Get Solution of These Packages & Learn by Video Tutorials on

Transcription

1 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on INTODUCTION The banch of physics which deals with electic effect of static chage is called electostatics.. ELECTIC CHAGE Chage of a mateial body o paticle is the popety (acuied o natual) due to which it poduces and expeiences electical and magnetic effects. Some of natually chaged paticles ae electon, poton, α-paticle etc.. Types of Chage (i) Positive chage : It is the deficiency of electons compaed to potons. (ii) Negative chage : It is the excess of electons compaed to potons.. Units of Chage Chage is a deived physical uantity. Chage is measued in coulomb in S.Ι. unit. In pactice we use mc ( C), µc ( 6 C), nc( 9 C) etc. C.G.S. unit of chage electostatic unit esu. coulomb 9 esu of chage Dimensional fomula of chage [MºLºT Ι ]. Popeties of Chage ( i ) Chage is a scala uantity : It adds algebically and epesents excess, o deficiency of electons. (ii) Chage is tansfeable : Chaging a body implies tansfe of chage (electons) fom one body to anothe. Positively chaged body means loss of electons, i.e., deficiency of electons. Negatively chaged body means excess of electons. This also shows that mass of a negatively chaged body > mass of a positively chaged identical body. (iii) Chage is conseved : In an isolated system, total chage (sum of positive and negative) emains constant whateve change takes place in that system. (iv ) Chage is uantized : Chage on any body always exists in integal multiples of a fundamental unit of electic chage. This unit is eual to the magnitude of chage on electon (e.6 9 coulomb). So chage on anybody ± ne, whee n is an intege and e is the chage of the electon. Millikan's oil dop expeiment poved the uantization of chage o atomicity of chage (v) (vi) (vii) ELECTOSTATICS Note : ecently, the existence of paticles of chage ± e and ± e has been postulated. These paticles ae called uaks but still this is not consideed as the uantum of chage because These ae unstable (They have vey shot span of life). Like point chages epel each othe while unlike point chages attact each othe. A chaged body may attact a neutal body o an oppositely chaged body but it always epels a similaly chaged body. Note : epulsion is a sue test of electification wheeas attaction may not be. Chage is always associated with mass, i.e., chage can not exist without mass though mass can exist without chage. The paticle such as photon o neutino which have no (est) mass can neve have a chage. As chage can not exist without mass, the pesence of chage itself is a convincing poof of existence of mass. Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

2 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on (viii) (ix) Chage is elativistically invaiant: This means that chage is independent of fame of efeence, i.e., chage on a body does not change whateve be its speed. This popety is woth mentioning as in contast to chage, the mass of a body depends on its speed and inceases with incease in speed. A chage at est poduces only electic field aound itself; a chage having unifom motion poduces electic as well as magnetic field aound itself while a chage having acceleated motion emits electomagnetic adiation also in addition to poducing electic and magnetic fields.. 4 Chaging of a body A body can be chaged by means of (a) fiction, (b) conduction, (c) induction, (d) themoinic ionisation, (e) photoelectic effect and (f) field emission. (a) Chaging by Fiction : When a neutal body is ubbed with othe neutal body (at least one of them should be insulato) then some electons ae tansfeed fom one body to othe. The body which gains electons becomes negatively chaged and othe becomes positively chaged. (b) Conduction (flow): Thee ae thee types of mateial in natue () Conducto : Mateials which have lage numbe of fee electons. () Insulato o Dielectic o Nonconductos : Mateials which do not have fee electons. When a chaged conducto is connected with a neutal conducto then chage flows fom one body to othe body. In case of two chaged conductos chage flows fom highe potential enegy to lowe potential enegy. The chage stops flowing when the potential of the two bodies become same. Note: If two identical shaped conductos kept at lage distance ae connected to each othe then they will have eual chages finally. (c) Induction : When a chaged paticle is taken nea to neutal object then the electons move to one side and thee is excess of electons on that side making it negatively chaged and deficiency on the othe side making that side positively chaged. Hence chages appea on two sides of the body (although total chage of the body is still zeo). This phenomenon is called induction and the chage poduced by it is called induced chage. A body can be chaged by in duction in following two ways. Method Ι : The potential of conducto A becomes zeo afte eathing. To make potential zeo some electons flow fom the Eath to the conducto A and now connection is emoved making it negatively chaged. Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

3 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & Method ΙΙ : The conducto which has induced chage on it, is connected to a neutal conducto which makes the flow of chage such that thei potentials become eual and now they ae disconnected making the neutal conducto chaged. (d) Themionic emission : When the metal is heated at a high tempeatue then some electons of metals ae ejected and the metal gets ionised. It becomes positively chaged. (e) Photoelectic effect : When light of sufficiently high feuency is incident on metal suface then some electons come out and metal gets ionized (f) Field emission : When electic field of lage magnitude is applied nea the metal suface then some electons come out fom the metal suface and hence the metal gets positively chaged. Ex. Chage consevation is always valid. Is it also tue fo mass? No, mass consevation is not valid. Mass can be conveted into enegy. Ex. What ae the diffeences between chaging by induction and chaging by conduction? Majo diffeences between two methods of chaging ae as follows : (i) (ii) (iii) In induction, two bodies ae close to each othe but do not touch each othe while in conduction they touch each othe. In induction, total chage of body emains unchanged while in conduction it changes. In induction, induced chage is always opposite in natue to that of souce chage while in conduction chage on two bodies is of same natue.. If a glass od is ubbed with silk it acuies a positive chage because : (A) potons ae added to it (B) potons ae emoved fom it (C) electons ae added to it (D) electons ae emoved fom it. Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

