1/22/2007 J.Velkovska, PHYS117B 1

PHYS117B: Lecture 6Electric 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 1. 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

2. Evaluate the flux through the surface and the enclosed charge

EA +EA = 2EA Qencl = σ A

3. 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 ?

f

i

W F dl ����������������������������

Well, the work is not just Force multiplied by displacement, it is the SCALAR Product between the two.

Charge q2 moves in the field of q1

1/22/2007 J.Velkovska, PHYS117B 10

Electric potential energy in a uniform field: a charge inside a parallel plate capacitor

b

a

W F dl ����������������������������

U qEy

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

2

0 /f

f

ii

W F dl kq q dr r ����������������������������

1/22/2007 J.Velkovska, PHYS117B 12

If we have a collection of charges:

1 2 3 .....F F F F ��������������������������������������������������������

f

i

W F dl ����������������������������

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