Lecture 2: May 20 th 2009

1. Lecture 2: May 20th 2009 Physics for Scientists and Engineers II Physics for Scientists and Engineers II , Summer Semester 2009

2. Electric Field due to a Continuous Charge Distribution • We can model a system of charges as being continuous (instead of discrete) if the distance between the charges is much smaller than the distance to the point where the electric field is calculated. • Procedure: - Divide charge distribution into small charge elements Dq. - Add contributions to E from all charge elements. Dq r P Physics for Scientists and Engineers II , Summer Semester 2009

3. Charge Density (a useful concept when calculating E from charge distribution) Physics for Scientists and Engineers II , Summer Semester 2009

4. Example: Electric Field due to a Uniformly Charged Rod y dq = l dx dx x x P a l Physics for Scientists and Engineers II , Summer Semester 2009

5. Example: Electric Field due to a Uniformly Charged Rod…..this is harder…. y P dq = l dx r a Q x x l Physics for Scientists and Engineers II , Summer Semester 2009

6. Example: Electric Field due to a Uniformly Charged Rod…..this is harder…. y P dq = l dx r a Q x x l Physics for Scientists and Engineers II , Summer Semester 2009

7. ….solving the integral for Ex Physics for Scientists and Engineers II , Summer Semester 2009

8. ….solving the integral for Ey a Qmax l Physics for Scientists and Engineers II , Summer Semester 2009

9. ….and the final result Physics for Scientists and Engineers II , Summer Semester 2009

10. + Visualizing Electric Fields with Electric Field Lines • The electric field vector is always tangent to the electric field line. • The electric field line has a direction (indicated by an arrow). The direction is the same as that of the electric field (same direction as force on a positive test charge). • The number of lines per unit area through a normal plane (perpendicular to field lines) is proportional to the magnitude of the electric field in that region. Example:Electric field lines of a point charge • N field lines • Surface density of field lines at an imagined sphere of radius r is Electric field strength is proportional to Physics for Scientists and Engineers II , Summer Semester 2009

11. Visualizing Electric Fields with Electric Field Lines • For a single positive point charge: Electric field lines go from the positive charge to infinity. • For a single negative point charge: Electric field lines go come from infinity and end at the negative point charge. • For multiple point charges: Lines can start at the positive charges and end at the negative charges. • Electric field lines can never cross (think about why that is so). • For two unequal point charges of opposite sign with charges Q1 and Q2 , the number N1 of field lines terminating at Q1 and the number N2 of field lines terminating at Q2 are related by the equation Physics for Scientists and Engineers II , Summer Semester 2009

12. Motion of a Charged Particle in a Uniform Electric Field • Assume particle has charge q, mass m. • Particle experiences a force • The force results in an acceleration (according to Newton’s second law): • For positive charges: Acceleration is in the same direction as electric field. • For negative charges: Acceleration is in a direction opposite to the electric field. • A uniform electric field will cause a constant acceleration of the particle.  You can use equations of motion for constant acceleration. • Work is done on the particle by the electric force as the particle moves. Physics for Scientists and Engineers II , Summer Semester 2009

13. - - - - - - - - - - + + + + + + + + + + Example (similar to Ex. 23.10 in book) Q - L = 0.100 m Electron: m = 9.11x10-31 kg ; q = 1.60x10-19 C Electric Field: E = 800 N/C The electron leaves the electric field at an angle of Q = 65 degrees. Q1: What was the initial velocity of the electron? Q2: What is the final velocity of the electron (magnitude)? Q3: How low would the electric field have to be so that the net force on the electron is zero? Q4: Were we justified in neglecting the gravitational force in Q1 and Q2? Physics for Scientists and Engineers II , Summer Semester 2009

14. Physics for Scientists and Engineers II , Summer Semester 2009

15. Physics for Scientists and Engineers II , Summer Semester 2009

16. Gauss’s Law – An alternative procedure to calculate electric fields of highly symmetric charge distributions The concept of “Electric Flux”: Area = A Physics for Scientists and Engineers II , Summer Semester 2009

17. Normal to green surface The electric flux through the two surfaces is the same Physics for Scientists and Engineers II , Summer Semester 2009

18. Normal to green surface The electric flux through the two surfaces is the same • To calculate the flux through a randomly oriented area you need to know the angle between the electric field and the normal to the area. Physics for Scientists and Engineers II , Summer Semester 2009

19. How to treat situations where the electric field is not constant over the area? • Divide area into small areas over which E is constant. • Calculate flux for each small area. • Add fluxes up. Area vector: magnitude = area direction = perpendicular to area “surface integral” Physics for Scientists and Engineers II , Summer Semester 2009

20. Flux through a closed surface: • Convention: Area vectors always point outwards. • Field lines that cross from the inside to the outside of the surface : (positive flux because cos Q is positive) • Field lines that cross from the outside to the inside of the surface: (negative flux because cos Q is negative) Physics for Scientists and Engineers II , Summer Semester 2009

21. Example: Cube in a uniform field dA3 dA1 dA6 dA2 dA5 dA4 Physics for Scientists and Engineers II , Summer Semester 2009