Review from Yesterday There are positive charges and negative charges Law of Charges Physics for Scientists Engineers 2 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 Spring Semester 2005 The total charge of an isolated system is strictly conserved Lecture 3 January 11 2005 Physics for Scientists Engineers 2 1 January 11 2005 Electrostatic Charging Physics for Scientists Engineers 2 2 Charging by Induction There are two ways to charge an object We can also charge an object without physically connecting to it Conduction First we charge a paddle with negative charge Induction Then we ground the object to be charged Charging by conduction We can charge an object by connecting a source of charge directly to the object and then disconnecting the source of charge The object will remain charged Connecting the object to the Earth provides an effectively infinite sink for charge We bring the charged paddle close to the object but do not touch it We remove the ground connection and move the paddle away Conservation of charge The object will be charged by induction January 11 2005 Physics for Scientists Engineers 2 3 January 11 2005 Physics for Scientists Engineers 2 4 1 Electric Force Coulomb Coulomb s Law Coulomb Coulomb s Law 2 Consider two electric charges q1 and q2 The electric force F between these two charges separated by a distance r is given by Coulomb s Law F k q1q2 r2 January 11 2005 N m 2 1 C 1 C F 8 99 10 9 8 99 10 9 N C2 1 m 2 which is the weight of 450 Space Shuttles at launch The constant k is called Coulomb s constant and is given by k 8 99 10 9 We can get a feeling for how big a coulomb of charge is if we calculate the force between two 1 C charges 1 meter apart The coulomb constant is also written as k N m2 C2 C2 N m2 Fundamental constant 5 Electric Force January 11 2005 Physics for Scientists Engineers 2 6 Example The Helium Nucleus q1q2 r2 The nucleus of a helium atom has two protons and two neutrons These four nucleons are bound together by the strong force What is the magnitude of the electric force between the two protons in the helium nucleus The electric force unlike the gravitational force can be positive or negative If the charges are the opposite sign the force is negative Each proton has charge q 1 602 10 19 C The distance between the two protons is approximately 2 0 10 15 m Attractive If the charges are the same sign the force is positive The force is given by Repulsive F k We can also write the electric force in vector form 2 19 q1q2 N m 2 1 602 10 C 8 99 10 9 58 N 2 2 15 r C 2 0 10 m 2 Considering that the mass of a proton is 1 67 10 27 kg this force is huge qq F2 1 k 1 3 2 r2 r1 r January 11 2005 0 8 85 10 12 0 is the electric permittivity of free space Physics for Scientists Engineers 2 The electric force is given by F k 1 4 0 Physics for Scientists Engineers 2 7 January 11 2005 Physics for Scientists Engineers 2 8 2 Example Equilibrium Position Example Equilibrium Position 2 Consider two charges located on the x axis x2 x1 We can see that the equilibrium point must be along the x axis The charges are described by q1 0 15 C x1 0 0 m Let s consider three regions along the x axis where we might place our third charge q2 0 35 C x2 0 40 m Where do we need to put a third charge for that charge to be at an equilibrium point x3 x1 x1 x3 x2 At the equilibrium point the force from each of the two charges will cancel January 11 2005 Physics for Scientists Engineers 2 x2 x3 9 January 11 2005 Example Equilibrium Position 3 x1 Physics for Scientists Engineers 2 10 Example Equilibrium Position 4 x2 x2 x1 x1 x3 x2 x3 x1 Here the forces from q1 and q2 can balance Here the forces from q1 and q2 will always point in the same direction to the left for a positive test charge k No equilibrium q1q3 q3q2 k q3 cancels x3 x1 2 x2 x3 2 q1 q2 x3 x1 2 x2 x3 2 x2 x3 q1 x2 x3 2 q2 x3 x1 2 Here the forces from q1 and q2 will always point in the same direction to the right for a positive test charge q1 x2 x3 q2 x3 x1 x3 No equilibrium January 11 2005 x2 x1 Physics for Scientists Engineers 2 11 January 11 2005 x3 q1 x2 q2 x1 q1 q2 0 15 C 0 4 m 0 15 C 0 35 C 0 16 m q1 x2 q2 x1 q1 q2 Physics for Scientists Engineers 2 12 3 Example Charged Balls Example Charged Balls 2 Consider two identical charged balls hanging from the ceiling by strings of equal length 1 5 m Each ball has a charge of 25 C The balls hang at an angle 25 with respect to the vertical Looking at the left ball q2 T sin d2 y directon mg T cos x direction k What is the mass of each ball Dividing these two equations we get The distance between the balls is d 2 sin 2 1 5 m sin 25 1 27 m kq 2 T sin tan mgd 2 T cos The coulomb force between the balls is m qq q2 Fc k 1 2 2 k 2 r d The gravitation force on each ball points down Fg mg January 11 2005 A similar analysis applies to the right ball Physics for Scientists Engineers 2 13 Electric Force and Gravitational Force q1q2 r2 Fgravity G 14 qe r2 m2 Fgravity G 2e r Felectric kqe2 8 99 10 9 N m 2 C2 1 602 10 19 C 2 4 2 10 42 2 Fgravity Gme 6 67 10 11 N m 2 kg 2 9 109 10 31 kg 2 Felectric k m1m2 r2 So the electric force is always very much larger than the gravitational force Macroscopic objects are usually uncharged so only gravity plays a role Gravitation is always attractive Motion of the planets k and G give the strength of the forces Physics for Scientists Engineers 2 Physics for Scientists Engineers 2 What is relative strength of the force of gravity compared with the electric force for two electrons 2 Both forces vary as the inverse square of the distance between the objects January 11 2005 January 11 2005 Example Forces between Electrons Coulomb s Law that describes the electric force and Newton s gravitational law have a similar functional form Felectric k kq 2 8 99 10 9 N m 2C 2 2 5 10 5 C 2 0 …
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