1. Electricity and Magnetism Review 1: Units 1-6

2. Review Formulas Coulomb’s Law Force law between point charges q1 q2
Electric Field
Force per unit charge
Property of Space
Created by Charges
Superposition
Gauss’ Law
Flux through closed surface is always proportional to charge enclosed
Gauss’ Law
Can be used to determine E field
Spheres
Cylinders
Infinite Planes
Electric Potential
Potential energy per unit charge
Electric Potential
Scalar Function that can be used to determine E 2

3. Applications for Conductors
Charges free to move
What Determines How They Move?
They move until
E = 0 !
E = 0 in conductor determines charge densities on surfaces
Spheres
Cylinders
Infinite Planes
Gauss’
Law
Field Lines & Equipotentials
Field
Lines
Equipotentials
Work Done By E Field
Change in Potential Energy 3

4. Example Consider two point charges q1 and q2 located as shown.
1. Find the resultant electric field due to q1 and q2 at the location of q3.
2. Find the resultant force on q3.
3. Find the electric potential due to q1 and q2 at the location of q3.
Use components
F = 9e9 x (1.6e-19 * 3e22)^2 / 100 = 2 e 15 Newtons
Equivalent to weight of 2e14 kg object near surface of the earth
paperclip 1e-3 kg
textbook 1 kg
person kg
car kg
Aircraft carrier tons = 1e5 x 1e3 = 1e 8 kg
Mt Everest m height,. Estimate volume as 1/3 h^3 = 1/3 * (9e3)^3 = 243 e 9 m^3. (if density = 1000 kg/m^3) = 2.43e14 kg.

5. Example: Spherical Symmetry
A solid insulating sphere of radius R has uniform charge density ρ and carries total charge Q. Find the Electric field everywhere. R x y Q r
Choose a suitable Gaussian surface: A sphere
Calculate the charge
enclosed within the Gaussian surface for r > R and for r < R
For r > R:
For r < R:

6. Example
A solid insulating sphere of radius R has uniform volume charge density ρ and carries total charge Q. Find the Potential difference between two points inside the sphere A and B at distances rA and rB . R x y Q
From the previous problem we know that for r < R:

7. Example: Cylindrical Symmetry
Find the electric field at a distance r from a line of positive charge of infinite length and constant linear charge density λ.
Choose suitable Gaussian surface: A cylinder
Calculate the charge
enclosed within the Gaussian surface

8. Example: Planar Symmetry
Find the electric field at a distance r due to an infinite plane of positive charge with uniform surface charge density σ.
Choose suitable Gaussian surface: A cylinder perpendicular to the plane
Calculate the charge
enclosed within the Gaussian surface

9. Example
Point charge +3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.
a) What is E everywhere?
Use Gaussian surface = sphere centered on origin
neutral
conductor r1 r2 y x
+3Q
r < r1
r > r2
r1 < r < r2

10. Example
Point charge +3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.
What is E everywhere?
b) What is charge distribution at r1?
neutral
conductor r1 r2 y x
+3Q r1 r2
+3Q
r1 < r < r2

11. Example Suppose we give the conductor a charge of -Q
a) What is E everywhere?
b) What are charge distributions at r1 and r2? r1 r2
-3Q +
+2Q
+3Q
r1 < r < r2
r < r1
r > r2

12. Example
Charge q1 = 2μC is located at the origin. Charge q2 = - 6μC is located at (0, 3) m. Charge q3 = 3.00 μC is located at (4, 0) m
Find the total energy required to bring these charges to these locations starting from infinity.

13. Example
Point charge q at center of concentric conducting spherical shell of radii a1, and a2. The shell carries charge Q.
What is V as a function of r?
cross-section
Charges q and Q will create an E field throughout space
metal +Q a1 a2 +q
1. Spherical symmetry: Use Gauss’ Law to calculate E everywhere
2. Integrate E to get V
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

14. Example Gauss’ law: r > a2: +Q a1 < r < a2 : +q r < a2 r
cross-section
metal +Q a1 a2 r +q
r > a2:
a1 < r < a2 :
r < a2
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

15. Example To find V: 1) Choose r0 such that V(r0) = 0, usual: r0 = ∞ +Q
cross-section
To find V:
1) Choose r0 such that V(r0) = 0, usual: r0 = ∞
2)Integrate! a4 a3 +Q +q
r > a2 :
a1 < r < a2 :
metal
r < a1 :
The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

16. Example
A rod of length ℓ has a total charge Q uniformly distributed. Find V at point P.

17. Example
Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? y P x a h r
dq = l dx
What is ?

18. Example
Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? x a P r q1 q
dq = l dx h y

19. Example
Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? x a P r q1 q
dq = l dx h y

20. Example
The spheres have the same known mass m and charge q and are in equilibrium.
Given the angle θ and the length L, find the charge q on each sphere.
Is equilibrium possible if the charges are different?
F = 9e9 x (1.6e-19 * 3e22)^2 / 100 = 2 e 15 Newtons
Equivalent to weight of 2e14 kg object near surface of the earth
paperclip 1e-3 kg
textbook 1 kg
person kg
car kg
Aircraft carrier tons = 1e5 x 1e3 = 1e 8 kg
Mt Everest m height,. Estimate volume as 1/3 h^3 = 1/3 * (9e3)^3 = 243 e 9 m^3. (if density = 1000 kg/m^3) = 2.43e14 kg.
Yes the force will be the same if the product of the charges is the same
Use x and y components

21. Example
A solid insulating sphere of radius R has uniform volume charge density ρ and carries total charge Q. Find the flux through the outside sphere (r ≥ a). What if r < R? R x y Q r