5 Pr a
Sir Lid 2) a

Eger) mi
Eee rar lubes

Silo, Force applied by Fu, Farkas (RTE)
Ir

= Kute eee |
we wer

= Kite E a)
Rew wa
¿[up 0 a
A et

Here, we can cleosly obsene that > Fo Fa ‚te: 3rd law of newton & valid In
dufrostatles alto,

Electric Field due to dipole on axis: AES Sri
o

Me juve doo pese chants send HE
nor le nich makes Ea à “pi a Fa, ae
's_a vondem prt on axis at a distance e
(rt from centre! of. dipole. bins ssa >

Now, Field dueto q! at À — Ey=—ky
Stilo Feld due to “4! ot À — Eye oe

2. So, Net Field > E= Eq tag
sk.
cae” ea
= RGO kg (el
[en Gn
= 25(24)Y y 2kb
eat

Non, ff work, we ten fjmose L* tothe denaminatos,

Ee y
oe
E> 2k J En
E pas ER,
# Electric ffeld dueto dipole on equftovtal Sines E th
Eye Be bye ke Pie DS
vr E ‘ of.
Gar JEG EE + q en ee) =,
+ 269 (0320
a GE an Tntite Ay 2
un CT W wor ña) A Co
ama re
- [2x9 , 2k%% «os20 an a?
CET TE Ch
El Dk g* [is 1os20'
ent ame,
= He [aos (ir 1405282 dod)
= 2kL (6) se
Qu)
puting 050 ere fora à,
Elda ERP
CLS

243% can be neglected Tn the denominator.

3. here -ve sign denotes: Sale fel Bs. arti-per
& Be CR pue (do ve)

quel

3% Toxque on dipole M external freldi

pe shat un die de oth hangs hoy
di ne plced in a ui pate

Hence,

af makes an angle © wlth electuic field. =%
À +42 — force on charge à
ge ze on change à

whih means “the “force acting on dipole fs equal fa magnitude and ite
ru RN
As coule fs acting on dipole ‚so ft produces fosque

itud
mat ri.)

= fx)
= 9€ x(lsine)
T=PEsmel (:P-900)
TAE ost
a
(sel: when 0-0" : smO=0 ; ¿ali means [T=0
this condition À called cause when the. dipole fs
displaced from His: cafentation, ft aluaye mes back sume coofiveaton
Gene nee O=Ig0" > + stnleo'=0, which mem

endiion 5 led ‚gelabk et ue one, displaced
pele eses co fentafion rated te alfgrs elf
pate field
i
OseD: when 0=46 3. info" whch means Tis mima. at BET
Sen N
= PE sty
To PE = Lun eer,
saz icine
Pye a

4. 3% Guess lave Verification using Qulombs Lawt

We Evene net elects lé Hough a closed surface bo) Hens
fhe net charge enclosed by she sffce- ©

WE = geil

Neviffcatfon s
pastes According to elechic Flux,

her § Ed = f Edocoso

We know, fntensil of electric field IE at same distance from
charge 9 will remain constant

alo for sprerical surface 0=0°
so dlechfe fluxt- À = Egas coso”
de = Ef ds
(hs gas means area = Any”)
5 ges Eur — ©

Sr

Noo, aceding do cales lad € Lo a,
n Fm

on In @, we get

here

o

5. ah Electvte ffeld due to a errata: Jong chayed conductos
leere old dhe do a strat ht erfor changed mida neo, Ka
seas part of fergth ‘Aon this untform =
Sep austen surface wil be elidel

lat a be te small areas ia ns, er
As conductor fs,

post rely
divecHon of E sed Fi Der
outwards, t
Now, dı- fedacoso — [o-00) pi da
d.-fedhuwo — [0-0]

4.2 ean aie le Grays Arad

Hence, Net flox dus gr dep,
= Jean cos + Jean cs dot fedn ast

+ 0 + feda

a [gee Lema Estatal ened nen of surface -2nxd]
Now, Au. de Gauss Lao à

ft
foro, Elem = Mr gue AL]
aL tot

Hee, we con clearly see, EX L

So, Gevaphfeally + € +
7

6. # Electyic Field due to finite Plane sheet of density. ‘c's
i

i long changed plat sheet
: french sole dur

H ! density (07). “

ame

D ssfan «ylíndex ea of rad£us “y! . Take 3 sample i

BORN, eh area of yadius a sample. smal

Thal flax, Aus Gt di tps
= fedAcaso’ + Seda coso" + fEda cos w
= Seda + fedA 40

But o

Ke de qu lay Bell à on

from © 49 > 264 > o-A

“
See

Independent of x

7. 3% Potential at a point due to point change:
==

¡AE
Lik ere be point P ob a distance st from +9 charge
Ékctrte potential (means workdone 40 bring a unit tre Charge fiom infintle
do the pote’ P. zi
Wee) > J few dr
= T KAW) de cose

ts
> ka [id [tots osteo)

E
y

Al
al
= +0 [1 sé)
Ve
PO ye
# Potentfal due + dipole:
@ M a point on axfal (inet
4 +e
A RG
en >
PAS

Consider a pole with changes 444-2 cepmated by a distance of '20',
A port P Rs on the axtal fine at a distance ‘x ‘from centre of po.

