![2 CHAPTER 1 [Topic 1] Coulomb's Law, Electrostatic Field And Electric .](/img/52/physics.jpg)
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CHAPTER 1 : Electric Charges and Fields2[Topic 1] Coulomb’s law, electrostatic field andelectric dipoleSummaryElectric Charge Electrostatic charge is a fundamental property ofmatter due to which it produces and experienceselectrical and magnetic effects. Properties of atoms, molecules and bulk matterare determined by electric and magnetic forces. It can be inferred from simple experiments basedon frictional electricity that there are two type ofcharges in nature: negative and positive; and likecharges repel and unlike charges attract. By convention, the charge on electron isconsidered as negative and the charge on protonis considered as positive and the charge present isequal. The S.I. unit of electric charge is coulomb.Its C.G.S unit is stat coulomb. The nature and amount of electric charge presentin a charged body is detected by Gold-leafelectroscope. Total charge on a body is expressed as q ne.Conductors and Insulators Objects that allow charges to flow through themare called Conductors (metals) and objects thatdo not allow charges to flow through are calledInsulators (rubber, wood, and plastic).Three basic properties of electriccharge Quantization: When the total charge of a body isan integral multiple of a basic quantum of charge,this is known as quantization of electric charge.i.e., q ne wheren 1, 2, 3, . Additivity: It means that the total charge of asystem is the algebraic sum (adding taking intoaccount negative and positive signs both) of allthe charges in the system. Conservation of charge: Conservation ofelectric charges means that there will be nochange in the total charge of the isolated systemwith time. There is transfer of the electric chargefrom one body to another, but no charge will becreated or destroyed.Coulomb’s lawThe force between two point charges q1 and q2is directly proportional to the product of the twocharges(q1q2) and inversely proportional to thesquare of the distance between them(r2)and it actsalong the straight line joining the two charges.F12 force on q2 due to q1 r 221r̂2114 πε 0 Objects that behave as an intermediate betweenconductors and insulators are called semiconductors, for example- silicon.where k The process of sharing charges with the earth,when we bring a charged body in contact with theearth is called grounding or earthing.8.854 10–12C2N–1m–2The experimental value of the constant ε0 isTherefore, the approximate value of k is9 109Nm2C–2Charging by Inductionr1F21q2r21 r2 – r1F12q1 Charging by induction means charging withoutcontact. If a plastic comb is rubbed with wool, it becomesnegatively charged.k ( q1 q 2 )r2oFig. Depiction of Coulomb’s law
CHAPTER 1 : Electric Charges and FieldsFacts about Coulomb’s law: Coulomb’s law is not valid for charges in motion;it should only be used for point charges in vacuumat rest. The electrostatic force obeys Newton’s third lawof motion and acts along the line joining the twocharges. Presence of other charges in the neighborhooddoes not affect Coulomb’s force. The ratio of electric force and gravitational forcebetween a proton and an electron is representedk e2by 2.4 1039G me mpSuperposition PrincipleThe presence of an (or more) additional charge doesnot affect the forces with which two charges attractor repel each other. Superposition principle statesthat the net force on any charge due to n number ofcharges at rest is the vector sum of all the forces onthat charges, taken one at a time.i.e. F0 F01 F02 F03 .F0n The force on a small positive test charge q placedat the point divided by the magnitude of thecharge is the electric field E at a point due tocharge configuration.Electric Field The space around a charge up to which its forcecan be experienced is called electric field. Electric field due to a point charge q has aqmagnitude E ( r ) rˆ4 πε 0 r 2 It is radially outwards if q is positive. It is radially inwards if q is negative. Electric field satisfies the superposition principle. The unit of electric field is N/C. Electric field inside the cavity of a chargedconductor is zero.Electric Field lines The tangent at each point on the curve of electricfield line, gives the direction of electric field atthat point. The relative strength of electric field at differentpoints is indicated by the relative closeness offield lines. In regions of strong electric field, they crowdnear each other. In regions of weak electric field, they are farapart.3In regions of constant electric field, the fieldlines formed are uniformly spaced parallelstraight lines. Field lines are continuous curves. There will beno breaks. qqFig. Electric field lines Field lines are not intersecting. They cannot crosseach other. Electrostatic field lines begin at positive chargesand terminate at negative charges. No closed loop can be formed by them.Electric Dipole A pair of equal and opposite charges q and–q separated by small distance 2a is knownas electric dipole. The magnitude of its dipolemoment vector is 2qa and is in the direction ofthe dipole axis from –q to q.E q2apE–qprq–qFig. Electric dipole Field of an electric dipole in its equatorial planeat a distance r from the center:E p124 πε o a r 2( p4 πε o r 3)32for r a Dipole electric field on the axis at a distance rfrom the center:E 2pr(4 πε o r 2 a 22p4 πε o r 3)2for r aThe 1/r3 dependence of dipole electric fields shouldbe noted in contrast to the 1/r2 dependence of electricfield due to a point charges. In a uniform electric field E, a dipole experiencesa torque t given byt p EBut no net force will be experienced by it.
