General Procedure for Calculating Electric Field of Distributed Charges

1. Cut the charge distribution into pieces for which the field is known

2. Write an expression for the electric field due to one piece(i) Choose origin(ii) Write an expression for E and its components

3. Add up the contributions of all the pieces(i) Try to integrate symbolically(ii) If impossible – integrate numerically

4. Check the results:(i) Direction(ii) Units(iii) Special cases

Distance dependence:

Far from the ring (z>>R):

Close to the ring (z<<R): Ez~z

Ez~1/z2

A Uniformly Charged Thin Ring

A total charge Q is uniformly distributed over a half ring with radius R. The total charge inside a small element dθ is given by:

1. Choice One2. Choice Two3. Choice Three4. Choice Four5. Choice Five6. Choice Six

θdθ

Q

R

dRQ

2

A.

dRQ

B.

RQ

C.

dQD.

dQ2

E.

Clicker Question

A total charge Q is uniformly distributed over a half ring with radius R. The Y component of electric field at the center created by a short element dθ is given by:

1. Choice One2. Choice Two3. Choice Three4. Choice Four

θdθ

Q

R

sin2RkQdA.

cos2RkQdB.

sin3RkQd

C.

cos3RkQd

D.

+yClicker Question

A Uniformly Charged Disk

Close to the disk (0 < z < R)

Along z axis

Approximations:

If z/R is extremely smallVery close to disk (0 < z << R)

Field Far From the Disk

Exact

For z>>R

Point Charge

Uniformly Charged Disk Edge On

Two disks of opposite charges, s<<R: charges distribute uniformly:

+Q-Q

s

A single metal disk cannot be uniformly charged: charges repel and concentrate at the edges

We will calculate E both inside and outside of the disk close to the center

Two uniformly charged metal disks of radius R placed very near each other

Almost all the charge is nearly uniformly distributed on the inner surfaces of the disks; very little charge on the outer surfaces.

Capacitor

Why must there be charge on the outer surfaces?

+Q-Q

s

We know the field for a single disk There are only 2 “pieces”

E-

E+Enet

Step 1: Cut Charge Distribution into Pieces

Step 2: Contribution of one Piece

Origin: left disk, center

E-

E+Enet

sz

0

Location of disks: z=0, z=sDistance from disk to 2

z, (s-z)

Left:

Right:

Step 3: Add up Contributions

E-

E+Enet

sz0

Location: 2 (inside a capacitor)

Does not depend on z

Step 3: Add up Contributions

E-

E+Enet

sz0

Location: 3 (fringe field)

For s<<R: E1=E30

Far from the capacitor (z>>R>>s): E1=E3~1/z3 (like dipole)

Fringe field is very small compared to the field inside the capacitor.

E-

E+Enet

sz

0Units:

Inside:

Fringe:

Step 4: check the results:

Electric Field of a Capacitor

Which arrow best represents the field at the “X”?

A) B)

C) E=0D)

E)

Clicker Question

Given: capacitor, radius R=50 cm, gap s=1 mm (air).Find: maximum charge before sparks are formed (Ecrit=3106 N/C)

Solution:

What Q would cause sparking if spacing s 2s?What is the attractive force between the plates?

F=QE= (2.110-5C)(3106 N/C)=63 N

F=Q(Ecrit/2)= (2.110-5C)(3106/2 N/C)=31.5 N

Exercise

Field inside:

Field outside: (like point charge)

Qualitative approachIntegration

Electric Field of a Spherical Shell of Charge

E=0: Implications

Fill charged sphere with plastic.Will plastic be polarized? No!

Solid metal sphere: since it is a conductor, any excess charges on the sphere arranges itself uniformly on the outer surface.There will be no field nor excess charges inside!

In general: there is no electric field inside metals

E of a Sphere Inside

What is electric field right at the surface?

Need to be >1000 atomic diameters away from surfacefor equations to work!

E of a Sphere Inside

Electric field at the surface is highly variable in magnitude and direction

A solid metal ball bearing a charge –17 nC is located near a solid plastic ball bearing a uniformly distributed charge +7 nC (on surface). Show approximate charge distribution in each ball.

What is electric field field inside the metal ball?

Metal-17 nC

Plastic+7 nC

Exercise

Two uniformly charged thin plastic shells.Find electric field at 3, and 10 cm from the center

3 cm: E=0

10 cm:

Exercise

What if charges are distributed throughout an object?

Step 1: Cut up the charge into shells

r E

RFor each spherical shell:

outside:

inside: dE = 0

Outside a solid sphere of charge:

for r>R

A Solid Sphere of Charge

Inside a solid sphere of charge:

E

R

r

for r<R

Why is E~r?

On surface:

A Solid Sphere of Charge

What is in the box?

no charges? vertical charged plate?

Patterns of Fields in Space

Box versus open surface

Seem to be able to tellif there are charges inside

…no clue…

Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.

Patterns of Fields in Space