80, fotential of P dueto ty, V,
due t0-4, V-

i i ra
Nek poten ot Py Ma =| te \

v)
be ee)

8. Hente, Ni” Ep
ort dipole (1>>0)- =
Shot died ©

do At a point on Sqvatovtal es
let here be a point ‘Plat adishence ‘x’ on equal fhe, | UN,

So, as fn dlagram: ELE re

ee
fotenbal at P dueto +9, Von EL
Te
dueto 4, V= kp)
[area
Se, Net potential at Py Y," )+(W)
- [de
er ar

Pes to

Hence, ekchfe potential due to dipole ot any point on eg. Kine wl be O,
© Ar ony avbFhory point à A

let A be any ovbitsary point at a distance's’ from
Cemte of qe mag an angle © with dipole ols
Observe the fine face ill % Tf we vesolve.
dipl memes") Pro! two weckaflgular Components — q, 2
as chown. a — Beg —
Then, print À Les on axfal Iine of dipole wth dipdle moment ‘prose’
so, potential at a due to tric component = Kpcoso)
a
‘

and potnt A ites on equatorial (fne of dipole uffn dipole moment ‘psin®
tu Fa dice megas A ecb of Ine er parc! de Lo opor
will be zero-

9. Hens, Va = Epes +0

y = Kets] od
5 an
& Relation between Ekctric field and Potential 7

MALE quo ejuipotental susfate A ard 8 separated
FR In:

4

a dan of x’, Att the poteital of surface ai
ui lar ved

Pre in be Vas Vedv Ep

Now, work done fe displace unit pesttfve charge from Bto Az

d= Fax cosiso”
du = fae

E er
ser

$ dw--Edk—®
Alo, we Brow di y (Va- Va)

dd) dv à
dw= dv 0
fom 9 1@ + -Eax=dv
e fel
Ede a LE?

% Potential Energy of System of +00 point charge (in eae of EF)
Trial there were no charge. al A and 8 : a
Eee qt ELA teva
Ds va=0, repleta oe
Now, well bring 9, from © 40 & (and tn this case 9, fs alveady at A)
©, eee at B dueto 4 at Ay Va= kor u -®

se Work done do plae 9, at B, Merken
je Li En Es) (+9 fond)
vie= kage

10. And as we know, Sum of work done Ps equal +o the potential energy of System
A Rental Energy) O + Kad
y

ME | le
Porental Energy of a System of duo chargés Sn an external ekchie fel >
let potential at A and B be Va ard Ve respectively au
Now, Work done to place 9,at A ASE
Wa = 4, a —O (à tal 4, was mot there) | Dr

Wirk done fo place 9, at ®
We= Ver k%% ®
Y

vo Net work We Wa + Wi
neous N ae MAA

And as ue know, this work done equal to potentfal energy of system

Ver] y
Ce
$ Capacitame of a parallel pute capacite’[wFthovt. electa]

es Gil plate capacitor of plate area À and

Let +07 be the surface charge densit
at td a opacity poet ae

Now, the electric Fred side the capacitor plate,
f= (4 application of qause law of change. plate)
&
Potentfal difference between the plates, V= Ed
awed

sv 24-0

11. We kw thet , C=

pets FR 4 rro)

# Capacttance of parallel plate cipacitor [with dielectste]
are a parallel ange capacitor of phte area

Partie R
N
I, 20, Le Ae, gute charge dent a
between th a pate gi ial ath area |
substance having dfel PUS constant K. +
The clectite fred behween plats wl be: a Le
Es
SK
e-@ —O (al
AGK A,
Se Potential difference between the phe

Ve Ed
3 ee —O (um ®
er (vs 0)
Wow, eMC

I

FER

C'= AGK
vy c= ck C'=kC]

C' = capacitance with dtelectric.
ce Gapactance stint dich
K = dfelectste constamt of ¿ne medfum.

Conclusion + A Tnserfng dfelechrfc. medium fn between, ne plates of a
capacttos, F's apacitance fncreases by°k' tres tal apache

ahere,,

12. 2% Capacttance In Parahel +
Constdex two capacitors connected fa parallel combination
as Sion in ne bes E
pa combtnaffon potential difference. acess all

capacttors vematns ¿Same but dishrtbukfen of huge
across each capacitor wi be dtftevent.

