1. 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)

2. Sketch the field lines coming from the charges below, if q is positive
Sample Problem
Sketch the field lines coming from the charges below, if q is positive
Let’s have four lines for each unit of q
Eight lines coming from red, eight going into green, four coming from blue
Most of the “source” lines from red and blue will “sink” into green
Remaining lines must go to infinity
+2q
-2q +q

3. Electric Fields From Continuous ChargesP
If you have a spread out charge, we can add up the contribution to the electric field from each part
To deal with this problem, you have to divide it up into little segments of length dl
Then calculate the charge dQ =  dl for each little piece
Find the separation r and the direction r-hat for each little piece
Add them up – integrate
For a 2D object, it becomes a double integral
For a 3D object, it becomes a triple integral r  dl

4. Sample Problem Divide the charge into little segments dl y
Because it is on the y-axis, dl = dy
The vector r points from the source of the electric field to the point of measurement
It’s magnitude is r = y
It’s direction is the minus-y direction
Substitute into the integral
Limits of integral are y= a and y = 
Pull constants out of the integral
Look up the integral
Substitute limits
y=a x y
What is the electric field at the origin for a line of charge on the y-axis with linear charge density (y) = Qy2/(y2+a2) stretching from y = a to y = ? dy
(y) r P

5. Acceleration in a Constant Electric Field
If a charged particle is in a constant electric field, it is easy to figure out what happens
We can then use all standard formulas for constant acceleration
A proton accelerates from rest in a constant electric field of 100 N/C. How far must it accelerate to reach escape velocity from the Earth ( km/s)?
Look up the mass and charge of a proton
Find the acceleration
Use PHY 113 formulas to get the distance
Solve for the distance

6. Acceleration in a Constant Electric Field
If a charged particle is in a constant electric field, it is easy to figure out what happens
We can then use all standard formulas for constant acceleration
A proton accelerates from rest in a constant electric field of 100 N/C. How far must it accelerate to reach escape velocity from the Earth ( km/s)?
Look up the mass and charge of a proton
Find the acceleration
Use PHY 113 formulas to get the distance
Solve for the distance

7. 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, you can just use multiplication to get the flux
When the surface is curved, or the fields are not constant, you have to perform an integration

8. 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

9. Total Flux Out of Various Shapes
A point charge q is at the “center” of a (a) sphere (b) joined hemispheres (c) cylinder (d) cube. What is the total electric flux out of the shape? b a q q a q

10. Gauss’s Law
No matter what shape you use, the total electric flux out of a region containing a point charge q is 4keq = q/0.
Why is this true?
Electric flux is just measuring how many field lines come out of a given region
No matter how you distort the shape, the field lines come out somewhere
If you have multiple charges inside the region their effects add
However, charges outside the region do not contribute q q4 q3 q1 q2

11. Using Gauss’s Law
Gauss’s Law can be used to solve three types of problems:
Finding the total charge in a region when you know the electric field outside that region
Finding the total flux out of a region when the charge is known
It can also be used to find the flux out of one side in symmetrical problems
In such cases, you must first argue from symmetry that the flux is identical through each side
Finding the electrical field in highly symmetrical situations
One must first use reason to find the direction of the electric field everywhere
Then draw a Gaussian surface over which the electric field is constant
Use this surface to find the electric field using Gauss’s Law
Works generally only for spherical, cylindrical, or planar-type problems

12. Sample Problem
A very long box has the shape of a regular pentagonal prism. Inscribed in the box is a sphere of radius R with surface charge density . What is the electric flux out of one lateral side of the box?
End view
Perspective view
The flux out of the end caps is negligible
Because it is a regular pentagon, the flux from the other five sides must be the same