CHAPTER 1 : Electric Charges and Fields8Topic 2: Electric FluxSummary Electric flux is proportional to number of linesleaving a surface, outgoing lines with positivesign, incoming lines with negative sign.EWhere, r is the radial (perpendicular) distance of thepoint from the wire and n̂ is the radial unit vector inthe plane normal to the wire passing through thepoint. Infinite plane sheet (thin) of uniform surfacecharge density sSurface chargedensityzSEEFig. Electric flux Through a small area element DS, the flux Df ofelectric field E is given byy21xExxFig. Infinite plane sheet (thin)Df E.DSAnd the vector area element DS is S SnˆE σnˆ2ε oWhere n̂ is a unit vector normal to the plane andgoing away from it. Thin spherical shell of uniform surface chargedensity sq Er( r R)Gauss’s Law and its application 4 πε o r 2 The flux of electric field through any closedGaussian surfaceSurface chargesurface S is 1/e0 times the total charge encloseddensityPby S.Where DS is the magnetic of the area element and n̂is normal to the area element, which can be consideredplanar for the sufficiently small DS. The law is mainly useful in determining electricfield E, when the source distribution has simplesymmetry: Thin infinitely long straight wire of uniformlinear charge density lPEr λn2πε o rFig.: Thin uniformly surface charged sphericalshell (r R)(For r R)E 0 (r R)Surface chargedensityRlrFig. Thin infinitely long Straight wireE rROq enclosed φ E dAε0GaussiansurfaceOrPFig.: Thin uniformly surface charged sphericalshell (r R)(For r R)Where r is the distance of the point from the centerof the shell whose radius is R with the total chargeq. The electric field outside the shell is the same asthe total charge is concentrated at the center. A solidsphere of uniform volume charge density shows thesame result. Inside the shell at all the points, thefield is zero.
