Charge and Electric Force

1. PHY 2054 Exam 1 Review Michael Teti | mteti@fau.edu February 1, 2016

2. Charge and Electric Force • Atoms consist of protons, neutrons, and electrons • Protons and electrons both have the same amount of charge, e • where e= • Coulomb is the standard unit of charge • Protons are +e and electrons are –e • Protons are not removed in electricity. Electricity is transfer of electrons from one atom to another. • Individual charge of an atom/object represented by q • Charge of an object containing multiple charges represented by Q

3. Charge and Electric Force Cont’d. • Since atoms can lose/gain electrons, we can calculate the charge of an atom (or even an object with many atoms) using the number of electrons and protons it has • number of protons; number of electrons • So charge of an object/atom = - . • E.g. an atom has 4 protons and 7 electrons • Total charge is e(4-7) = e(-3) = (-3) = • Charges typically given in C (micro-Coulombs) • 1C = C • 1 C = C

4. Insulators and Conductors • Atoms in insulators don’t share electrons easily • Atoms in conductors do easily share electrons • Thus, electrical wire is made up of some kind of metal like Copper, where its atoms can easily accept/lose electrons. • Electricity is basically electrons being transferred from one atom to another down the length of the wire

5. Electric Charge and Electric Force • We know that objects and atoms can have a total charge based on the number of protons and electrons • e.g. 7e, -3e, 1e, etc. • Two objects, both with positive/negative charges, repel one another • Two objects with opposite charges attract one another. • A positive object/charge does not affect and is not affected by an object with a neutral charge • This principle explains how an insulator can be used to charge a conductor by induction

6. Electric Charge • electrons can be transferred from one object to another (one object loses some amount of electrons, and the other object gains those same electrons) • Each object’s individual charge changes as a result • However, the total charge (total number of electrons and protons) between the two objects is the same before and after the transfer • Ex., -5e; 1e • Total charge initially = -4e • transfers 2 electrons to • Total charge is now -4e, but and

7. Coulomb’s Law • We know how to determine the direction of the force on a charge caused by another charge (like charges repel, opposite charges attract) • We can use Coulomb’s Law to determine the amount of force on a charge as a result of another charge • , where k=8.99 x First number tells us what charge we are calculating the force on** Second number tells us what charge is causing the force we are calculating ** The force calculated by this equation is the same on both charges involved (in this case charge 1 and 2), so . This is just the notation used during calculation.

8. Example 1 • Suppose we have and ; • is at +5cm on the x axis, and is at +13 cm on the x axis • What is the magnitude of the force on charge 1 (same as the amount of force on charge 2)? • 14.0 N • What is direction of force? Must be in coulombs Must be in meters

9. Example 2 • What is magnitude of electric force on charge 2? (-.5m, .4m) C C C (0 m, -.3m)

10. Example 2 Cont’d. – Finding • or • Needs to be in meters, so .41m • = 0.22N Don’t need to square it because this is already The m^2 is the unit.

11. Example 2 Cont’d. – Calculating • Now that we have the magnitude of force on charge 2 from each other charge, let’s find the total force on charge 2: • To do this, we need to find out how much force in the x and y direction each force we just calculated is: • is easy • Since they are opposite charges, they attract • is only in the –y direction: C C (0 cm, -.3 m)

12. Example 2 Cont’d. – Calculating Components • To find direction of first sketch the vector based on the sign and direction of each charge: • Next, find : • D, but we can find opp. angle (φ) • φ = (opp./adj.) • φ = (.4/.5) = 38.66 • φ = , so =38.66 • Then, use cos or sin to find x and y components (-.5m, .4 m) C φ C

13. Example 2 Cont’d. – Calculating Components • sin(38.66) = .22Nsin(38.66) = .14N • And we remember that: • To find the magnitude of the total force • Add up the x and y components of each vector • + (-.5m, .4m) C C

14. Example 2 Cont’d. – Calculating • To find the magnitude of the total force on charge 2: • = = 1.37N • What about the angle from the positive x axis? • First, sketch out the x and y component vectors found previously • Since the x component is positive, draw a line in the +x direction • y is negative, so draw a vector in the –y direction (green arrow) • Connect the two vectors from the point of the charge (orange arrow) • Need to find θ • θ= (1.36N/.17N) = 82.9 • The angle in this case would be 360- 82.9 • .17N • θ • 1.36N

15. Coulomb’s Law and Proportions • By just looking at the equation, , we can tell how the force would change if either or both charges changed, or the distance between them changed. • How would the force between two charges change if one charge was doubled? • What about if one charge went up by a factor of 6 and the other went down by a factor of 2? • How would it change if the distance increased by a factor of 8 and one charge decreased by a factor of 2?

