2. + - q b b a b s r  a z r  50 kV Dirty air Clean air

3. Electric forces affect only objects with charge Charge is measured in Coulombs (C). A Coulomb is a lot of charge Charge comes in both positive and negative amounts Charge is conserved – it can neither be created nor destroyed Charge is usually denoted by q or Q There is a fundamental charge, called e All elementary particles have charges that are simple multiples of e Particleq Protone Neutron0 Electron-e Oxygen nuc.8e  ++ 2e Red dashed line means you should be able to use this on a test, but you needn’t memorize it Electricity Electric Fields Electric Charge

4. Charge may be at a point, on a line, on a surface, or throughout a volume Linear charge density units C/m Multiply by length Surface charge density  units C/m 2 Multiply by area Charge density  units C/m 3 Multiply by volume Charge Can Be Spread Out

5. Concept Question A box of dimensions 2 cm  2 cm  1 cm has charge density  = 5.0  C/cm 3 throughout and linear charge density = – 3.0  C/cm along one long diagonal. What is the total charge? A) 2  CB) 5  CC) 11  C D) 29  CE) None of the above 5.0  C/cm 3 2 cm 1 cm 2 cm – 3.0  C/cm

6. The Nature of Matter Matter consists of positive and negative charges in very large quantities There are nuclei with positive charges Surrounded by a “sea” of negatively charged electrons ++++ ++++ ++++ ++++ To charge an object, you can add some charge to the object, or remove some charge But normally only a very small fraction 10 -12 of the total charge, or less Electric forces are what hold things together But complicated by quantum mechanics Some materials let charges move long distances, others do not Normally it is electrons that do the moving Insulators only let their charges move a very short distance Conductors allow their charges to move a very long distance

7. Some ways to charge objects – – – – – + + + + + By rubbing them together Not well understood By chemical reactions This is how batteries work By moving conductors in a magnetic field Get to this in March By connecting them to conductors that have charge already That’s how outlets work Charging by induction Bring a charge near an extended conductor Charges move in response Separate the conductors Remove the charge + – – – – – + + + + +

8. Coulomb’s Law Like charges repel, and unlike charges attract The force is proportional to the charges It depends on distance q1q1 q2q2 Other ways of writing this formula The r-hat just tells you the direction of the force When working with components, often helps to rewrite the r-hat Sometimes this formula is written in terms of a quantity  0 called the permittivity of free space

9. Concept Question +2.0  C 5.0 cm –2.0  C 5.0 cm –2.0  C What is the direction of the force on the purple charge? A)Up B) Down C) Left D) Right E) None of the above The separation between the purple charge and each of the other charges is identical The magnitude of those forces is identical The blue charge creates a repulsive force at 45  down and left The green charge creates an attractive force at 45  up and left The sum of these two vectors points straight left 7.2 N

10. Sample Problem 2.0 mC 1.0 m 2.0 m Three charges are distributed as shown at right. Where can we place a fourth charge of magnitude 3.0 mC such that the total force on the 1.0 mC vanishes? -4.0 mC 1.0 mC  ? 3.0 mC 1.16 m

11. Forces From Continuous Charges If you have a spread out charge, it is tempting to start by calculating the total charge Generally not the way to go The charge of the line is easy to find, Q = L But the distance and direction is hard to find To deal with this problem, you have to divide it up into little segments of length dl Then calculate the charge dQ = dl for each little piece Find the separation r for each little piece Add them up – integrate For a 2D object, it becomes a double integral For a 3D object, it becomes a triple integral q dl r

12. The Electric Field Suppose we have some distribution of charges We are about to put a small charge q 0 at a point r What will be the force on the charge at r? q0q0 r Every term in the force is proportional to q 0 The answer will be proportional to q 0 Call the proportionality constant E, the electric field It is assumed that the test charge q 0 is small enough that the other charges don’t move in response The electric field E is a function of r, the position It is a vector field, it has a direction in space everywhere The electric field is assumed to exist even if there is no test charge q 0 present The units for electric field are N/C

13. Electric Field From a Point Charge From a single point charge, the electric field is easy to find qq0q0 It points away from positive charges It points towards negative charges + -

14. Electric Field from Two Charges Electric field is a vector We must add the vector components of the contributions of multiple charges + + + -

15. Electric Fields From Continuous Charges If you have a spread out charge, we can add up the contribution to the electric field from each part To deal with this problem, you have to divide it up into little segments of length dl Then calculate the charge dQ = dl for each little piece Find the separation r and the direction r-hat for each little piece Add them up – integrate For a 2D object, it becomes a double integral For a 3D object, it becomes a triple integral P dl r

16. (y) P What is the electric field at the origin for a line of charge on the y-axis with linear charge density (y) = Qy 2 /(y 2 +a 2 ) stretching from y = a to y =  ? dy r Divide the charge into little segments dl –Because it is on the y-axis, dl = dy The vector r points from the source of the electric field to the point of measurement –It’s magnitude is r = y –It’s direction is the minus-y direction Substitute into the integral –Limits of integral are y= a and y =  Pull constants out of the integral Look up the integral Substitute limits Sample Problem y=a x y

17. P What is the electric field at the point P for a line with constant linear charge density and the geometry sketched above? dx r a b c Divide the line charge into little segments Find the charge dQ = dx for each piece Find the separation r for each little piece Add them up – integrate x Look up integrals Sample Problem

18. Electric Field Lines Electric field lines are a good way to visualize how electric fields work They are continuous oriented lines showing the direction of the electric field + - They start on positive charges and end on negative charges (or infinity) They never cross Where they are close together, the field is strong The bigger the charge, the more field lines come out

19. Sample Problem Sketch the field lines coming from the charges below, if q is positive +2q+2q -2q-2q +q Let’s have four lines for each unit of q Eight lines coming from red, eight going into green, four coming from blue Most of the “source” lines from red and blue will “sink” into green Remaining lines must go to infinity

20. Acceleration in a Constant Electric Field If a charged particle is in a constant electric field, it is easy to figure out what happens We can then use all standard formulas for constant acceleration A proton accelerates from rest in a constant electric field of 100 N/C. How far must it accelerate to reach escape velocity from the Earth (11.186 km/s)? Look up the mass and charge of a proton Find the acceleration Use PHY 113 formulas to get the distance Solve for the distance