CHAPTER 2 : Electrostatic Potential and Capacitance16Topic 1: Electrostatic Potential and Electrostatic PotentialEnergySummaryq from a point R to a point P is VP – VR , which isthe difference in potential energy of charge q between the final and initial points.Electrostatic potential:Potential difference: The amount of work done by an external force inmoving a unit positive charge from one point toanother in electrostatic field is called electricalpotential. Such that V When the work is done upon a charge to change itspotential energy then the difference between the final and the initial location is called electric potentialdifference.1 q4 πε rElectric Potential due to a dipole: Where, q charge causing the field, e permittivity, r separation between centre of charge point. Electrostatic force is a conservative force. Work done by an external force (equal and opposite to the electrostatic force) in bringing a charge The electrostatic potential at a point with distancer due dipole at a point making an angle q withdipole moment p placed at the origin is given by1p rˆV(r) 2 .4 πε o rPr1q ra2ar2Opa––qFig. Electrical potential due to dipole It is a scalar quantity. Let A and B be the initial and final location for asingle charge q then the potential difference between A and B is given by:BBBAAA V VB VA E ds Eds cos θ E dsWhere, E is the field due to a point charge, ds dr,so thatrBVB VA rAq drq 24 πε 0 r4 πε 0rBq 11 1 r rA 4 πε 0 rB rA The result is true also for a dipole (with charges–q and q separated by 2a for r a.Dipole and System of charges For a charge configuration q1, q2, ., qn with position vectors r1, r2, r3, ., rn, then the potentialV1 at point P due to charge q1 will be,1 q1V1 4 πε 0 r1And the sum of potentials due to individual chargesis given by the superposition principle,V q 1 q1 q 2 . n 4 πε o r1P r2PrnP
CHAPTER 2 : Electrostatic Potential and Capacitance In this system the two charges q1 and q2 whenseparated by distance r, will either repel orattract each other. Electrical potential of charges q1 and q2 is given by:q5q4q1r5Pr4Pr1P17U r3Pq3Pr2Pq2 In a uniformly charged spherical shell, the electricfield outside the shell with outside potential isgiven by,V 1 q4 πε 0 rEquipotential surfaces A surface over which potential has a constant value is known as an equipotential surface. The amount of work done in moving a charge overan equipotential surface is zero. Concentric spheres centered at a location of thecharge act as equipotential surfaces for a pointcharge. The electric field E, at a point and equipotentialsurface are mutually perpendicular to each otherthrough the point. The direction of the steepestdecrease of potential is in E. Regions of strong and weak fields are located because of the spacing among equipotential surfaces.Potential Energy of a System ofCharges:Potential energy stored in a system of charges isthe work done by an external agency in assemblingthe charges at their locations. Total work donein assembling the charges is given by1 q1 q 2 q1 q 3 q 2 q 3 U where r12 is distance4 πε o r12r13r23 between q1 and q2, r13 is distance between q1 & q3 andr23 is distance between q2 & relabel q3.r23q3q2r13r12q1Fig. Potential energy due to System of chargesElectric potential energy of systemof two point charges Here the work done doesn’t depend on path.1 2 q i Vi2 i 1Potential Energy in an External Field: The potential energy of a charge q in an externalpotential V(r) is qV(r). The potential energy of a dipole moment p in a uniform electric field E is –p.E. Electric dipole in an electrostatic field: Electricpotential due to a dipole at a point at distance rand making an angle q with the dipole moment pis given byV 1 p cos θ4 πε 0 r 2Electrostatics of conductors: Electrostatic field is zero inside a conductor. Electrostatic field at the surface of a chargedconductor must be normal to the surface at everypoint. In the static situation, there cannot be any excesscharge in the interior of a conductor. Throughout the volume of the conductor, theelectrostatics potential is constant and has samevalue on its surface. Electrostatics field E is zero in the interior of aconductor; just outside the surface of a chargedconductor, E is normal to the surface given byσE nˆ where n̂ is the unit vector along theεooutward normal to the surface and σ is the surfacecharge density. Electrostatic shielding: A field which is insidethe cavity of a conductor is always zero and itremains shielded from the electric field, which isknown as electrostatic shielding.Dielectrics and Polarization: Dielectrics: A non-conducting substance whichhas a negligible number of charge carriers unlikeconductors is called dielectrics. Electric polarization: The difference betweeninduced electric field and imposed electric field indielectric due to bound and free charges is knownas electric polarization. It is written as:P D E4πNote: Polarisation can also be written as polarization (with ‘z’ in place of ‘s’)