16. Example 3 • A positive point charge of +3𝜇C rests on a surface. Its weight is 5.2kg. A negative point charge of -1𝜇C is placed directly above the charge on the surface, and the charge on the surface is in equilibrium. What is the magnitude of the normal force on the charge on the surface if the two charges are .2m apart? • First, sketch problem: • Next, draw all forces on charge (in orange) C C

17. Example 3 Cont’d. • After sketching forces, divide them into x and y direction (not applicable in this example) • Since the charge is in equilibrium, all of the forces add up to 0. • In y dir.: • We can find the weight and the electric force • w = mg = (5.2kg)(9.81m/N • F = .67N C C

18. Example 3 Cont’d. • Now just plug in and find the normal force: • We know that • = 51.01N - .67N = 50.34N

19. Electric Fields from Point Charges • Single point charges can create electric fields • If the point charge is positive, the electric field at an arbitrary point around the charge is travelling away from the charge • If the point charge is negative, the electric field at a point near the charge is travelling toward the charge

20. Electric fields from point charges -q +q These points are always positive, so they are repelled by positive charges and attracted by negative charges.

21. Calculating Electric Fields • The magnitude of an electric field created by a point charge at a point can be calculated using the equation , where r is the distance between the point and the charge and q is the charge of the object in micro-coulomb’s • This doesn’t give us the direction, just the size of the electric field • For example, the electric field at a point .2m away from a -4C point charge would have a magnitude of Electric fields are usually very large

22. Calculating Electric Field at a Point due to Multiple Point Charges • To find the total electric field at a point from two or more point charges, it is basically the same procedure as finding the total electric force on a charge • The only thing different is the equation used to find the magnitude • For electric field, use instead of • After calculating the magnitude of electric field for each charge using the first equation above, • Sketch the direction of the electric field based on whether the charge is positive or negative. • Find the x and y direction of each electric field at the point • Sum all x’s and all y’s • Plug into to find the magnitude of the total force

23. Finding Electric Force on a Charge given electric field • If you know the total electric field at a point is N/C in the +x direction, and you place a point charge of +2 micro-coulombs, what is the magnitude of the electric force on the charge? • What is the direction of this force? • The positive x direction. It is a positive charge. Positive charges move in the direction of the electric field • Negative charges move the opposite way as the electric field

24. Electric fields and electric force +q -q

25. Uniform Electric Fields • If a charge is in a uniform electric field, the force and acceleration on the charge from the field is constant • This is because the electric field is the same magnitude everywhere • F=ma

26. Electric Flux • Measures how much electric field is going through a given surface or object • Where E is the magnitude of the electric field, A is the area of the surface the electric field is passing through, and θ is the angle between the normal vector (which will be given) and the electric field vector • If flux is positive, more electric field entering the object • If it’s negative, there is more electric field leaving the object • If it’s zero, there is no electric field passing through the object • When given an object (such as a cube) and asked to find the electric flux through the object, find the flux going through each side first

27. Electric Flux

28. Electric Flux Through an Object/Surface • If you know how much charge is contained in an object or surface, you can easily find the electric flux through the object • Where is the total amount of charge contained in that particular surface/object and is a constant, 8.85 x 10-12 C2/(N m2)

29. Infinite Sheet Charges • Suppose we have an infinite sheet which is positively charged with charge spread out evenly all over it. This is called a uniform charge distribution when the charge is spread out evenly. This means if we randomly choose two different, but equal areas of the sheet, the amount of charge in each area is the same. • Suppose we wanted to find the charge per unit area of the sheet, σ • σ = Q/A, where Q is amount of charge in the area given • For a sheet with +5 micro-coulombs in a .2 area • σ = = 25C/

30. Finding Electric Field Created by a Sheet • What’s the magnitude of the electric field produced by the sheet in the previous slide? • C

31. Questions/Example Problems