20-08
» ot sy C3 o=ev)
exe (e 6)

des?

> ECG) pos
The effective capacitance of a combinalfon of ‘n/ copacitors
Fe lich era En tet cele our of apa. of

pais

% Capacttoss In Sexfes mi

a a. e
Consider two capacitors ave Commected en sestes 3] | ‘| E
combination fa a Urt with capacitance inme E

I Ab E

Conc vespecivel as Shaun th figure.

jener

In sestes combination, the potential differene as 6 2
each capacitor fs different but distribution of y y
charge "semuias Samt.
se Ve MM
+4 1\g-
a2. (AS

TSS rae
a Ci €

Which means the effective capacitance of a combination of ‘n’ capacttore
In sevtes ts:
Lelyirtr---.L
Cle Ca ca Cm

3% Energy Stored fn Copacitos And Expression for Energy density’

a Small amount of change tumsferred by the ace frm ne

2 Then work done by the source ts dW=vdg [1 v- de]
> do-L4 fr ga]

13. ++ Total work done by the source ts trmnsferving amount of change

Now, the. work done fs in the form of potential energy , ire: +

w-fdw

E] w= [2 dy
> Carat
DAC E à)
a Wiy

se ll
a wel a [i
a | We deve

Us y
CA ve LE
7)
> [aw wo

ENERGY DENSITY +

ac energy per unit volume of a
do Energy density,

4
pe

volume

capacitor Es konoion

= cv?

Meer a
gee
= 28

Axd

= 1 CMG ad
2 ALA

as Ereny

2

14. Met LE?

Tf any medium ?s there between plates of a Capacitor,
MeL 66

15. 3% Obtain on expression for Deift Velocity of Electrons +

Drift velodty fs the | with which electrons in a Conductor awe
datfted towards tne positive terminals of the potential source.

We know that fra condictos there are N number of electrons.

InfHally y without any electric field the electoons fn the corductor
trove “randomly with Same velocity’ Vr)

Ber Y &u-o -®

‚Nom, when an electafe Held ts applied, Pat se conductor; The fore
electric freld fs +

applied en a electron by the

where, a= acceleration of e® Awards +e terminal.
ro= mass of the electron.

Tt we take 'T’ tobe the average elaxaHon #me the Hime Paberval
between any two successive liston)

then by Fast ewation, of molfon,
ils Gis alo
M= 0 4 (28) (1)
» [Me Er] whee we Msg dript velocity

eres
3% Ratten behveen Current ond duft velocity + e
e
Considera, tonductos of Lena Lond area of cross-secion un
‘A! and ‘n° be the al sent per unit volume eme

A Nen
Total charge, Q=nAle
À custtné fa the Conductor,
re

+
= nAle

Em et)

16. 2% Sesíes combination of Resistance?

Tuo reststors of vesfstance Ri and Ra ae Comcel
Mn series.

Le
un
As e know, Pn sexfes ombinatton,cumert fs zi
same but yoltage fs diffesent across the components
pe ui. bes law,
IR> IR, +IR,
ERAS pb?
à for ‘n’ ro of veste An seres, KR RR == Rn
% Prsallel combination of Recfstance? 4
Mie eer AE mat GE A
la ASE mm me
Components of the cPrcult I;
à 1=1+1,
ung, er
R Re
MIA à)
Ki E ta po Ls
for in aesfstors fr parallel, te 4 + E + &* EE

Relation between Internal vesistance, terminal potential dif}. and Enr?

Constdes a cel of emf 'E' nn
tonnected +0 en

‘the external sesPstance (R). The current
in the dseuft Ps?

fntemmal vesielance.'s'

Inf 0

Terminal potential di ference

IR ©

17. Now, © con be mile as > MRE E (coots-moliyy)
IR+Iv=€
yizy-E (fm)
V=E-1r] (For vse)

fos v>€ [V-E+Ir] which fs the relation bin €,vandy.

# Cells tn series:

Considers two cells with er El Enr as =
frternal Hee, rid an respectively.
Comected fn

M=E-Iy (for ve)
deo, Woes In
We ka, jeta u Guarent fs same but potential across component Ts deft
Yas | SEVER)
= (6+€2)- (Inte)
= (E+£) - (n+a)1 —0
Noo, we know Ve = Ey” 19, —0

Comparing © LO,

% Cells fa Pevalel +

Lorsidex twa cells of emf Er and Es chin interndl
rasighance 4 ond 1, vetpecively connected M
parallel

ví Komo, a patel combi, polen)
der ts fale bt Cuvier wth be dihierer i
Pare ea ety
A I-Irn
1 ty} fey (E ve za]
e -
la)
pus AA
ae

18. bar Een (5% )
Tate ER

a Vie
Comparing nis with V= by - Ly

Get Es Y

Es Ê VA
ond | Ye = Ute reat
mate.