CHAPTER 2 : Electrostatic Potential and Capacitance25Topic 2: CapacitanceSummaryCapacitor and Capacitance Capacitor: The system of two conductorsseparated by an insulator is called capacitor.The device which is used to store charge is knownas capacitor. The applied voltage and size ofcapacitor decides the amount of charge that canbe stored i.e., Q CVTwo similar connecting plates are placed incapacitor in the front of each other where oneplate is connected to the positive terminal andother plate is connected to the negative terminal. Capacitance: The ratio of magnitude of chargestored on the plate to potential difference betweenthe plates is called capacitance. It is written as:QC VSize, shape, medium and other conductors insurrounding influence the capacitance of aconductor.Its S.I. unit is farad.1F 1CV–1 For a parallel plate capacitor (withAvacuum between the plates), C ε owhere Adis the area of each plate and d in the separationbetween the parallel plates.Area AI1 E–d––––––––––2IIFig. CapacitorEffect of Dielectric on Capacitance: If the medium between the plates of a capacitoris filled with an insulating substance (dielectric),the electric field due to the charged plates inducesa net dipole moment in the dielectric. This effect,called polarization, gives rise to a field in theopposite direction. The dielectric is polarised by the field and alsothe effect is equivalent to two charged sheets withsurface charge densities sp and –sp. The net electric field inside the dielectric andhence the potential difference between the platesis thus reduced. Consequently, the capacitanceC increases from its value Co when there is nomedium (vacuum),εC KCo where K is the dielectric constantε0of the insulating substance.Types of capacitor: Parallel plate capacitor: C Kε 0Ad Cylindrical capacitor: C 2πKε 01ln ( b a ) ab Spherical capacitor: C 4 πKε 0 b a Combination of Capacitors For capacitors in the series combination, the totalcapacitance C is given by11111 .C C1 C2 C3Cn In the parallel combination, the total capacitanceC is C C1 C2 C3 . Cn, where C1, C2, C3 .are individual capacitances. Capacitors connected in series have the samecharges and when connected in parallel have thesame voltage. Potential across capacitor remains same if thebattery is connected but if it is disconnectedthen charge remains the same which is stored incapacitor.Electrical Energy Stored in aCapacitor: The energy U stored in a capacitor of capacitanceC, with charge Q and voltage V is111 Q2 .U QVCV 2 222 C The electric energy density (energy per unit vol1ume) in a region with electric field is ε o E2 .2 Electric density is alternatively known aselectrostatic pressure.
CHAPTER 2 : Electrostatic Potential and Capacitance26Van De Graaff Generator: A Van de Graaff generator consists of a largespherical conducting shell (a few meters in diameter). There are two pulleys, one at ground level andone at the center of the shell. Both of them arewounded around by a long and narrow endlessbelt of insulating material. The motor drives the lower pulley which keepsmoving this belt continuously. At ground level to the top, it continuously carriesthe positive charge and sprayed on to it by a brush.Then the positive charge is transferred by it to another conducting brush connected to the large shell. After the transferring of the positive charge isdone, it spreads out uniformly on the outer surface. It can build the voltage difference of as muchas 6 to 8 million volts.plasticcharge taken by rollerpointed electrodepointed electrodeproduces chargeby friction or highvoltagesphericalmetal coverrubber beltplasticrollerFig. Yande Graff GeneratorHow will the (i) charge and (ii) potentialdifference between the plates of the capacitorsbe affected after the slabs are inserted?PREVIOUS YEARS’EXAMINATION QUESTIONSTOPIC 21 Mark Questions1. Why should electrostatic field be zero inside aconductor?[All INDIA 2012]2. A capacitor has been charged by a dc source.What are the magnitudes of conduction anddisplacement current, when it is fully charged?[All INDIA 2013]3. Define dielectric constant of a medium. What isits S.I. unit?[DELHI 2015]4. Predict the polarity of the capacitor in thesituation described below:SNABSN[All INDIA 2017]2 Mark Questions5. Figure shows two identical capacitors, C1 and C2each of 1µF capacitance connected to a battery of6V. Initially switch ‘S’ is closed. After sometime‘S’ is left open and dielectric slabs of dielectricconstant K 3 are inserted to fill completely thespace between the plates of the two capacitors.[DELHI 2011]6. A slab of material of dielectric constant K hasthe same area as that of the plates of a paralleldplate capacitor but has the thickness 2 , where3d is the separation between the plates. Find outthe expression for its capacitance when the slabis inserted between the plates of the capacitor.[DELHI 2011]7. A capacitor of unknown capacitance is connectedacross a battery of V volts. The charge stored init is 360 μC. When potential across the capacitoris reduced by 120 V, the charge stored in itbecomes 120 μC.Calculate:(i) T h e p o t e n t i a l V a n d t h e u n k n o w ncapacitance C.(ii) What will be the charge stored in thecapacitor, if the voltage applied had increasedby 120 V?[DELHI 2011]8. A parallel plate capacitor of capacitance C ischarged to a potential V. It is then connected toanother uncharged capacitor having the samecapacitance. Find out the ratio of the energystored in the combined system to that storedinitially in the single capacitor[All INDIA 2014]
CHAPTER 3 : Current Electricity38Topic 1: Electricity conduction, Ohm’s law and resistanceSummaryElectric Current: Net charge flowing across a givenarea of conductor per unit time is defined as electriccurrent.ρl, ρ being the resistivity of the material of theAR conductor.lq, S.I. unit of current is Ampere (A).I tA steady current is generated in a closed circuitwhere electric charge moves from lower to higherpotential. Electromotive force or emf is the workdone by the source in taking the charge from higherto lower potential energy.Drift velocity: The free electrons drift with somevelocity towards the positive terminal when apotential difference is applied across the ends. Theaverage velocity with which the electrons move istermed as drift velocity.Drift velocity, vdeEτ eVτ mmlWhere e charge on electronE Electric field intensityV Potential difference across the ends of theconductort Relaxation timem Mass of electronRelation between current and drift velocity:Current is directly proportional to the drift velocity.I vdWhen the number of electrons are less, current is lessso the drift velocity is small.When the number of electrons are large, high currentflows so the drift velocity is large.Ohm’s law: The voltage across the ends of theconductor is directly proportional to the electriccurrent flowing through the conductor.V IOr V IR, where R is the electrical resistance of theconductorResistance: The property that resists the flow ofcurrent through any conductor is called the resistanceof the conductor.VR IIt varies directly with the length of the conductorwhile depends inversely on the area of cross sectionof the conductor.AFig.: Resistance in a conductorResistivity: It depends on the nature of the materialand temperature. It is also termed as specificresistance.ρ m gives the relation between resistivity andne 2 τrelaxation time.There is an increasing order of resistivity as we gofrom metal to insulator.ρmetals ρsemiconductors ρinsulatorsConductivity and conductance: The reciprocal ofresistivity is conductivity (s).1and its S.I. unit is W–1m–1.ρThe reciprocal of resistance is the conductance of theconductor. Its S.I. unit is mho.σ Current Density: The amount of charge flowing perunit area per second is called the current density.J mqvd , where vd is the drift velocity of the chargecarriers, n is the number of charge carriers and q isthe charge.The relation between current density and conductivityisJ sEMobility: Mobility is the ratio of drift velocity to theapplied electric field. Mobility is symbolized by m. µv d qτ EmIts S.I. unit is m2s–1V–1.Resistors: The objects which resist the flow of chargeare called resistors which can be of two types, i.e. wirebound resistors and carbon resistors.Resistors can combine in two different ways; either inseries or in parallel. Consider n number of resistors connected in series,then the combined resistance will be as follows:R eqv R1 R 2 R 3 . R n
CHAPTER 3 : Current Electricity39Same amount of current will flow through each resistorconnected in series while the potential differencewould be different for every resistor.Internal resistance: It is the resistance on the current offered by the electrolyte and the electrodes. Itis symbolize by r. Consider n number of resistors connected inparallel, then the combined resistance will be asfollows:Let us assume a cell with 2 electrodes connected byεan external resistance R. Then current is, I R rR eqv 1111 . R1 R 2 R 3Rnwhere e emf, r Internal resistanceThe current flowing through each resistor would bedifferent in this case while the potential differencewould be same for all the resistors.PREVIOUS YEARS’EXAMINATION QUESTIONSTOPIC 1[DELHI 2014][ALL INDIA 2012]2. Show on a graph, the variation of resistivity withtemperature for a typical semiconductor.[ALL INDIA 2012]3. The graph shown in the figure representsa plot of current versus voltage for a givensemiconductor. Identify the region, if any,over which the semiconductor has a negativeresistance.B9. Graph showing the variation of current versusvoltage for a material GaAs is shown in thefigure, Identify the region of:(i) Negative resistance(ii) Where Ohm’s law is obeyedCurrent I1 Mark QuestionsCurrent (mA)[DELHI 2014]8. Show variation of resistivity of copper as afunction of temperature in a graph.1. When electrons drift in a metal from lower tohigher potential, does it mean that all the freeelectrons of the metal are moving in the samedirection?O7. Define the term ‘electrical conductivity’ of ametallic wire. Write its S.I. unit.ACDEBVoltage V[DELHI 2015]10. V-I graph for a metallic wire at two differenttemperature T1 and T2 is as shown in the figure.Which of the two temperatures is higher andwhy ?CAVoltage (V)T1[ALL INDIA 2013]4. Define the term ‘Mobility’ of charge carriers ina conductor. Write its S.I. unit.[DELHI 2014]5. Plot a graph showing variation of current versusvoltage for the material Ge.[DELHI 2014]6. Define the term ‘drift velocity’ of charge carriersin a conductor and write its relationship withthe current flowing through it.[DELHI 2014]VT2θ1θ2I[ALL INDIA 2015]
CHAPTER 3 : Current Electricity45Topic 2: Kirchhoff’s Laws, cells and their combinationsSummaryKirchhoff’s law:Cells in series and in parallel The equivalent emf of a series combination of ncells is just the sum of their individual emfs The equivalent internal resistance of a seriescombination of n cells is the sum of their internalresistances.ε2ε1IBr1Cr2 Loop Rule: The sum of changes in potentialaround any loop that is closed should be zero.Wheatstone bridge: It is an arrangement of fourresistors in a way so that a galvanometer is placedbetween the two opposite arms.There is a null-point condition in the wheatstone bridge where current is zero which can berepresented as follows:R1 R 3 R2 R4e e1 e2 In a parallel connection,Bε eq ε1ε111 . nand . rrrnreq Dr2I4 UStaar ndam rdA Junction Rule: The sum of currents entering ajunction would be equal to the sum of currentsleaving the junction.I3εFig.: Wheastone bridgePREVIOUS YEARS’EXAMINATION QUESTIONSTOPIC 23. Two identical cells, each of emf E, havingnegligible internal resistance, are connectedin parallel with each other across an externalresistance R. What is the current through thisresistance?[ALL INDIA 2013]1 Mark Questions1. A cell of emf E and internal distance r draws acurrent ‘I’. Write the relation between terminalvoltage ‘V’ in terms of E, I, r.[DELHI 2013]2. A heating element is marked 210 V, 630 W. Whatis the value of current drawn by the elementwhen connected to a 210 V, dc source?[DELHI 2013]2 Marks Questions4. A cell of emf E and internal resistance r isconnected to two external resistances R1 andand R2 a perfect ammeter. The current in thecircuit is measured in four different situations:(i) without any external resistance in the circuit(ii) with resistance R1 only(iii) with R1 and R2 in series combination(iv) with R1 and R2 in parallel combination
