1/10/ Lecture 31 PHY 184 Spring 2007 Lecture 3 Title: The Coulomb Force

1/10/ Lecture 32 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.

1/10/ Lecture 33 Outline 1 – Review 2 – Electrostatic charging 3 – Coulomb’s Law

1/10/ Lecture 34 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× 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.

1/10/ Lecture 35 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

1/10/ Lecture 36 Charging by Conduction We brought charge onto the electrode by contact Electroscope

1/10/ Lecture 37 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

1/10/ Lecture 38 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

1/10/ Lecture 39 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 Charging by induction ground

1/10/ Lecture 310 Coulomb’s Law

1/10/ Lecture 311 Electric Force - Coulomb’s Law  Consider two electric charges: q 1 and q 2  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

1/10/ Lecture 312 Coulomb’s Law (2)  The coulomb constant is also written as   0 is the “electric permittivity of vacuum” A fundamental constant of nature

1/10/ Lecture 313 Example: What is the force between two charges of 1 C separated by 1 meter? Answer: 8.99 x 10 9 N, i.e., huge!

1/10/ Lecture 314 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

1/10/ Lecture 315 Electric Force Vector Electric force in vector form x y q1q1 q2q2 r2r2 r1r1 r

1/10/ Lecture 316 Superposition Principle  The net force acting on any charge is the vector sum of the forces due to the remaining charges in the distribution.

1/10/ Lecture 317 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.

1/10/ Lecture 318  Consider two charges located on the x axis  The charges are described by q 1 = 0.15  Cx 1 = 0.0 m q 2 = 0.35  Cx 2 = 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. Example - Equilibrium Position

1/10/ Lecture 319 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 x 3 < x 1 x 1 < x 3 < x 2 x 3 > x 2

1/10/ Lecture 320 Example - Equilibrium Position (3)  x 3 <x 1 Here the forces from q 1 and q 2 will always point in the same direction (to the left for a positive test charge) No equilibrium  x 2 <x 3 Here the forces from q 1 and q 2 will always point in the same direction (to the right for a positive test charge) No equilibrium

1/10/ Lecture 321 Example - Equilibrium Position (4) x 1 < x 3 < x 2 Here the forces from q 1 and q 2 can balance. Answer: x 3 = 0.16 m q3q3 Check the signs!!

1/10/ Lecture 322 Example - Charged Pendulums  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. x y

1/10/ Lecture 323 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! Answer: m = 0.76 kg A similar analysis applies to the ball on the right. d=2 l sin 

1/10/ Lecture 324 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.

1/10/ Lecture 325 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?

1/10/ Lecture 326 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 q 4 resulting from the electric force from the remaining three charges? Set up an xy-coordinate system with its origin at q 2.

1/10/ Lecture 327 Example - Four Charges (2) Answer: F ( on q 4 ) = N … and the direction?