We tan also witie these He fh simple ways,

“arte
ae i
ond, | Ye, = er

%& Wheatstone Païdge :
Wheatstone bridge fs an arrangement
DAP re nar O Y

one vestsiors fn erms of other Ahnee eststos.

For a balanced bridge,
Ve= Vo (es M fige)

Now, appli. kivchopfs wule on loop ADBAS

L-R-1P=0
+ LP LR —@

Now, pp ing Körchoffs wle on loop BB

28-1,5=0
10-15 — O

3

dida © 40 ; we get * 8 NE
>

LE

u = oe

Coren be bt
mis % he condo far Cae

= pr hy

19. ES Fadiny unknown vetistance. asa slide wire bridge:
Primiple of Meteibridge and Finding unknown resistance
3 s
fatale * Wheat stone bxlge wp Ye dy
As shown a figure, A € I
Re unknown zeststance nun)

S= known resistance

Move, the Sockey G) on vofve AC of Lneth A do obtain the null point (i-
Tage FE tear Bd pan Me RES

fc the bridge Ts balanced therefore by Whensone bufdge perce ©

Rees
@ 9
As

Sure

MEE a

Y æ(00-€) À
(EE ape à

20. HK Maqnelic Field at the centre of q chreulay loop crying coment:
Consider a cceular current carrying loop cry
¿rre 1, Weave do Eva. agil Fld at Ma
centre af Als loop.

sE
lonstder a small current element d on checumferene
of ffs Loop . Clearly angle between d and y Bs 90

Applying Biar Savarts Ia, we get
dB= Ho (tat sin
Am je
> d8= wo IM
Ce Sa
Trtepsativg. both des, we get:
{ole = [ue Lae
hn y
a 6 = Lu
an
3 B=H 1,207 ( Sat means total ciscomfooa

SER Se

Conde q clreulas loop of vadius ‘a’ the axis of the circulas Loop at which
PAR ANA o quete tel doe de chow lead e
ie distance befween the Coop und the Joint Pr

Acording ty Biot Savavts law,

AB= Me Tdi sin
aK w

21. So, the magnelfc. field at P due do cussert element dl:
de = Me Idk sing" [dl
papa dee er)

a Bq po Ide
Ur (ara?)

Magnet feld at P due do cursent element 148
del = Me dt sai
aS

Un
ade = de 14
an (ano)
ve Can see — Here, de-d8'

Resolving dB En two Components ‚we find Hat cos® component
dinmetiicaly opposite. ere eh qe
S Hor ng

netic Feld intensity ot P win be only che do sino Component
Mt foe etal rngnete fr due do the. thee el
B = Sde se

Be (ue lat sine
on ee

B= MoT sno
B+ aie

Sr oe
Um (ae) Fee

E- lata
aaa Be 7
- lo Lae (as

E My

Hov»a ‚dena Ps neglegibe

B- WI ,
29°

= wt
2x.

22. x Amperes Cheeuital Low’

Je states thet the line fr \ Hic field intensi losed |
Hf lt rc En fon eT os et ek

Gte feral = pot

faut orstdes a stralght conductos: crying. as shown in ne
la ne Ay al ree

round the conductor . E
ts Band dÜ ove in same. direction so ongle Lehacen them 50.

jedi
= [edt csv’
= fat
= Bide
BEE Ls gat al JA2 means iscomesena 2706)
Baur] 0

e

al

Bao gata
Application of Ampere's CErceital Law ++ sotenotc

Toxotd

3% Magnetic Feld due to an Mnkinttely fog stafght coment cavnying Conducta

Were gin a dary shalght cite of a Cross-techinal rads ‘a’ or, stead
ca dl CRT ae lat 7

Now, we have de calculate magnetic field at a distance + from centre
But here welll have Bcasess- 0,

Q WA, fe: point lies wide wire

VISA; ie poil Lies on the wie a

do t<aÿ He point lier side He coe à

LasE(D+ v>a at point hi.
Now, to find the magnetic Held as I Po

outstde the wire make a circular Joop
made of vadive iz! as shown in figure

23. Using Anperés Lavo,

> seal = pl

> $Odl cos0"= Mas

> Bhdt = Wot

3 Ben) = WI (3 fas means Urtumferene -2m)
B= Mol ® free 7 te dkance of point fon]

2nt
px + (for 7a)

+ =a at pont Po

1d he dic fied fntenstly at he surface of the wine.
aber a Mad sl iil r

“a Similnly ke Ost well ger — | B= LeT

(ose I
Noo,
Ma.

