1. Electric Fields II Electric fields are produced by point charges and continuous charge distributions Definition: is force exerted on charge by an external field Note: q produces a field but it is not external

2. Example: Electric Dipole Derive an expression for at point B. y aa x B : (0,y) A : (x,0) -q+q

3. Dipole solution:

4. For fun: find E at point A, and show that it is approximately proportional to x  3, at large distances x.

5. E = ? + + + + + + + + + + + Source + Continuous Charge Distributions Cut source into small (“infinitesimal”) charges dq Each produces dE dq or

6. Steps: Draw a coordinate system on the diagram Choose an integration variable (e.g., x ) Draw an infinitesimal element dx Write r and any other variables in terms of x Write dq in terms of dx Put limits on the integral Do the integral or look it up in tables.

7. Example: Uniformly-Charged Thin Rod (length L, total charge Q ) L d Charge/Length = “Linear Charge Density”  = constant = Q/L + + + + + + + + +

8. Solution:

9. In 2D problems, integrate components separately to obtain the electric field: { x-component of

10. Example: Uniformly-Charged Ring Total charge Q, uniform charge/unit length, radius R Find: E at any point (x, 0) x y (x,0) R

11. Solution:

12. Example: Uniformly-Charged Semicircle Charge/unit length,, is uniform Find: at origin R y x + + + +

13. Solution:

14. Summary Field of several point charges q i : Field of continuous charge distribution: