1January 11, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 3January 11, 2005 Physics for Scientists&Engineers 2 2Review from YesterdayReview from Yesterday There are positive charges and negative charges Law of Charges• Like charges repel and opposite charges attract The unit of charge is the coulomb defined as• 1 C = 1 A•s Law of charge conservation• The total charge of an isolated system is strictlyconservedJanuary 11, 2005 Physics for Scientists&Engineers 2 3Electrostatic ChargingElectrostatic Charging There are two ways to charge an object• Conduction• Induction Charging by conduction• We can charge an object by connecting a source ofcharge directly to the object and then disconnecting thesource of charge• The object will remain charged– Conservation of chargeJanuary 11, 2005 Physics for Scientists&Engineers 2 4Charging by InductionCharging by Induction We can also charge an object without physicallyconnecting to it• First we charge a paddle with negative charge• Then we ground the object to be charged• Connecting the object to the Earth provides an effectivelyinfinite sink for charge• We bring the charged paddle close to the object but donot touch it• We remove the ground connection and move the paddleaway• The object will be charged by induction2January 11, 2005 Physics for Scientists&Engineers 2 5Electric Force - CoulombElectric Force - Coulomb’’s Laws Law Consider two electric charges: q1 and q2 The electric force F between these two chargesseparated by a distance r is given by Coulomb’s Law The constant k is called Coulomb’s constant and isgiven byF = kq1q2r2k = 8.99 !109 N ! m2C2January 11, 2005 Physics for Scientists&Engineers 2 6CoulombCoulomb’’s Law (2)s Law (2) We can get a feeling for how big a coulomb of charge is ifwe calculate the force between two 1 C charges 1 meterapart The coulomb constant is also written as ε0 is the electric permittivity of free space• Fundamental constantk =14!"0F = 8.99 !109 N !m2C2"#$%&'1 C !1 C1 m( )2= 8.99 !109 N which is the weight of 450 Space Shuttles at launch!0= 8.85 "10#12 C2N " m2January 11, 2005 Physics for Scientists&Engineers 2 7Electric ForceElectric Force The electric force is given by The electric force, unlike the gravitational force,can be positive or negative• If the charges are the opposite sign, the force isnegative• Attractive• If the charges are the same sign, the force is positive• Repulsive We can also write the electric force in vector formF = kq1q2r2 !F2!1= kq1q2r3(!r2"!r1)January 11, 2005 Physics for Scientists&Engineers 2 8Example - The Helium NucleusExample - The Helium Nucleus The nucleus of a helium atom has two protons and two neutrons. Thesefour nucleons are bound together by the strong force. What is themagnitude of the electric force between the two protons in the heliumnucleus?Each proton has charge q = 1.602 !10"19 CThe distance between the two protons is approximately 2.0 ! 10-15 m The force is given byF = kq1q2r2= 8.99 !109 N ! m2C2"#$%&'1.602 !10(19 C( )22.0 !10(15 m( )2= 58 NConsidering that the mass of a proton is 1.67 ! 10-27 kgthis force is huge3January 11, 2005 Physics for Scientists&Engineers 2 9x1x2 Consider two charges located on the x axis The charges are described by• q1 = 0.15 µC x1 = 0.0 m• q2 = 0.35 µC x2 = 0.40 m Where do we need to put a third charge for thatcharge to be at an equilibrium point?• At the equilibrium point, the force from each of the twocharges will cancelExample - Equilibrium PositionExample - Equilibrium PositionJanuary 11, 2005 Physics for Scientists&Engineers 2 10Example - Equilibrium Position (2)Example - Equilibrium Position (2) We can see that the equilibrium point must bealong the x-axis Let’s consider three regions along the x-axiswhere we might place our third charge• x3<x1• x1<x3<x2• x2<x3x1x2January 11, 2005 Physics for Scientists&Engineers 2 11Example - Equilibrium Position (3)Example - Equilibrium Position (3) x3<x1• Here the forces from q1 and q2 will always point in thesame direction (to the left for a positive test charge)• No equilibrium x2<x3• Here the forces from q1 and q2 will always point in thesame direction (to the right for a positive test charge)• No equilibriumx1x2January 11, 2005 Physics for Scientists&Engineers 2 12Example - Equilibrium Position (4)Example - Equilibrium Position (4) x1<x3<x2• Here the forces from q1 and q2 can balancex1x2kq1q3(x3! x1)2= kq3q2(x2! x3)2 q3 cancelsq1(x3! x1)2=q2(x2! x3)2"q1(x2! x3)2= q2(x3! x1)2"q1(x2! x3) = q2(x3! x1) "x3=q1x2+ q2x1q1+ q2x3=q1x2+ q2x1q1+ q2=0.15 µC ! (0.4 m)0.15 µC + 0.35 µC= 0.16 m4January 11, 2005 Physics for Scientists&Engineers 2 13Example - Charged BallsExample - Charged Balls Consider two identical charged balls hanging fromthe ceiling by strings of equal length 1.5 m. Eachball has a charge of 25 µC. The balls hang at anangle θ = 25° with respect to the vertical. What is the mass of each ball? The distance between the balls isd = 2! sin!= 2(1.5 m)sin 25° = 1.27 mThe coulomb force between the balls isFc= kq1q2r2= kq2d2The gravitation force on each ball points downFg= mgJanuary 11, 2005 Physics for Scientists&Engineers 2 14Example - Charged Balls (2)Example - Charged Balls (2)Looking at the left ballx direction: kq2d2= T sin!y directon: mg = T cos!Dividing these two equations we getkq2mgd2=T sin!T cos!= tan!"m =kq2tan!gd2=(8.99·109 N m2C#2)(2.5·10#5 C)2tan25°(9.81 m/s2)(1.27 m)2= 0.76 kgA similar analysis applies to the right ballJanuary 11, 2005 Physics for Scientists&Engineers 2 15Electric Force and Gravitational ForceElectric Force and Gravitational Force Coulomb’s Law that describes the electric forceand Newton’s gravitational law have a similarfunctional form Both forces vary as the inverse square of thedistance between the objects Gravitation is always attractive k and G give the strength of the forcesFelectric= kq1q2r2Fgravity= Gm1m2r2January 11, 2005 Physics for Scientists&Engineers 2 16Example - Forces between ElectronsExample - Forces between Electrons What is relative strength of the force of gravity comparedwith the electric force for two electrons? So the electric force is always very much larger than thegravitational force Macroscopic objects are usually uncharged so only gravityplays
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