Electric field Revisited

1. Electric field Revisited

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

3. Electricity Electric Fields Electric Charge • 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 thatare simple multiples of e Particleq Proton e Neutron 0 Electron -e Oxygen nuc. 8e ++ 2e

4. Charge Can Be Spread Out • 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/m2 • Multiply by area • Charge density units C/m3 • Multiply by volume

5. Concept Question A box of dimensions 2 cm 2 cm  1 cm has charge density  = 5.0 C/cm3 throughout and linear charge density  = – 3.0 C/cm along one long diagonal. What is the total charge? A) 2 C B) 5 C C) 11 C D) 29 C E) None of the above – 3.0 C/cm 2 cm 5.0 C/cm3 1 cm 2 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 negativelycharged 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 Conductors allow their charges to move a very long distance Insulators only let their charges move a very short 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 • 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 q1 q2 • 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 aquantity0 called the permittivity of free space

9. Concept Question • What is the direction of the force on the purple charge? • Up B) Down C) Left • D) Right E) None of the above +2.0 C 5.0 cm 5.0 cm –2.0 C 5.0 cm 7.2 N –2.0 C • The separation between the purple charge and each of the other charges is identical • The magnitude of those forces is identical 7.2 N • 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

10. The Electric Field • Suppose we have some distribution of charges • We are about to put a small charge q0 at a point r • What will be the force on the charge at r? • Every term in the force is proportional to q0 • The answer will be proportional to q0 • Call the proportionality constant E, the electric field q0 r The units for electric field are N/C • It is assumed that the test charge q0is 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 q0 present

11. Electric Field From a Point Charge q q0 • From a single point charge, the electric field is easy to find • It points away from positive charges • It points towards negative charges - +

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

13. 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 never cross • Where they are close together, the field is strong • The bigger the charge, the more field lines come out • They start on positive charges and end on negative charges (or infinity) -

14. Gauss’s Law Electric Flux • Electric flux is the amount of electric field going across a surface • It is defined in terms of a direction, or normal unit vector,perpendicular to the surface • For a constant electric field, and a flat surface, it is easy to calculate • Denoted by E • Units of Nm2/C • When the surface is flat, and the fields are constant, youcan just use multiplication to get the flux • When the surface is curved, or the fields are not constant,you have to perform an integration

15. Electric Flux For a Cylinder A point charge q is at the center of a cylinder of radius a and height 2b. What is the electric flux out of (a) each end and (b) the lateral surface?  top s b b  r r z • Consider a ring of radius s and thickness ds a q b a lateral surface

16. Conductors and Gauss’s Law • Conductors are materials where charges are free to flow in response to electric forces • The charges flow until the electric field is neutralized in the conductor Inside a conductor, E = 0 • Draw any Gaussian surface inside the conductor In the interior of a conductor, there is no charge The charge all flows to the surface

17. Electric Field at Surface of a Conductor • Because charge accumulates on the surface of a conductor, there can be electric field just outside the conductor • Will be perpendicular to surface • We can calculate it from Gauss’s Law • Draw a small box that slightly penetrates the surface • The lateral sides are small and have no flux throughthem • The bottom side is inside the conductor and has no electric field • The top side has area A and has flux through it • The charge inside the box is due to the surface charge  • We can use Gauss’s Law to relate these

18. Sample problem An infinitely long hollow neutral conducting cylinder has inner radius a and outer radius b. Along its axis is an infinite line charge with linear charge density . Find the electric field everywhere. b end-on view perspective view a • Use cylindrical Gaussian surfaces when needed in each region • For the innermost region (r < a), the total charge comes entirely from the line charge • The computation is identical to before • For the region inside the conductor, the electric field is always zero • For the region outside the conductor (r > b), the electric field can be calculated like before • The conductor, since it is neutral, doesn’t contribute

19. Where does the charge go? + – + – – + + – – + – + – + – + + + + – – – – + – + – + How can the electric field appear, then disappear, then reappear? + • The positive charge at the center attracts negative charges from the conductor, which move towards it • This leaves behind positive charges, which repel each other and migrate to the surface end-on view • In general, a hollow conductor masks the distribution of the charge inside it, only remembering the total charge • Consider a sphere with an irregular cavity in it cutaway view q