CHAPTER 3 : Current Electricity54Topic 3: Electrical devicesSummaryMeter Bridge: Meter Bridge is the simplest formof the Wheatstone bridge which is used for accuratecomparison of resistances.In order to find out an unknown resistance R with thehelp of a standard known resistance S:SRBAl1Potentiometer: It is a device which is used tocompare potential differences and emf’”s. It alsomeasures the internal resistance of a cell.ε113ε22ABN1RGDC100 - l1CGN2K1Metre scaleFig.: Potentiometerεε1l 1ε 2 l2K1Fig.: Meter bridgeR Sl1, l1 being the distance of the jockey from100 l1Potentiometer does not draw any current fromthe voltage source being measured. The internalresistance of a given cell can be measured by: l r R 1 1 l2 end A at the balance point.PREVIOUS YEARS’EXAMINATION QUESTIONSTOPIC 34. Two electric bulbs P and Q have their resistancesin the ratio of 1:2. They are connected in seriesacross a battery. Find the ratio of the powerdissipation in these bulbs.[DELHI 2018]1 Mark Questions1. A resistance R is connected across a cell of emf εand internal resistance r. A potentiometer nowmeasures the potential difference between theterminals of the cell as V. Write the expressionfor ‘r’ in terms of ε, V and R.[ALL INDIA 2011]2 Marks Questions2. Use Kirchhoff’s rules to obtain conditions for thebalance condition in a Wheatstone bridge.[DELHI 2015]3. Describe briefly, with the help of a circuit diagram,how a potentiometer is used to determine theinternal resistance of a cell.[ALL INDIA 2013]5. In a potentiometer arrangement for determiningthe emf of a cell, the balance point of the cell inopen circuit is 350cm.When a resistance of 9Ωis used in the external circuit of the cell, thebalance point shifts to 300 cm. Determine theinternal resistance of the cell.[ALL INDIA 2018]3 Marks Questions6. A potentiometer wire of length 1.0 m has aresistance of 15 Ω. It is connected. o a 5V batteryin series with a resistance of 5 Ω. Determine theemf of the primary cell which gives a balancepoint at 60 cm.[DELHI 2016]
CHAPTER 4 : Moving Charges and Magnetism66[Topic 1] Magnetic Field Laws and theirApplicationsSummary Magnetic field at centre of the coil isµ 0 NiB ( x 0) The Oersted’s law states that an electric current 2Rcreates a magnetic field. Magnetic field due to current carrying circular The Biot Savart’s law states that, the magnitudeµ iof magnetic field dB is proportional to the currentarc with centre O is B 04rI, the element length dl and inversely proportional to the square of the distance r. Its direction is If we curl the palm of our right hand around theperpendicular to the plane containing dl and r.circular wire with the fingers pointing in the diµThus in vector notation, dB Idl r , where 0rection of the current, the right hand thumb rule4πr3gives the direction of the magnetic field.is the constant of proportionality and is equal to Ampere’s circuital law: The line integral of the10–7 Tm/A.magnetic field around some closed loop is equal toCurrent elementYθIdlloop, C,rApplications of Ampere’s LawMagnetic field due to current carrying solenoid, B m0nIFig.: Biot Savart’s lawApplications of Biot-Savart’s Law: Magnetic field at a point in circular loop will beµ 0 IR 2(µ 0IP dBX2 R 2 x2 B.dl CIB the times the algebraic sum of the currents whichpass through the circular loop. For some circuital)32At the end of a short solenoid, B µ 0 nI2 The magnetic force produced by a Solenoid asstated by Ampere’s law is given as F m0nI,where n is the number of turns of the wire perunit length, I is the current flowing through thewire and the direction is given using the righthand thumb rule. Due to a toroid a magnetic field is given as,µ NIwhere ‘N’ is the number of turns of theB 02πrYdldB1rROIxdBθXP dBxdlZFig.: Magnetic field at a point in circular looptoroid coil, I is the amount of current flowing andr is the radius of the toroid. Antiparallel currents repel and parallel currentsattract. Magnetic moment on a rectangular current loopin a uniform magnetic field, m NIA where mis the magnetic moment and N is the number ofclosely wounded turns and A is the area vector.
CHAPTER 4 : Moving Charges and Magnetism73Topic 2: Lorentz Force and CyclotronSummary The electric field, E produced by the source of theQrˆfield Q, is given as E , where r̂ is the( 4π 0 ) r 2Magnetic field outof the paperDeflection plateExit Portunit vector and the field E is a vector field. Acharge ‘q’ interacts with this field and experiencesa force F, expressed as F qE qQrˆ( 4π 0 ) r 2 In the presence of both electric field E(r) and magnetic field B(r)
Potential difference: When the work is done upon a charge to change its potential energy then the difference between the fi-nal and the initial location is called electric potential difference. Electric Potential due to a dipole: The electrostatic potential at a point with distance r due dipole at a point making an angle q with
