Tue. Jan. 27 – Physics Lecture #23 Electric Field, Continued (and Continuous!) 1. Electric Field due to Continuous Charge Distributions Warm-up: Consider a very thin rod, length 2 L, that has a total charge Q uniformly distributed along its length and is oriented as shown. How can we determine the electric field at the point P ? Brainstorm with your neighbors and come up with some strategies. P -L-L L

ConceptCheck: Consider the very thin rod, length 2 L with a total charge Q uniformly distributed along its length as shown. What is the direction of the electric field at the point P ? 1. No field 2. Down 3. Up 4. Right 5. Left 6. Not enough information P -L-L L

P -L-L L (0, y P ) dQdQ ( x, 0)

A line of charge, length 2L, with uniform linear charge density = Q /2L is oriented along the x-axis as shown. A field point P is on the y-axis at the point (0, y P ). The electric field at the point P is given by Consider the case where y P is very large compared to L or alternatively where the field point is very far away from the line charge. What is a good approximation for E y ? P -L-L L Consider the case where L is very large compared to y P, or alternatively where the field point is very close to the line charge. What is a good approximation for E y ?

Example: Consider the semicircular arc shown in the figure, with radius a. The arc has charge uniformly distributed along its length with uniform linear charge density. Set up the integral that will allow you to calculate the x -component of the electric field at the origin, due to this charge distribution.