4 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on A positively chaged body 'A' attacts a body 'B' then chage on body 'B' may be: (A) positive (B) negative (C) zeo (D) can't say Answes :. D. B, C. COULOMB S LAW (INVESE SUAE LAW) Ex. On the basis of expeiments Coulomb established the following law known as Coulamb's law. The magnitude of electostatic foce between two point chages is diectly popotional to the poduct of chages and invesely popotional to the suae of the distance between them. i.e. F (i) (ii) (iii) (iv) (v) (vi) (vii) (x) F Ex.4 F F Impotant points egading Coulamb's law : It is applicable only fo point chages. K F The constant of popotionality K in SI units in vacuum is expessed as medium expessed as chage is ε and in any othe. If chages ae dipped in a medium then electostatic foce on one. ε and ε ae called pemittivity of vacuum and absolute pemittivity of the medium espectively. The atio ε / ε ε is called elative pemittivity of the medium, which is a dimensionless uantity. The value of elative pemittivity ε vaies between to. Fo vacuum, by definition it is eual to. Fo ai it is nealy eual to and may be taken to be eual to fo calculations. Fo metals the value of ε is. The value of 9 9 Nm C. The foce acting on one point chage due to the othe point chage is always along the line joining these two chages. It is eual in magnitude and opposite in diection on two chages, iespective of the medium, in which they lie. The foce is consevative in natue i.e., wok done by electostatic foce in moving a point chage along a close loop of any shape is zeo. Since the foce is a cental foce, in the absence of any othe extenal foce, angula momentum of one paticle w..t. the othe paticle (in two paticle system) is conseved, In vecto fom fomula can be given as below. F ε ε ˆ hee is position vecto of the test chage with espect to the souce chage. If the distance between two eual point chages is doubled and thei individual chages ae also doubled, what would happen to the foce between them?...() ; Again, F' ()() () 4 o F' F 4 So, the foce will emain the same. A paticle of mass m caying chage is evolving aound a fixed chage in a cicula path of adius. Calculate the peiod of evolution and its speed also. 4π m mω ' T Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 4 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

5 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & T ( ) (4π m) and also we can say that o T 4π πεm mv V m 4. PINCIPLE OF SUPEPOSITION The electostatic foce is a two body inteaction, i.e., electical foce between two point chages is independent of pesence o absence of othe chages and so the pinciple of supeposition is valid, i.e., foce on chaged paticle due to numbe of point chages is the esultant of foces due to individual point chages, theefoe, foce on a point test chage due to many chages is given by F F F.... F 5. ELEC TOSTATIC EUILIBIUM Ex. 5 The point whee the esultant foce becomes zeo is called euilibium position. 5. Stable Euilibium : If chage is initially in euilibium position and is displaced by a small distance. If the chage ties to etun back to the same euilibium position then this euilibium is called position of stable euilibium. 5. Unstable Euilibium : If chage is displaced by a small distance fom its euilibium position and the chage has no tendency to etun to the same euilibium position. Instead it goes away fom the euilibium position. Two eual positive point chages '' ae placed at points A(a, ) and B( a, ). Anothe test chage is also placed at O(, ). Show that the euilibium at 'O' is (i) stable fo displacement along X-axis. (ii) unstable fo displacement along Y-axis. (i) Initially F AO F BO FAO K FBO a When chage is slightly shifted towads x axis by a small distance x, then. FAO < FBO Theefoe the paticle will move towads oigin (its oiginal position) hence the euilibium is stable. (ii) When chage is shifted along y axis Afte esolving components net foce will be along y axis so the paticle will not etun to its oiginal position so it is unstable euilibium. Finally the chage will move to infinity.. In example numbe 5 if is negative point chage then pove that the euilibium at 'O' is Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 5 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

6 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on Ex. 6 (i) stable fo displacement in Y-diection. (ii) unstable fo displacement in X-diection. 5. Neutal Euilibium : If chage is displaced by a small distance and it is still in euilibium condition then it is called neutal euilibium. Two point chages of chage and (both of same sign) and each of mass m ae placed such that gavitation attaction between them balances the electostatic epulsion. Ae they in stable euilibium? If not then what is the natue of euilibium? K In given example : Ex. 7 Ex. 8 Gm We can see that iespective of distance between them chages will emain in euilibium. If now distance is inceased o deceased then thee is no effect in thei euilibium. Theefoe it is a neutal euilibium. Two eually chaged identical metal sphee A and B epel each othe with a foce 5 N. Anothe identical unchaged sphee C is touched to B and then placed at the mid point between A and B. What is the net electic foce on C? Let initially the chage on each sphee be and sepaation between thei centes be ; then accoding to given poblem. F 5 N When sphee C touches B, the chage of B, will distibute eually on B and C as sphee ae identical conductos, i.e., now chages on sphees; B C (/) So sphee C will expeience a foce (/ ) F CA ( / ) F along AB due to chage on A (/ )(/ ) and, F CB ( / ) F along BA due to chage on B So the net foce F C on C due to chages on A and B, F C F CA F CB F F 5 N along AB. Five point chages, each of value ae placed on five vetices of a egula hexagon of side L. What is the magnitude of the foce on a point chage of value coulomb placed at the cente of the hexagon? Method : I If thee had been a sixth chage at the emaining vetex of hexagon foce due to all the six chages on at O would be zeo (as the foces due to individual chages will balance each othe), i.e., F Now if f is the foce due to sixth chage and F due to emaining five chages. F f i.e. F f o, F f L L Ans. Method : II In the diagam we can see that foce due to chage A and D ae opposite to each othe Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 6 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