(seis v<a, at point Pa
ind th He Hell
Ti et et à
rd wire — make a Grcvlar J
bnade ef “radíus x (co)

Rat ji this ee the enclosedl current I, & not I but less
valle. Since the event dishfbstion & ff: the. curvent enclaed

m

Je= Int
GE
ving Anpues lan, BEE ~ Uy Te
> feat- Ue Lat
BG dete i

à B(2m)= Me 1 ar
» B2n= ur 3 [8- Lote
ar ETS y
>

24. 9 Magneke Preld due do Solenoid:

fern, =
¡E

Ne of fume"

Let a solenotd consists of tums, per unit Jey and carry Covent I.
Pee Do al ae li IR
Consider à close loop ABCD.
Gee alee ale [84 És
&
Here, af prdt-0 [B ovkside=0]
[ple fado (23,4)

GEM - {50,00
$84 - fee ceso”
$541 - 8 [al

Edi- BL] —©
Aeorltng do Ampere! aw +

bear = Wor

Here, M ae 4 Jens, are pont

a St fol — fon

Hence,

25. B= pont
where ,1= no. of downs ger unit: Length de,

a
[Fl

Gogh

# Using Ampere cscoal Laa], rel AR. magnetic ee pu a tue en
Case) Inside
from Ampere's kw
FE um Ge A)
hee, [1:0
$5 di-0 » [8-0
(ose 1) Between he forro +

trom Ampere law:- GB AE = Mo di GER)

fai (850 be
eae HN L
ret) = MONT

= MNT Men I] [rene N
SENT) or [ou [my Be

Qu D Ovlside + (at A)
$Bdt = Ue thy
6-0

% Foree acting on a current covsying Conductos placed fn MP: 7
Confides a conductos of Ah 2 ond osea, As section A
caer, covert I placed tr a pages ad of an
shown. Dors ity of elechsons in tre ==
ductor Ps n, then total no, of elechvore tn the conductos fe Al

26. 4e force acting one decken % FM stn® uhese Va fs the cette velocity of
Alan

E tal Hing on the forts = A
the fotal force acting on the conduc na
=(AneVa)Lb sino pe
E= W680] „we
STE an m determined by Fling’
Left and rule.

2% Fone bekueen two parallel st ductors. Onrying current:
too pra tit dans ying a

Consider duo foltrite Jong strafght contactar coments Tard Tin the
same divechon -

“Fey ose held pæallel 4 each otter at a distance '

Stace magnetic Held Ts paul te due to cuwert Dom “each conductor ,Herfar
each cond tor expexientes a force

and, the force will be ZLB Sind,

New, mings Heid at P due +o cvasent I,
Mok, —©

As the coment my conductos Y. Mes Pa the
Het ee unit length of Y will experien
are ‘given by-
> f= Lob stg" [orton tite

E.= Bilaxl (2 shr40-))

Fr = Mol Ta
27%

¿Y

Magnetic Held due to eurent L at point @
8.= Not: _®

Bt ndlr X_alalee paie a a
cf E dueto Ten ii BR

%

F-8 1,50 [des (ont le]

E = Ba L sido"
Fe Mo LL

21%.

27. We com obstwe that E, ach perpendicular do X and directed towards Y.

Rene X and Y attract each “other.

bok when cent will be fh opposite directions,
the conductors wil fepel each ofhes arch magnitudı
will be same as derived abe E 8.
CT
‘jane coment divecton — otraction 2 a
opposite. coment dine — zepulston
3% TOROVE octing on a Cumsent cossyieg Looplotl fn uniform M-F. (xectang ylar)
4 yg Ka "i

he ta Curent Careying (ofl fs placed fe 1 Ae
a a forge Ce epa

Magnetic Farce on a Curvert carsying conductor.
F= Ib sn ——on arms ADL CD onl

but field exerts no forte on the fwo cums AD and BL of loop hecause
Bis onttpasallel to Y E
0) so

A AAA
ee a PT perio E ret esc
unten ts cvected Foto the plane of the Jo Ve, fe)

Fiz LABsPng0 + 149
Simflariy, the Fe field exerts a force am CO, which is divectal oot
le Ben cats a force Gon which ts die 4

f= IN = A
Tis, the net fore on the np ?s zer (os sald easkier)

28. Bot, as ue can see there wll a taque on the Loop die the pal of forces Ford Fi

ert
GROHE a, constdes the cage when e plae of the Jap, % nd
along the magnetic. feld and makes an angle with ft:

Let the angle between the field and 4he poimal to the
oth be he 8.

