Law of Electrical Force Charles-Augustin Coulomb (1785) " The repulsive force between two small spheres charged with the same sort of electricity is in the inverse ratio of the squares of the distances between the centers of the spheres" q2 q1 r 7

What We Call Coulomb's Law The force from 1 acting on 2 q1 q2 MKS Units: r in meters q in Coulombs in Newtons is a unit vector pointing from 1 to 2 We call this group of constants “k” as in:  1 4 = 9 · 109 N-m2/C2 This force has same spatial dependence as the gravitational force, BUT there is NO mention of mass here!! The strength of the force between two objects is determined by the charge of the two objects, and the separation between them. 8

Coulomb Law Qualitative r What happens if q1 increases? F (magnitude) increases What happens if q1 changes sign ( + - )? The direction of is reversed What happens if r increases? F (magnitude) decreases

1a: A) The charge of Q3 has the same sign of the charge of Q1 A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1. When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at distance d23 (< d12) directly above Q3. Q2 Q1 g d12 d23 Q3 1a: A) The charge of Q3 has the same sign of the charge of Q1 B) The charge of Q3 has the opposite sign as the charge of Q1 C) Cannot determine the relative signs of the charges of Q3 & Q1 1b: A) The magnitude of charge Q3 < the magnitude of charge Q1 B) The magnitude of charge Q3 > the magnitude of charge Q1 C) Cannot determine relative magnitudes of charges of Q3 & Q1

1a: A) The charge of Q3 has the same sign of the charge of Q1 A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1. When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at distance d23 (< d12) directly above Q3. Q2 Q2 d12 d23 g Q1 Q3 1a: A) The charge of Q3 has the same sign of the charge of Q1 B) The charge of Q3 has the opposite sign as the charge of Q1 C) Cannot determine the relative signs of the charges of Q3 & Q1 To be in equilibrium, the total force on Q2 must be zero. The only other known (from 111) force acting on Q2 is its weight. Therefore, in both cases, the electrical force on Q2 must be directed upward to cancel its weight. Therefore, the sign of Q3 must be the SAME as the sign of Q1

1b: A) The magnitude of charge Q3 < the magnitude of charge Q1 A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1. When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at distance d23 (< d12) directly above Q3. Q2 Q2 d12 d23 g Q1 Q3 1b: A) The magnitude of charge Q3 < the magnitude of charge Q1 B) The magnitude of charge Q3 > the magnitude of charge Q1 C) Cannot determine relative magnitudes of charges of Q3 & Q1 The electrical force on Q2 must be the same in both cases … it just cancels the weight of Q2 . Since d23 < d12 , the charge of Q3 must be SMALLER than the charge of Q1 so that the total electrical force can be the same!!

Gravitational vs. Electrical Force q1 m1 q2 m2 F elec = 1 4 pe q 2 r grav = G m ® F elec grav = q 1 2 m 4 pe G * smallest charge seen in nature! For an electron: * |q| = 1.6 ´ 10-19 C m = 9.1 ´ 10-31 kg ® F elec grav = ´ + 4 17 10 42 .

Notation For Vectors and Scalars Vector quantities are written like this : F, E, x, r To completely specify a vector, the magnitude (length) and direction must be known. r ˆ For example, the following equation shows specified in terms of , q1, q2, and r : The magnitude of is ; this is a scalar quantity The vector can be broken down into x, y, and z components: Where Fx, Fy, and Fz (the x, y, and z components of F ) are scalars. A unit vector is denoted by the caret “^”. It indicates only a direction and has no units.

Vectors: an Example q1 and q2 are point charges, q1 = +2mC and q2 = +3 mC. q1 is located at and q2 is located at Find F12 (the magnitude of the force of q1 on q2). y (cm) x (cm) 1 2 3 4 3 2 1 q1 q2 r To do this, use Coulomb’s Law: Now, plug in the numbers. F12 = 67.52 N

Vectors: an Example continued y (cm) x (cm) 1 2 3 4 3 2 1 q1 q2 r q Now, find Fx and Fy, the x and y components of the force of q1 on q2. Symbolically Now plug in the numbers Fx = 47.74 N Fy = 47.74 N