7 Get Solution of These Packages & Lean by Video Tutoials on F DO F AO...(i) FEE Download Study Package fom website: & Similaly So F BO F EO Using (i) and (ii) F Net...(ii) F AO F BO F CO F DO F EO F Net F CO L along CO. Ex.9 Two identical chaged sphees ae suspended by stings of eual length. Each sting makes an angle θ with the vetical. When suspended in a liuid of density σ.8 gm/cc, the angle emains the same. What is the dielectic constant of the liuid? (Density of the mateial of sphee is.6 gm/cc.) Initially as the foces acting on each ball ae tension T, weight mg and electic foce F, fo its euilibium along vetical, T cos θ mg...() and along hoizontal T sin θ F...() Dividing En. () by (), we have Ex. F tan θ mg...() When the balls ae suspended in a liuid of density σ and dielectic constant K, the electic foce will become (/K) times, i.e., F' (F/K) while weight mg' mg F B mg Vσg [as F B Vσg, whee σ is density of mateial of sphee] σ i.e., mg' mg as V m ρ ρ So fo euilibium of ball, tan θ' F ' F mg' Kmg[ ( σ / ρ)]... (4) Accoding to given infomation θ' θ; so fom euations (4) and (), we have K ρ ( ρ σ).6 Ans. (.6.8) A paticle of mass m and chage is located midway between two fixed chaged paticles each having a chage and a distance l apat. Pove that the motion of the paticle will be SHM if it is displaced slightly along the line connecting them and eleased. Also find its time peiod. Let the chage at the mid-point the displaced slightly to the left. The foce on the displaced chage due to chage at A, F ( l x) The foce on the displaced chage due to chage at B, F ( l x) Net foce on the displaced chage. o F F F F o F ( l x) ( l x) ( l x) ( l x) ( l x 4lx ) Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 7 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

8 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & lx Since l >> x, F 4 πεl x o F πεl We see that F x and it is opposite to the diection of displacement. Theefoe, the motion is SHM. T m π, hee k k π l mπ π.4 A paticle of mass m and chage is located midway between two fixed chaged paticles each having a chage and a distance l apat. Pove that the motion of the paticle will be SHM if it is displaced slightly along pependicula bisecto and eleased. Also find its time peiod. Ex. ml Ans. π, whee K K. A thin staight od of length l caying a unifomly distibuted change is located in vacuum. Find the magnitude of the electic foce on a point chage '' kept as shown in the figue. As the chage on the od is not point chage, theefoe, fist we have to find foce on chage due to chage ove a vey small pat on the length of the od. This pat called element of length dy can be consideed as point chage. Chage on element d λdy dy l K.d. Electic foce on '' due to element y All foces ae along the same diection K...dy y F df This sum can be calculated using integation, theefoe F K l y K Note : If a >> l then F a al behaviou of the od is just like a point chage. a.l l K. l a a l K a(a l) a l y a Kdy.5 Thee identical sphees each having a chage (unifomly distibuted) and adius, ae kept in such a way that each touches the othe two. Find the magnitude of the electic foce on any sphee due to othe two. Ans..6 Two chages of each ae placed at two opposite cones of a suae. A chage is placed at each of the othe two cones. (a) Ιf the esultant foce on is zeo, how ae and elated? (b) Could be chosen to make the esultant foce on each chage zeo? Ans. (a), (b) No. y l Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 8 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

9 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on 6. ELECTIC FIELD Electic field is the egion aound chaged paticle o chaged body in which if anothe chage is placed, it expeiences electostatic foce. Ex. Ex. Ex.4 6. Electic field intensity E : Electic field intensity at a point is eual to the electostatic foce expeienced by a unit positive point chage both in magnitude and diection. If a test chage is placed at a point in an electic field and expeiences a foce F, the electic field F intensity at that point is given by E ; If the E is to be detemined pactically then the test chage should be small othewise it will affect the chage distibution which is poducing the electic field and hence modify the uantity which is measued. A positively chaged ball hangs fom a long silk thead. We wish to measue E at a point in the same hoizontal plane as that of the hanging chage. To do so, we put a positive test chage at the point and measue F/. Will F/ be less than, eual to, o geate than E at the point in uestion? When we ty to measue the electic field at point P then afte placing the test chage at P it epels the souce chage (suspended chage) and the measued value of electic field E measued than the actual value E act. 6. Popeties of electic field intensity E : F will be less (i) It is a vecto uantity. Its diection is the same as the foce expeienced by positive chage. (ii) Electic field due to positive chage is always away fom it while due to negative chage always towads it. (iii) I t s S. Ι. unit is Newton/Coulomb. (iv) Its dimensional fomula is [MLT A ] (v) Electic foce on a chage placed in a egion of electic field at a point whee the electic field intensity is E is given by F E. (vi) Electic foce on point chage is in the same diection of electic field on positive chage and in opposite diection on a negative chage. It obeys the supeposition pinciple, that is, the field intensity at a point due to a point chage distibution is vecto sum of the field intensities due to individual point chages. E E E... E Calculate the electic field intensity which would be just sufficient to balance the weight of a paticle of chage µc and mass mg. As foce on a chage in an electic field E is F E So accoding to given poblem F W i.e., E mg i.e., E mg N/C., in downwad diection. Electostatic foce expeienced by µc chage placed at point Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 9 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

