PHY 184

1. PHY 184 Spring 2007 Lecture 3 Title: The Coulomb Force 184 Lecture 3

2. Announcements • Don’t forget to register your clicker! • Homework Set 1 is due Tuesday, January 16, at 8:00 am • We will soon post the complete SLC schedule. Strosacker Learning Center Room 1248 B. P. S. 184 Lecture 3

3. Outline 1 – Review 2 – Electrostatic charging 3 – Coulomb’s Law 184 Lecture 3

4. Review • There are two types of charge: negative and positive. • Most objects are electrically neutral; they have equal numbers of negative and positive charges (net charge is 0). • An object becomes charged by adding or removing electrons. • An electron carries negative charge of magnitude e = 1.602×10-19 C. • Law of Charges • Like charges repel and opposite charges attract. • Law of charge conservation • The total charge of an isolated system is strictly conserved. • Conductors are materials where some of the electrons can move freely. • Insulators are materials where none of the charges can move freely. 184 Lecture 3

5. Electrostatic Charging • There are two ways to charge an object • Conduction • Induction • 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 • Conservation of charge 184 Lecture 3

6. Charging by Conduction + + + + + + + + + Electroscope We brought charge onto the electrode by contact. 184 Lecture 3

7. Induction - - - - - - - - - + + + + + + + + + Induction The presence of the positively charged rod leads to a redistribution of charge (a kind of polarization). It pulls electrons up to the electrode. 184 Lecture 3

8. Charging by Induction • We can also charge an object without physically connecting to it • First we charge a rod positively • Then we ground the object to be charged • Connecting the object to the Earth provides an effectively infinite sink for charge • We bring the charged rod close to the object but do not touch it • We remove the ground connection and move the rod away • The object will be charged by induction 184 Lecture 3

9. Charging by induction - - - - - - - - - ground + + + + + + + + + Induction The presence of the positively charged rod leads to a redistribution of charge. Grounding pushes positive charge to Earth (or rather pulls electrons from Earth!) leaving the electroscope negative. 184 Lecture 3

10. Coulomb’s Law 184 Lecture 3

11. Electric Force - Coulomb’s Law • 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 • The constant k is called Coulomb’s constant and is given by 184 Lecture 3

12. Coulomb’s Law (2) • The coulomb constant is also written as • 0 is the “electric permittivity of vacuum” • A fundamental constant of nature 184 Lecture 3

13. Example: What is the force between two charges of 1 C separated by 1 meter? Answer: 8.99 x 109 N, i.e., huge! 184 Lecture 3

14. Electric Force • The electric force is given by • The electric force, unlike the gravitational force, can be positive or negative • If the charges have opposite signs, the force is negative • Attractive • If the charges have the same sign, the force is positive • Repulsive 184 Lecture 3

15. Electric Force Vector q1 r y r1 q2 r2 x Electric force in vector form 184 Lecture 3

16. Superposition Principle • The net force acting on any charge is the vector sum of the forces due to the remaining charges in the distribution. 184 Lecture 3

17. Example - The Helium Nucleus Part 1: The nucleus of a helium atom has two protons and two neutrons. What is the magnitude of the electric force between the two protons in the helium nucleus? Answer: 58 N Part 2: What if the distance is doubled; how will the force change? Answer: 14.5 N Inverse square law: If the distance is doubled then the force is reduced by a factor of 4. 184 Lecture 3

18. Example - Equilibrium Position • Consider two charges located on the x axis • The charges are described by • q1 = 0.15 C x1 = 0.0 m • 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? • At the equilibrium point, the forces from the two charges will cancel. 184 Lecture 3

19. Example - Equilibrium Position (2) • The equilibrium point must be along the x-axis. • Three regions along the x-axis where we might place our third charge x3 < x1 x1 < x3 < x2 x3 > x2 184 Lecture 3

20. Example - Equilibrium Position (3) • x3<x1 • Here the forces from q1 and q2 will always point in the same direction (to the left for a positive test charge) • No equilibrium • x2<x3 • Here the forces from q1 and q2 will always point in the same direction (to the right for a positive test charge) • No equilibrium 184 Lecture 3

21. Example - Equilibrium Position (4) q3 x1 < x3 < x2 Here the forces from q1 and q2 can balance. Check the signs!! Answer: x3 = 0.16 m 184 Lecture 3

22. Example - Charged Pendulums y x • Consider two identical charged balls hanging from the ceiling by strings of equal length 1.5 m (in equilibrium). Each ball has a charge of 25 C. The balls hang at an angle  = 25 with respect to the vertical. What is the mass of the balls? Step 1: Three forces act on each ball: Coulomb force, gravity and the tension of the string. 184 Lecture 3

23. Example - Charged Pendulums (2) Step 2: The balls are in equilibrium positions. That means the sum of all forces acting on the ball is zero! d=2 l sin q Answer: m = 0.76 kg A similar analysis applies to the ball on the right. 184 Lecture 3

24. Electric Force and Gravitational Force • Coulomb’s Law that describes the electric force and Newton’s gravitational law have a similar functional form • Both forces vary as the inverse square of the distance between the objects. • Gravitation is always attractive. • k and G give the strength of the force. 184 Lecture 3

25. Example - Forces between Electrons • What is relative strength of the electric force compared with the force of gravity for two electrons? • Gravity is irrelevant for atomic and subatomic processes – the electric force is much much stronger. • But sometimes gravity is most important; e.g, the motion of the planets. Why? 184 Lecture 3

26. Example - Four Charges Consider four charges placed at the corners of a square with sides of length 1.25 m as shown on the right. What is the magnitude of the electric force on q4 resulting from the electric force from the remaining three charges? Set up an xy-coordinate system with its origin at q2. 184 Lecture 3

27. Example - Four Charges (2) Answer: F (on q4) = 0.0916 N … and the direction? 184 Lecture 3