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

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

Dipole solution:

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

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

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.

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

Solution:

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

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

Solution:

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

Solution:

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