10 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on P due to a point chage system S as shown in figue is F î 9ĵ N. (i) Find out electic field intensity at point P due to S. (ii) If now µc chage is placed and µc is emoved at point P then foce expeeniced by it will be. (i) F E Ex.5 (ii) î 9ĵ µ C(E) E 7 î ĵ Since the souce chages ae not distubed the electic field intensity at 'P' will emain same. F µc ( E ) ( 7 î ĵ ) 4 î 6 ĵ N Find out electic field intensity at point A (, m, m) due to a point chage µc situated at point B( m,, m). K K E ˆ P.V. of A P.V. of B Ex.6 (- î ĵ kˆ ) ( ) () () E ( 8 ) (P.V. Position vecto) ( î ĵ kˆ ).5 ( î ĵ kˆ ) N/C. Two point chages µc and µc ae placed at point A and B as shown in figue. Find out electic field intensity at points C and D. [All the distances ae measued in mete]. Electic field at point C (E A, E B ae magnitudes only and aows espesent diections) Electic field due to positive chage is away fom it while due to negative chage it is towads the chage. It is it is clea that E B > E A. E Net (E B E A ) towads negative X-axis K (µ c) K(µ c) towads negative X-axis ( ) ( ) 8 ( î ) N/C Electic field at point D : SInce magnitude of chages ae same and also AD BD So E A E B Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

11 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & Veticle components of diection..k(µ c) So, E net E A cosθ E A and E B cancel each othe while hoizontal components ae in the same cos45 K 6 9 î.7 Thee chages, each eual to, ae placed at the thee cones of a suae of side a. Find the electic field at the fouth cone. Ans. ( ) 8 πε a. Ex.7 Ans. Ex.8 Ex.9 Positive chage is distibuted unifomly ove a cicula ing of adius. A point paticle having a mass m and a negative chage, is placed on its axis at a distance x fom the cente. Find the foce on the paticle. Assuming x <<, find the time peiod of oscillation of the paticle if it is eleased fom thee. (Neglect gavity) When the negative chage is shifted at a distance x fom the cente of the ing along its axis then foce acting on the point chage due to the ing: F E E (towads cente) ( Kx x ) / if >>x then x ~ x F E (Towads cente) Since estoing foce F E x, theefoe motion of chage the paticle will be S.H.M. Time peiod of SHM. T π k m π m 4 πε 6π εm Find out electic field intensity at the cente of unifomly chaged semicicula ing of adius and linea chage density λ. λ linea chage density. The ac is the collection of lage no. of point chages. Conside a pat of ing as an element of length dθ which substends an angle dθ at cente of ing and it lies between θ and θ dθ de de x î de y ĵ E x de x (due to symmaty) π E y de y desinθ E y Kλ π sin θ.dθ Kλ Deive the expession of electic field intensity at a point 'P' which is situated at a distance x on the axis of unifomly chaged disc of adius and suface chage density σ. Also deive esults fo (i) x >> (ii) x << The disc can be consideed to be a collection of lage numbe of concentic ings. Conside an element of the shape of ing of adius and of width d. Electic field due to this ing at P is K. σπ.d.x de / ( x ) Put x y d ydy / N/C. Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

12 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & K. σπy.dy.x ydy de Kσπ.x y y Electic field at P due to all ings is along the axis. E de x E Kσπx dy y x Cases : (i) If x >> Kσπx σ ε E σ ε ε x x Kρπx. x, along the axis x y x x x Kσπ x / x σ [ ] [ ε ] x x x σ [ σ 4ε x σπ x x i.e. behaviou of the disc is like a point chage. (ii) If x << E σ σ ε [ ] ε i.e. behaviou of the disc is like infinite sheet. highe ode tems] 4 πε x.8 Find out electic field intensity at the cente of unifomly chaged cicula ac (uate ing) of adius and linea chage density λ. Kλ Ans. (î ĵ) Ex. Ex. A point chage is placed at a distance fom a vey long chage thead of unifom linea chage density λ. Find out total electic foce expeienced by the line chage due to the point chage. (Neglect gavity). Foce on chage due to the thead, Kλ F. By Newton's ΙΙΙ law, evey action has eual and opposite eaction so foce on the thead (away fom point chage) Kλ. A block having mass m and chage is esting on a fictionless plane at a distance L fom fixed lage non-conducting infinite sheet of unifom chage density σ as shown in Figue. Discuss the motion of the Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

