1. PHYS117B: Lecture 6 Electric field in planar geometry
PHYS117B: Lecture 6 Electric field in planar geometry. Electric potential energy.
Last lecture:
Properties of conductors and insulators in electrostatic equilibrium
E = 0 inside the conductor and all excess charges are on the surface
Used Gauss’s law to find the field in and out of spheres (conductors and insulators) … and similarly we can do spherical shells and spheres inside spherical shells
The electric field outside a sphere is = to the E of a point charge located in the center
We “played’ with cylinders the previous time
1/22/2007
J.Velkovska, PHYS117B

2. The electric field of an infinite charged plane
Use symmetry:
The field is ┴ to the surface
Direction: away from positive charge, and toward a negative charge
Use Gauss’s law to determine the magnitude of the field
1/22/2007
J.Velkovska, PHYS117B

3. Here’s how we do it: … as easy as 1,2,3
Choose a Gaussian surface:
a cylinder would work: the field is ┴ to the area vector on the sides and ║ to the area vector on the top and the bottom of the cylinder
a cube or a parallelogram with sides ║to the surface would work, too
Evaluate the flux through the surface and the enclosed charge
EA +EA = 2EA
Qencl = σ A
Apply Gauss’s law:
E = σ/ 2ε0
The electric field of an infinite plane of charge does NOT depend on the distance from the plane, but ONLY on the surface charge density
1/22/2007
J.Velkovska, PHYS117B

4. Now add a second plane with opposite charge: parallel plate capacitor
For the negatively charged plane:
Flux : - 2EA
Charge: -σ A
E = σ/ 2ε0 , pointing towards the plane
Use superposition to find the field between the plates and outside the plates:
E=0 , outside the plates
E = σ/ ε0
Direction : from + to -
1/22/2007
J.Velkovska, PHYS117B

5. Use the properties of conductors and Gauss’s law: expel the field from some region in space
When a lightening strikes
You are safe inside your car
1/22/2007
J.Velkovska, PHYS117B

6. Electric field shielding has multiple uses
If you want to measure the gravitational force between 2 objects (Cavendish balance), you need to make sure that electric forces don’t distort your measurement
Put the one of the objects in a light weight metal mesh (Faraday cage) to screen any stray electric fields
Use a coaxial cable ( has a central conductor surrounded by a metal braid which is connected to ground) to transmit sensitive electric signals
1/22/2007
J.Velkovska, PHYS117B

7. OK, we know how to get the Electric field in almost any configuration, but what does this tell us about how objects in nature interact ?
Well, we know the definition:
Electric field = Force/unit charge
So if we know E, we can find the force on a charge that is placed inside the field
We can use F= ma and kinematics to find how this charge will move inside the field ( we did this for homework)
Today: we will use conservation of energy – a very powerful approach !
1/22/2007
J.Velkovska, PHYS117B

8. Electric potential energy
The potential energy is a measure of the interactions in the system
Define: the change in potential energy by the WORK done by the forces of interaction as the system moves from one configuration to another
Electric force is a conservative force: the work doesn’t depend on the path taken, but only on the initial and final configuration => Conservation of energy
1/22/2007
J.Velkovska, PHYS117B

9. How can the path not matter ?
Charge q2 moves in the field of q1
Well, the work is not just Force multiplied by displacement,
it is the SCALAR Product between the two.
1/22/2007
J.Velkovska, PHYS117B

10. Electric potential energy in a uniform field: a charge inside a parallel plate capacitor
1/22/2007
J.Velkovska, PHYS117B

11. The potential energy of two point charges
The force is along the radius
The work ( and the change in the potential energy) depends only on the initial and final configuration
The potential energy depends on the distance between the charges
1/22/2007
J.Velkovska, PHYS117B

12. If we have a collection of charges:
1/22/2007
J.Velkovska, PHYS117B

13. Graph the potential energy of two point charges
U depends on 1/r and on the relative sign of the charges
Defined up to a constant. We take U = 0 when the charges are infinitely far apart. Think of it as “no interaction”.
1/22/2007
J.Velkovska, PHYS117B

14. Conservation of Energy in 2 charge system
Total mechanical energy Emech = const
Emech > 0 , the particles can escape each other
Emech < 0, bound system
1/22/2007
J.Velkovska, PHYS117B

15. 2 examples ( done on the blackboard)
Distance of closest approach for 2 like charges
Escape velocity for 2 unlike charges
1/22/2007
J.Velkovska, PHYS117B