Theforee on a de
FB before on Im Aa De

mea (Here k-1)
> [m-14) for N mh turns, MNIR
2% Conversion of Galvanometes fo Ammeter +
Galvanometex Can be. converted Ant ammeter by

Gnrecting a Small Resistance S(shurt) fn
onallel afin the galvaronete

Eqs (shunt)

= max Current Anvough galvanomedes la
Z = ammeter yange
& = Galvaromeles ? Resistance

As Sand & ave. cormected M pmallel, SCI) = LR

s= IR
I-ly

%# Conversion of Galvanometes into Voltmeter =

Galvanometer can be omested toto voltmeter by ©
Connecting high seststance fn Series, ©
Tg= urgent Anough. gehononete
hh resistant
Y= External potential
Ry> Galvanometes vesistame
total vesistone = Rr Ry

Noe, acc fo ohms lato, Ve 1g (rR)
Rss

[ESTA
AE

29. # Motfonal em os Induced Emp:

A
Consider a xectaguls conducting fe Pags Tr — m
So ple of Phe pa In Dont he aie peal RE
conductor PQ ts frée more x x ¥ [kt x yy
Ve} tre va £0 $ moved dond fm [> 3 8 2 ELSE
with & Constant af dica fe A Sus

there is no loss of eneafiy due da friction RA
Le pe od 2. distance donde gt, the area encosed by Loop
MEYOas és. ren e al +;
herefore, tre Sy a ae flue Linked vith the Joop Mncrensen.
flue arcos Aue BA
GAO
e Mdveed Emp nm the wil fs >
c= -dd
Gel fot [from @]
Es BL (da) i
A te)
Forte on the wire (external)
aus
À
igor

£. 0)
e

A Induced Emb due aotaton of Rad fn Magreffe Fade a

Consider à metallfe Lergth'£ fs placed Ina i
Ono mage ala an teo > |

Area covered by ae voter Y ana , Ne aes Hi

30. 2 fo Lu leon ede A L I entier mél opel)

> for 0 angle rotation = Le
a Pen ll le, A- 10 O
Ar A BR
Noo, flue traough area A, 8 ER
9-22)
emp b= -
Indveed me Inne rod, dé
& Herd)
£- pide
E= ES RI)
to
% Self- Induction of Jolensid :
Consider « solenafd having N° dor with Length L and cross-section area A,
I tsthe evrsent ai) Aro Te: So these so be a magnette Feld at a
quen paint tn the. sol dd, Yvepuaent it by 8.
Nouyfhe magnetic flint per turn wil be equ to produce Y Band grea of etch
= UNIT XA |
n

yeu

Total magnetic oc wil be gen by product 4 fm Be sehr and
he tae
G= Me NE —©

And, we ako know, g= LI —O
3 fom@ Od Lf- seg

E od

Co más Re elf Reluctance of a soenaid.

31. 2k Mutual frductance of two Solenoids DEN

toneider 40 long solengids $ ardlS2 each of ders

À: Ni and N, ave the ro. of tums Pnthe solenet
Sand 5 upeehteh-

S ls wound closely over Sy, so both the solaris
de considered Hd hoe the some. gata of cross
Section ‘A

I fe the crsrent flowing Hrevgh Si.

Ke, ie magnetic field 6, pduced at ony point inside soleneid Si dve +o curt
1 ds

= Ment —
Hey ©

“Sm tums
———

And, the magnetic flue Gated with each tum of S ie equal to BA.
Total rage fc flux Linked with solenoid Sa having Ne turns fs
r= BAN
> (lena) an, ve
5 & ={ = partida
f= mt, —® pee m se e ofi of mrbal Srducho

aaa MI,» ices ia
= MeN, A y
iol

And, PF the cove % filled vith a vragnete imaleial of permeability M
M> MNMA
Y

32. 3% AC Voltage applted +o a Resistance :
Consides a vesistor of-yosistor, of sesistante R ts
Connected fh seres with a ene Contatning
Alemating EME — Esinwt —©

°° (nt Hvovgh the revit,

Te 5 1 Gstout
E

T= Es oF} ed
te

tomparing DI we can say that ‚there % no phase difference behaeen In:

Wave form biaußAm > NOAA a
E Ol
pee ap RE ‘
x € Et Ei
fob EEE
+
SE PS
Phoor diagram
2% AC village applied do an Inductow! mm
Consides an tnductor ot inductance ‘L’ =
connected fh seres with a dixcutt ra
Allematiny EMP — fasin wt — O) € État
An eme waft Frduce fn the irductor due do the Current I.
= da
kein de Lenz low, the red emf til oppose the alternating eme.
se ah sys
+ Cog)
= Ldi
a Fe, a
> die Ede

> die fosinot dt
L

33. for total unse fntegrating both side,
(Az - [ee soot E
12 de (us)
a Tele est 22 sino) tose |
als a (rg >) | sine) = nd
„Te E sin(wt=p) —®

then stn (dtp) if be 2 yt Lu be pak vee + (Es

©» |1- to sn(ot-y d qe
\ Da
on Comparing OE@ ue see that Land E have different: phase
te: phase difference behueen Lard E
de w-wh

apes vr feads current.