13 Get Solution of These Packages & Lean by Video Tutoials on 6. Electic field intensities due to vaious chage distibutions ae given in table. FEE Download Study Package fom website: & Note : Name/Type Fomula Note Gaph Point chage Infinitely long line chage Infinite non- K ˆ λ ˆ πε K Kλˆ * is souce chage. * is vecto dawn fom souce chage to the test point. * Electic field is nonunifom. * λ is linea chage density (assumed unifom) * is pependicula distance of point fom line chage. * ˆ is adial unit vecto dawn fom the chage to test point. σ is suface chage density. conducting (assumed unifom) ε thin sheet * nˆ is unit nomal vecto. * Electic field intensity is independent of distance. Kx Unifomly chaged ing E / x E cente ( ) is total chage of the ing. * x distance of point on the axis fom cente of the ing. * electic field is always along the axis. Infinitely lage chaged σ nˆ * σ is the suface chage. conducting sheet ε density (assumed unifom) Unifomly chaged hollow conducting/ nonconducting /solid conducting sphee σ (i) fo k E ˆ (ii) fo < E nˆ * nˆ is the unit vecto pependicula is the suface. * Electic field intensity is independent of distance * is adius of the sphee. * is vecto dawn fom cente of sphee to the point. * Sphee acts like a point chage. placed at cente fo points outside the sphee. * E is always along adial diection. * is total chage ( σ4π ). (σ suface chage density) Unifomly chaged (i) fo * is vecto dawn fom cente solid nonconducting of sphee to the point sphee (insulating k * Sphee acts like a point chage mateial) E ˆ placed at the cente fo points outside the sphee * E is always along adial di n ρ 4 π. (ii) fo * is total chage (. ) K E ρ ε (ρ volume chage density) * Inside the sphee E. * Outside the sphee E /. (i) Net chage on a conducto emains only on the oute suface of a conducto. This popety will be discussed in the aticle of the conducto. (aticle no.5) (ii) On the suface of spheical conductos chage is unifomly distibuted. Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

14 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on : Ex. block assuming that collision of the block with the sheet is pefectly elastic. Is it SHM? The situation is shown in Figue. Electic foce poduced by sheet will acceleate the block towads the sheet poducing an acceleation. Acceleation will be unifom because electic field E due to the sheet is unifom. F E a, whee E σ/ε m m As initially the block is at est and acceleation is constant, fom second euation of motion, time taken by the block to each the wall L at i.e., t L a ml ae 4 As collision with the wall is pefectly elastic, the block will ebound with same speed and as now its motion is opposite to the acceleation, it will come to est afte tavelling same distance L in same time t. Afte stopping it will be again acceleated towads the wall and so the block will execute oscillatoy motion with 'span' L and time peiod. T t ml ae 4 mlε aσ Howeve, as the estoing foce F E is constant and not popotional to displacement x, the motion is not simple hamonic. Detemine and daw the gaph of electic field due to infinitely lage nonconducting sheet of thickness t and unifom volume chage density ρ as a function of distance x fom its symmety plane. (a) x t (b) x t We can assume thick sheet to be made of lage numbe of unifomly chaged thin sheets.conside an elementy thin sheet of width dx at a distance x fom symmaty plane. Chage in sheet ρadx (A : assumed aea of sheet) suface chage density ρadx s A so, electic field intensity due to elementy sheet. (a) (b) de When x < t When x > t ρdx ε x E Net t / ρ dx ε ε t / ρ t / ρ ρt E Net dx ε ε t / x ρx dx ε.9 In the pevious uestion if left half of the sheet contains chage density ρ and ight half contains chage density ρ then find the electic field at the symmety plane. Ans. E net Ex. ρt 4ε (towads left) Figue shows a unifomly chaged sphee of adius and total mlε aσ Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 4 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

15 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on chage. A point chage is situated outside the sphee at a distance fom cente of sphee. Find out the following : (i) Foce acting on the point chage due to the sphee. (ii) Foce acting on the sphee due to the point chage. (i) Electic field at the position of point chage K K E ˆ so, F ˆ K F (ii) Since we know that evey action has eual and opposite eaction so F K ˆ sphee F K sphee. Ex.4 Figue shows a unifomly chaged sphee of total chage and adius. A point chage is also situated at the cente of the sphee. Find out the following : (i) Foce on chage (ii) Electic field intensity at A. (iii) Electic field intensity at B. (i) Electic field at the cente of the unifomly chaged hollow sphee So foce on chage (ii) Electic field at A E K A E Sphee E ; CA E due to sphee, because point lies inside the chaged hollow sphee. (iii) Electic field E B at point B E Sphee E K K K( ).ˆ. ˆ. ˆ ; CB Note : Hee we can also assume that the total chage of sphee is concentated at the cente, fo calculation of electic field at B.. In the given example if point chage is situated at point A which is at a distance < fom the cente of the sphee then find out following (i) Foce acting on chage. (ii) Electic field at cente of sphee. (iii) Electic field at point B. K Ans. (i) (ii) (AC) K AC, AC (iii) CB CB K AB. AB Ex.5 Two concentic unifomly chaged spheical shells of adius and ( > ) have total chages and espectively. Deive an expession of electic field as a function of fo following positions. (i) < (ii) < (iii) (i) fo <, theefoe point lies inside both the sphees E net E Inne E oute (ii) fo <, theefoe point lies outside inne sphee but inside oute sphee: K K E net E inne E oute ˆ ˆ (iii) fo point lies outside inne as well as oute sphee theefoe. Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 5 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