What happens when you consider more than two charges? If q1 were the only other charge, we would know the force on q due to q1 . F 1 ® 2 q +q1 +q2 If q2 were the only other charge, we would know the force on q due to q2 . F ® What is the force on q when both q1 and q2 are present?? The answer: just as in mechanics, we have the Law of Superposition: The TOTAL FORCE on the object is just the VECTOR SUM of the individual forces. F ® = 1 + 2 11

(a) (b) (c) The force on Q3 can be zero if Q3 is positive. Two balls, one with charge Q1 = +Q and the other with charge Q2 = +2Q, are held fixed at a separation d = 3R as shown. Another ball with (non-zero) charge Q3 is introduced in between Q1 and Q2 at a distance = R from Q1. Which of the following statements is true? +2Q (a) The force on Q3 can be zero if Q3 is positive. (b) The force on Q3 can be zero if Q3 is negative. (c) The force on Q3 can never be zero, no matter what the (non-zero!) charge Q3 is. 12

(a) (b) (c) The force on Q3 can be zero if Q3 is positive. Two balls, one with charge Q1 = +Q and the other with charge Q2 = +2Q, are held fixed at a separation d = 3R as shown. Another ball with (non-zero) charge Q3 is introduced in between Q1 and Q2 at a distance = R from Q1. Which of the following statements is true? (a) (c) (b) The force on Q3 can be zero if Q3 is positive. The force on Q3 can be zero if Q3 is negative. The force on Q3 can never be zero, no matter what the (non-zero) charge Q3 is. The magnitude of the force on Q3 due to Q2 is proportional to (2Q Q3 /(2R)2) The magnitude of the force on Q3 due to Q1 is proportional to (Q Q3 /R2) These forces can never cancel, because the force Q2 exerts on Q3 will always be 1/2 of the force Q1 exerts on Q3!!

Another Example x (cm) y (cm) 1 2 3 4 5 4 3 2 1 qo q2 q1 q qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC. Their locations are shown in the diagram. What is the force acting on qo ( ) ? We have What are F0x and F0y ? Decompose into its x and y components

Another Example continued x (cm) y (cm) 1 2 3 4 5 4 3 2 1 qo q2 q1 qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC. Their locations are shown in the diagram. Now add the components of and to find and

Another Example continued x (cm) y (cm) 1 2 3 4 5 4 3 2 1 qo q2 q1 qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC. Their locations are shown in the diagram. Let’s put in the numbers . . . The magnitude of is

Electric field, introduction One problem with the above simple description of forces is that it doesn’t describe the finite propagation speed of electrical effects. In order to explain this, we must introduce the concept of the electric field. What is a Field? A FIELD is something that can be defined anywhere in space A field represents some physical quantity (e.g., temperature, wind speed, force) It can be a scalar field (e.g., Temperature field) It can be a vector field (e.g., Electric field) It can be a “tensor” field (e.g., Space-time curvature)

A Scalar Field 77 82 83 68 55 66 75 80 90 91 71 72 84 73 88 92 64 These isolated temperatures sample the scalar field (you only learn the temperature at the point you choose, but T is defined everywhere (x, y)

A Vector Field It may be more interesting to know which way the wind is blowing... It may be more interesting to know which way the wind is blowing… 77 82 83 68 55 66 75 80 90 91 71 72 84 73 57 88 92 56 64 That would require a vector field (you learn both wind speed and direction)

Summary F ® = + 1 2 Charges come in two varieties Coulomb Force negative and positive in a conductor, negative charge means extra mobile electrons, and positive charge means a deficit of mobile electrons Coulomb Force bi-linear in charges inversely proportional to square of separation central force F ® = 1 + 2 Law of Superposition Fields

Appendix A: Electric Force Example Suppose your friend can push their arms apart with a force of 100 lbs. How much charge can they hold outstretched? +Q -Q 2m F= 100 lbs = 450 N = 4.47•10-4 C That’s smaller than one cell in your body!

Appendix B: Outline of physics 212 Coulomb’s Law gives force acting on charge Q1 due to another charge, Q2. superposition of forces from many charges Electric field is a function defined throughout space here, Electric field is a shortcut to Force later, Electric field takes on a life of it’s own! Q1 Q2 Q3