Wave. dix TadE

Phaser dagxam for I mel €

34. FAC Voltage applied fo a Capacitor =
Const ton france SC” ts ls Yet
a oe clay ad of dmg Pe anne
€> E shat —@ e-&sinwt
Tre maximum voltage of the capacttos wih be equal to EMP of the AC.

Mso ‚chnge on capacitor,
EAS Wedestnat Five)
Instantaneous cuwent Tn the creutts
I= de
dt
I» d (cfosnot)
de
= Co +)
a 1-0 de Ghoi)
+ T- Clow coswok
ai Cato (cho + ct) —®
Now, 1 u be mox(peak) when (nz wt) will become 1.
li 1,60

Ss, @> IL mr) yA —®
ot

Compastr d e acom Tine
Goi, od Dow at ica bado

= Phase difference between Lande, $= w+ Ew

ng behfnd

35. *% Impedene fn sestes LCR circuit a

Consider a LER veut Comected 4 an AC sure fr sens [EA
e ke

Hexe, voltage drop across resistance , capacitos and An

Kr nische fer pee

Ver TR di

E where,
Ver 1X [0 | Kal
MIX Ara

Phosor dfogram for LCR chreutt >
Grsides, Ea % the dotal voHaye Supplied fn the

civeutt!
In the above phasor Afagsum , fet Y, > \e

2 We) —O

Now, volage across all the Components 4 V= [UD FRE
ve Jar -IR Y + (m)?
V= Jr (eae
Vel HO-x) +R
Le RT

E AAA,
Zit fh impedance poe

36. % Resonating frequency fo sexfes LeR elreuît +

Resonance occurs when inductive veactance becomes equal to cpact ve
veactance. >
x = Xe

ale En
wc
wre L

a Ma Nes
A+ y, ie will become. equal 40 Xe ind tesonance will

# Mau $5 brown as
re Fe
TEL Feequenty. E

= Average Power fo LCR Chreuit +
We know that a lie E-£ saut applied fo à states ALL deut drives a
Covent An the ciswit given by
EL So(ut-8) y whee R= Le
os fnslantaneous power by tne souxe iS Beim“ ei
p= El > shot x, dnfut-#
Fe Gh [usg- as (aut +8] —©
Hog de aja, mae cycle 15 fuen by Ahe overage of the hoo teams th

Gut we com See that only the second term Ps Ame Be Bann Bee stil
be zero“ eu Half of the Cosme camas the

me
-4:4)

37. ES Energy stored fn an Inductos +
Consider an fnductor of induchnee L comected toa FT
voltage source E ab shown In Fipae - ===}
As we know, P= EL
Po Lid H 2% 4)
aye = paa di THE de)

did=LId1 Ma
nteuating both cts, (ou FLrdz [Lemaire ie encuït)
W=L fou

wee fey
Wel [2-0]
aon in
RES nee ser the dra as migra poked
mA gem], et ie
x

38. CHAPTER #4 = Ray Optics —
2 Relation between Csfial angle and vefiaciive dex of a medium:
bes gi ay travelling from denser medium (4)

Randiny do Snells (aw: [sin = (1) sin 90"

M she =I
MERE

% Refraction at a sphevical Surface:

Prgure chows xefrachon bi Ang Hi
Bp soin item
let of,P ond Y be He angle made by ©
Rede tal ni wg =>
‘may wlth the pane axe

Te a rol drawn from the Comex vefraching Susface passes thous
ve a Co. AN RUN une rom pole. a
tector! of Incident say ls taken ve

low, In Aone 30 In Amı,
ur ore” il
a> B-y
, Ray Snell law?
Me 14800) = He nO,
4 0,£ 8, axe very Small, «> SinO/2:8, and Sn, XO,
> MO= MO,
wulxee) = HE (BE) —O
Here, ape Kar pr small

av
imb- 4 =P
tan Y= hav

39. A
X Lens Makes Formula“

(order a convex Lens (thick) » Let an noe
placed es E onen axts at ‘0
Te Image fox

med by the convex thick ‘hers
ue Ba tinough fist sufare (Rec):

Y surface ADC fs not present then Image will
ve formed at I, 04 Shown fr the Aue
2 Aconding Ho refraction formula:

E E
Ri v u

2) fusthes, sefracton Helen: seund surface (Apc) +
Ti the sveface ABC % not
jmüge 1 wih behave. tke

tmale by cecon ena we if eke at
E in wn In

Now, Actarding de vefsackon formol,
ef N e
pena ®

Am

Adding © 4D + nu ela a Mote ne m,n nó

ol een men mln
R (oy

> hn (mon). (4-4)

40. 4)"

leeren
nee mene aes a
He
¥ Refraction trough Pom

Considex, a triangulos prism, Jet a rayo:

Pg chutes on tHe fade. AB of dhe iat
lsm and then refracted by the.