16 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on K K E Net E inne E oute ˆ ˆ. Figue shows two concentic sphee of adius and ( > ) which contains unifomly distibuted chages and espectively. Find out electic field intensities at the following positions : (i) < (ii) < (iii) K( ) ˆ K Ans. (i) (ii) ˆ (iii). Ex.6 A solid non conducting sphee of adius and unifom volume chage density ρ has its cente at oigin. Find out electic field intensity in vecto fom at following positions : (i) (/,, ) (ii),, (iii) (,, ) (i) at (/,, ) : Distance of point fom cente ( / ) / <, so point lies inside the sphee so E ρ ρ ε ε [ î ] (ii) At,, ; distance of point fom cente ( / ) ( / ), so point (iii) lies at the suface of sphee, theefoe E K 4 K π ρ The point is outside the sphee so E K 4 K π ρ ( ) î ĵ [ î ĵ] ρ 6 ε ρ ε î ĵ [ î ĵ]. A solid non conducting sphee of adius and unifom volume chage density ρ has cente at oigin. Find out electic field intensity in vecto fom at following positions. (i) (,, ) (ii) (,, ) (iii) (,, ) Ans. (i) E Ex.7 ρî ε (ii) E ρkˆ 6ε (iii) E ρ(î ĵ kˆ) A Unifomly chaged solid nonconducting sphee of unifom volume chage density ρ and adius is having a concentic spheical cavity of adius. Find out electic field intensity at following points, as shown in the figue : (i) Point A (ii) Point B (iii) Point C (iv) Cente of the sphee Method I : (i) Fo point A : We can conside the solid pat of sphee to be made of lage numbe of spheical shells which have unifomly distibuted chage on its suface.now since point A lies inside all spheical shells so electic field intensity due to all shells will be zeo. 9 ε Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 6 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

17 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & Ex.8 E A (ii) Fo point B : All the spheical shells fo which point B lies inside will make electic field zeo at point B. So electic field will be due to chage pesent fom adius to OB. (iii) So, E B 4 K π(ob OB ) ρ ρ OB ε [ OB Fo point C, similaly we can say that fo all the shells point C lies outside the shell K[ 4 π( )] ρ E C [OC] OC ε OB Method : II We can conside that the spheical cavity is filled with chage density ρ and also ρ, theeby making net chage density zeo afte combining. We can conside two concentic solid sphees one of adius and chage density ρ and othe of adius and chage density ρ. Applying supeposition pinciple. (i) (ii) (iii) (iv) E A E ρ E B E ρ E ρ E ρ ρ(oa) ε [ ] ρ(oa) ε [OC] 4 K ρ(ob) π ( ρ) ρ ρ OB OB ε (OB) (OB) ρ OB ε ε OB ε E C E ρ ε (OC) E O E ρ 4 K π OC E ρ ρ [ ] OC E ρ ρ 4 K π ( ρ) OC OC OC In above uestion if cavity is not concentic and cented at point P then epeat all the steps. Again assume ρ and ρ in the cavity, simila to the pevious example. (i) E A E ρ E ρ ρ[oa] ε ρ ( ρ)pa ε [ OA PA ] ρ [ OP] ε ε Note : Hee we can see that the electic field intensity at point P is independent of position of point P inside the cavity. Also the electic field is along the line joining the centes of the sphee and the spheical cavity. ] OB OC Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 7 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

18 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on Ex.9 (ii) (iii) (iv) E B E ρ E C E ρ E O E ρ E ρ E ρ ρ(ob) K ε K E ρ [ 4 π ρ] [OC] K [ 4 π ( ρ) ] [PB] PB K[ 4 π ( ρ) ] OC [ 4 π ( ρ) ] [PO] A nonconducting solid sphee has volume chage density that vaies as ρ ρ, whee ρ is a constant and is distance fom cente. Find out electic field intensities at following positions. (i) < (ii) Method I : (i) (ii) fo < The sphee can be consideed to be made of lage numbe of spheical shells. Each shell has unifom chage density on its suface. So the pevious esults of the spheical shell can be used. Conside a shell of adius x and thickness dx as an element. Chage on shell d (4πx dx)ρ x Electic field intensity at point P due to shell PO Kd de x Since all the shell will have electic field in same diection [PC] E de de de Due to shells which lie between egion < x, electic field at point P will be zeo. Method II : (i) (ii) E Kd K.4πx E de K.4πx dxρx dxρ x 4πKρ ρ ˆ 4ε 4 4 x 4 PC ρ ˆ 4ε The sphee can be consideed to be made of lage numbe of spheical shells. Each shell has unifom chage density on its suface. So the pevious esults of the spheical shell can be used. we can say that all the shells fo which point lies inside will make electic field zeo at that point, so E ( < ) K (4π x dx) ρx ρ ˆ 4ε similaly fo, all the shells will contibute in electic field, theefoe E ( < ) K (4π x dx) ρx 4 ρ ˆ 4ε 7. ELECTIC POTENTIAL In electostatic field the electic potential (due to some souce chages) at a point is defined as the wok done by extenal agent in taking a point unit positive chage fom a efeence point (geneally taken at infinity) to that point without acceleation. 7. Mathematical epesentation : If (W P ) ext is the wok euied in moving a point chage fom infinity to a point P, the electic potential Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 8 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

19 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on Ex. of the point P is V p W p ) ext acc Note (i) W can also be called as the wok done by extenal agent against the electic field poduced by the souce chage. (ii) Wite both W and with pope sign. 7. Popeties : (i) (ii) Potential is a scala uantity, its value may be positive, negative o zeo. joule S.Ι. Unit of potential is volt and its dimensional fomula is [M L T Ι ]. coulmb (iii) Electic potential at a point is also eual to the negative of the wok done by the electic field in taking the point chage fom efeence point (i.e. infinity) to that point. (iv) Electic potential due to a positive chage is always positive and due to negative chage it is always negative except at infinite. (taking V ). (v) Potential deceases in the diection of electic field. 7. Use of potential : If we know the potential at some point ( intems of numeical value o intems of fomula) then we can find out the wok done by electic foce when chage moves fom point 'P' to by the fomula W el ) V p p A chage µc is taken fom infinity to a point in an electic field, without changing its velocity. If wok done against electostatic foces is 4µJ then find the potential at that point. V W ext 4µ J µ C V Ex. When chage µc is shifted fom infinity to a point in an electic field, it is found that wok done by electostatic foces is µj. If the chage is doubled and taken again fom infinity to the same point without acceleating it, then find the amount of wok done by electic field and against electic field. W ext ) p w el ) p w el ) p µj because KE V p W ) ext p µ J µ C V So if now the chage is doubled and taken fom infinity then w ) ext p µ C W ext ) P µj W el ) P µj Ex. A chage µc is eleased at est fom a point P whee electic potential is V then its kinetic enegy when it eaches to infinite is : W el K K f W el ) P V P 6 µj so, K f 6 µj. A chage µc is taken fom infinity to a point in an electic field without acceleation. If wok done by electostatic foces is µj then find out potential at that point. Ans. volt Ex. Two point chages µc and 4µC ae situated at points ( m, m) and ( m, m) espectively. Find out potential at point C. (4 m, m) and. D ( m, 5 m). Potential at point C V C V V Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 9 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

20 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & Similaly K (µ C) K ( 4µ C) 6 5 V K(µ C) V D V V ( 5) K (µ C) K ( 4µ C) 6 V. 9 K( 4µ C) ( 5) Ex.4 A point chage is placed at the cente of unifomly chaged ing of total chage and adius. If the point chage is slightly displaced with negligible foce along axis of the ing then find out its speed when it eaches to a lage distance. Only electic foce is acting on Ex.5 W el K mv K Now W el ) c V c. K mv v K m Two concentic spheical shells of adius and ( > ) ae having unifomly distibuted chages and espectively. Find out potential (i) at point A (ii) at suface of smalle shell (i.e. at point B) (iii) at suface of lage shell (i.e. at point C) (iv) at (v) at (vi) at Using the esults of hollow sphee as given in the table 7.4. Ex.6 (i) V A (ii) V B (iii) V C K K K K K K (iv) fo V K K (v) fo V (vi) fo V K K K K Two hollow concentic nonconducting sphees of adius a and b (a > b) contains chages a and b espectively. Pove that potential diffeence between two sphees is independent of chage on oute sphee. If oute sphee is given an exta chage, is thee any change in potential diffeence? Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

21 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & Electic Potential due to vaious chage distibutions ae given in table. Name/Type Fomula Note Gaph Point chage * is souce chage. K * is the distance of the point fom the point chage. ing at cente * is souce chage. (unifom/nonunifom K chage distibution) at the axis * x is the distance of the point K fom cente. Unifomly chaged fo * is adius of sphee k hollow conducting/ V * is the distance fom cente of sphee nonconducting / fo to the point solid conducting k * is total chage σ4π. sphee V Unifomly chaged fo * is adius of sphee solid nonconducting k V * is distance fom cente. sphee (insulating fo to the point K( ) mateial) * V cente Vsuface. ρ 4 6ε ( ) * is total chage ρ π. * Inside sphee potentail vaies paabolically * outside potential vaies hypebolically. Line chage Not defined * Absolute potential is not defined. * Potential diffeence between two points is given by fomula v B v A Kλ ln ( B / A ) Infinite nonconducting Not defined * Absolute potential is not defined. thin sheet x * Potential diffeence between two points is given by fomula v B v A σ ε ( ) Infinite chaged Not defined * Absolute potential is not defined. conducting * Potential diffeence between two thin sheet points is given by fomula v B v A σ ε B A ( ) B A Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

22 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & V inne sphee V oute sphee K b K a b a K b K a a a V inne sphee V oute sphee V K b b a K b K b b a Which is independent of chage on oute sphee. If oute sphee in given any exta chage then thee will be no change in potential diffeece Potential diffeence The potential diffeence between two points A and B is wok done by extenal agent against electic field in taking a unit positive chage fom A to B without acceleation (o keeping Kinetic Enegy constant o K i K f )) (a a ) Mathematical epesentation : If (W BA ) ext wok done by extenal agent against electic field in taking the unit chage fom A to B V A V B (W BA ) ext acc, o keeping KE constant o K i K f Note : Take W and both with sign (b ) Popeties : (i) The diffeence of potential between two points is called potential diffeence. It is also called voltage. (ii) Potential diffeence is a scale uantity. Its S.I. unit is also volt. (iv) If V A and V B be the potential of two points A and B, then wok done by an extenal agent in taking the chage fom A to B is (W ext ) AB (V B V A ) o (W el ) AB (V A V B ). (v) Potential diffeence between two points is independent of efeence point. (c c ) Potential diffeence in a unifom electic field : V B V A E AB V B V A E AB cos θ Special Cases : Case. Case. E d Ed d effective distance between A and B along electic field. o we can also say that E V d Line AB is paallel to electic field. Line AB is pependicula to electic field.. V A V B Ed Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