Face AB towards the base

prs BC and again R Es vefracted 2

‘oy the fase AC duay

Se
mn the normal A

e
Te Angle of fnddence

=
af o
and 1 = Hon b:
damien [I Krane
A= ongle fe
ER Pom \
In AQNR, Zu + Lu +/@NR = 180 —@

In quadiflateral AQNR , LA+ 40° + JQNR +90" = 360°

A LA+LONR = 360" -180"

> LA LONA = 180° —©
fom 020) > Lyx Cx, + Lede - Lae agi
Loran = LA
ow [An 3+ %|—®
Also, $= S1+ da

a b> int (e-%)
+ 6- (Pre) - Gan)

41. à $= (te -A (from 4)
ov [fre=§+A] —®

when , 6= Span ythen f=e
Dead

la en
AU

©

And, ey © becomes > f+ i= Aura
ee fut
xo

Na, Aunding jo weils Loo,
ail here ju ls refaite Index d He)
eit at IE st

(ESA) y (fn 10)
& A) MTL
Ma in (de)

n/m.) wy?

42. 2% fesft¥on ond width of the Frünge th Interfmenge“
f Ave bright

Teepe Opel ER dn LAS
aa the distance kehocen any tuo consecune fi
k Fringe is equal tothe width of a aight fant

lonsider {tye from two shit Siard S cprimpaed
Of pont fe eaten ipod do pane |

Let d be the distance between quo shit Sands |
on D be the distance between dit and screen.

Now, at point P the path difference of fo waves fs

Ate 8P-SP —O
ge Pope s pape InbSor,
a à P= SAT: In P, +
Dr (y- dy Pe
wae aw) % ug 2
Nov, D-4D

SPS Poe (NB (ya)
GP-S.) (SP SP) = yx
(Gr-s (rs > 2 yd
Aaurimg P vay dae 49 0, such that SP S2P=D
2 O+Dsx = 2yel
Apo yA

ax yd
(Esel

Axeni
yd = nA
AUS
when y m=o ,y=0 (Central bright fringe)
Lid ap (ist BF)
nen y yo DAD (nh FD

Ge 1) for Maxima,

43. Coreg) fix O
yd = (20-1) Ye
rap]
when, n=) , ra Cist dr]
me y ys aap. (24 or]
veal Bets Gyan (Cn de)

+2 Alternate Dark 2 Bright fringes appear:
Now, Expression for linge width ir
a Sires ee TETE batgrt finger gives dre füge width
Bou © does He
= (ntl), An nd

Pap]
B=

Gilly, for bart >

a

44. — CHAPTER FELL? Dual Nature of Radiation and Mattes
*e de- Brogife equations

ne eee A Ff fre pen) 4 wavelength A) propaga, fin vacum,

E-hv] —®
Awording to Einstein mass- energy equivalenie ,
[€= mc] —®

Comparing OLO,

Now, momentum of each photon Ts, P=mc

RE

7 c
(ENS
P= af (: 5)
Æ $] fe de-bnoyle.ey”.
Le us fake on example, eS aceleroted Fr a pent dif. then th
KE cam be writen ass Teel
The Loewy momentom $ Ke of e moving with ebay ‘ye! aves
l?= mw
4 k= Lm
mulkplyi ing ‘mn! bath sides: mk= L mv”
2mk= ee
Square Yoot both side: [ImK = mir
my = 2mk |
Now, from de-broglie eg": A= +

>

45. Subditoting

he 663x107 Js
m= 9.1 X 107
e= HEXIONS

}

El

46. — CHAPTER #12 + Atoms
hi e
5 Min rare IA ae was

ue an electron of, mass m and char pe Pis >
De with velocity Y avound a ret Bi E
Fan number CZ. Then the crt Ra \
fred by he elechon I po vided e
Bechrosföhe ox - q otacten De hae
and elechon “aco y de puttone \

Fe- Fi fa eee
> kede _ mw
v y

Ree. do Boho hostulates + mvr = O)

Nowy ze ras he,

mes‘ Ans | Kzet

Hessen

”
DER ®
| pels” —@

ORadivs of nt orbit
Novo, Velocity of e7 In stalfonayy osbitst

revi = nh
El
hé - oh
mnzer an

ve zee get
Bone | WE
Sot of © fn n° energy leve]

47. Now, Energy of O in StaHonary cbr

8 KE = Im“

Dre nude
= Zee No = Ke) (ze)
en el nthe ee
a mmzer
KE = mice MEE
Ent ht ES Per nés
D TE= ker PE

Te. me
CS