23 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & V A V B V A V D Note : In the diection of electic field potential always deceases. Ex.7 µc chage is shifted fom A to B and it is found that wok done by an extenal foce is 4µJ in doing so against electostatic foces then, find potential diffeence V A V B (W AB ) ext (V B V A ) 4 µj µc (V B V A ) V A V B 4 Ex.8 A unifom electic field is pesent in the positive x-diection. Ιf the intensity of the field is 5N/C then find the potential diffeence (V B V A ) between two points A (m, m) and B (5 m, m) V B V A E. AB (5 î ). (5 î ĵ ) 5V. Ex.9 V The electic field intensity in unifom electic field, E d Whee V potential diffeence between two points. d effective distance between the two points. (pojection of the displacement along the diection of electic field.) Find out following (i) V A V B (ii) V B V C (iii) V C V A (iv) V D V C (v) V A V D (vi) Aange the ode of potential fo points A, B, C and D. (i) V AB Ed.4 so, V A V B.4 V because In the diection of electic field potential always deceases. (ii) so, (iii) V BC Ed.4 V B V C.4 V V CA Ed 4.8 so, V C V A.8 V because In the diection of electic field potential always deceases. (iv) V DC Ed so, V D V C because the effective distance between D and C is zeo. (v) V AD Ed 4.8 so, V A V D.8 V because In the diection of electic field potential always deceases. (vi) The ode of potential V A > V B > V C V D..5 A unifom electic field of N/C exists in the vetically downwad diection. Find the inceases in the electic potential as one goes up though a height of 5cm. Ans. 5V.6 An electic field of N/C exists along the x-axis in space. Calculate the potential diffeence V B V A whee the point A and B ae given by (a) A (,) ; B (4m, m) (b) A (4m,m) ; B (6m, 5m) (c) A (,) ; B (6m, 5m) Answe : (a) 8V (b) 4V (c) V Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

24 FEE Download Study Package fom website: & Get Solution of These Packages & Lean by Video Tutoials on Euipotential Suface : Ex.4 If potential of a suface is same thoughout then such suface is known as a euipotential suface. (i) Popeties of euipotential sufaces : (a) When a chage is shifted fom one point to anothe point on an euipotential suface then wok done against electostatic foces is zeo. (b) Electic field is always pependicula to euipotential sufaces. (c) Two euipotential sufaces do not coss each othe. (ii) Examples of euipotential sufaces : (a) Point chage : Euipotential sufaces ae concentic and spheical as shown in figue. In figue we can see that sphee of adius has potential V thoughout its suface and similaly fo othe concentic sphee potential is same. (b) Line chage : Euipotential sufaces have cuved sufaces as that of coaxial cylindes of diffeent adii. (c) Unifomly chaged lage conducting / non conducting sheets Euipotential sufaces ae paallel planes. Note : In unifom electic field euipotential sufaces ae always paallel planes. Some euipotential sufaces ae shown in figue. What can you say about the magnitude and the diection of the electic field? Hee we can say that the electic will be pependicula to euipotential sufaces. Also E V d whee V potential diffeence between two euipotential sufaces. d pependicula distance between two euipotential sufaces. So E (sinº ) V/m Now thee ae two pependicula diections eithe diection o diection as shown in figue, but since we know that in the diection of electic field electic potential deceases so the coect diection is diection. Hence E V/m, making an angle with the x-axis Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 4 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.

25 Get Solution of These Packages & Lean by Video Tutoials on FEE Download Study Package fom website: & Ex.4 Figue shows the lines of constant potential in egion in which an electic field is pesent. The values of potentials ae witten in backets. The electic field is geatest E is lage whee euipotential sufaces ae close. ELOF ae to euipotential sufaces. In the figue we can see that fo point B they ae close so E at point B is maximum.7 Some euipotential sufaces ae shown in figue. What can you say about the magnitude and the diection of the electic field? Ans. adially outwad, deceasing with distance as 6 V m E. 8. ELECTOSTATIC POTENTIAL ENEGY 8. Electostatic potential enegy of a point chage due to many chages : Ex.4 Ex.4 The electostatic potential enegy of a point chage at a point in electic field is the wok done in taking the chage fom efeence point (geneally at infinity) to that point without acceleation (o keeping KE const.o K i K f )). Its Mathematical fomula is U V Hee is the chage whose potential enegy is being calculated and V is the potential at its position due to the souce chages. Note : Always put and V with sign. 8. Popeties : (i) Electic potential enegy is a scala uantity but may be positive, negative o zeo. (ii) Its unit is same as unit of wok o enegy that is joule (in S.Ι. system). Some times enegy is also given in electon-volts. ev.6 9 J (iii) Electic potential enegy depends on efeence point. (Geneally Potential Enegy at is taken zeo) The fou identical chages each ae placed at the cones of a suae of side a. Find the potential enegy of one of the chages due to the emaining chages. The electic potential of point A due to the chages placed at B, C and D is V a a a Potential enegy of the chage at A is V a. a A paticle of mass 4 mg and caying a chage 5 9 C is moving diectly towads a fixed positive point chage of magnitude 8 C. When it is at a distance of cm fom the fixed point chage it has spped of 5 cm/s. At what distance fom the fixed point chage will the paticle come momentaily to est? Is the acceleation constant duing the motion? If the paticle comes to est momentaily at a distance fom the fixed chage, then fom consevation of enegy' we have Teko Classes, Maths : Suhag. Kaiya (S.. K. Si), Bhopal Phone : , page 5 Successful People eplace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.