1. Chapter 16 Electric Field of Distributed Charges

2. Question 1 Which of the following statements is true?
A charged object is always repelled by a neutral object?
A charged object is always attracted to a neutral object?
A charged object is sometimes attracted and sometimes repelled
by a neutral object?
D) A charged object is not affected by a neutral object? B

3. Metal in Electric Field
Enet inside the conductor will be:
A. Uniform positive
B. Uniform negative
C. Zero
D. will have complex pattern
Metal polarizes!
Polarized is nor equal to charged

4. Metal in Electric Field
Metal polarizes!
Polarized is nor equal to charged
Note: It is not charged!
Net charge is still zero
Simplified diagram of polarized metal

5. Electric Field inside Metal
In static equilibrium:
Enet= 0 everywhere inside the metal!
Mobile charges on surface rearrange to achieve Enet= 0
Actual arrangement might be very complex!
It is a consequence of 1/r2 distance dependence
Enet= 0 only in static equilibrium!

6. Excess Charge on Conductors
Excess charges in any conductor are always found on an inner or outer surface!
Polarized insulator: collection of tiny dipoles
Polarized metal: forms a giant dipole
Do demo of charge on outside of sphere.

7. Conductors versus Insulators
Mobile charges
yes no
Polarization
entire sea of mobile charges moves
individual atoms/molecules polarize
Static equilibrium
Enet= 0 inside
Enet nonzero inside
Excess charges
only on surface
anywhere on or inside material
Distribution of excess charges
Spread over entire surface
located in patches

8. Charging and Discharging
Discharging by contact:
On approach: body polarizes
On contact:
charge redistributes over larger surface
An object is CHARGED: net charge is nonzero
Grounding: connection to earth (ground) – very large object

9. Charging by Induction

10. Charging by Induction

11. Yes, this can be done without changing the order of blocks B & C.
You have three metal blocks marked A, B, and C, sitting on insulating stands. Block A is charged +, but blocks B and C are neutral. Without using any additional equipment, and without altering the amount of charge on block A, can you make block B be charged + and block C be charged -?
Do demo. Ask Tammi for electronic electrometer. Ask students what is in the blue box.
Yes, but one must change the ordering of blocks B & C as an intermediate step.
Yes, this can be done without changing the order of blocks B & C.
No way!

12. Exercise An object can be both charged and polarized +
On a negatively charged metal ball excess charge is spread uniformly all over the surface.
What happens if a positive charge is brought near?

13. When the Field Concept is not Useful
Splitting universe into two parts does not always work. +q
If q is so small that it does not appreciably alter the charge distribution on the sphere, then we can still use the original field:
Otherwise, we must calculate new field:
If even the smallest charge affects original charge distribution use small polarizable atom to probe the field or measure for positive and negative charges.

14. Distributed Charges

15. Uniformly Charged Thin Rod
Length: L
Charge: Q
What is the pattern of electric field
around the rod?
Cylindrical symmetry
Rod – dielectric, not metal! (Ask: can it be metal?)
Could the rod be a conductor and be uniformly charged?

16. Step 1: Divide Distribution into Pieces
Apply superposition principle:
Divide rod into small sections Dy with charge DQ
Assumptions:
Rod is so thin that we can ignore its thickness.
Ask: Why do we require the rod to be so thin that we can disregard its thickness?
If Dy is very small – DQ can be considered as a point charge

17. Step 2: E due to one Piece What variables should remain in our answer?
⇒ origin location, Q, x, y0
What variables should not remain in our answer?
⇒ rod segment location y, DQ
y – integration variable
Vector r from the source to the observation location:
Remind class that we do not need to know if Q is + or – at this point.

18. Step 2: E due to one Piece Magnitude of r: Unit vector r:
Magnitude of E:

19. Step 2: E due to one Piece
Vector ΔE:

20. Step 2: E due to one Piece
DQ in terms of integration variable y:

21. Step 2: E due to one Piece
Components of DE:

22. Step 3: Add up Contribution of all Pieces
Simplified problem: find electric field at the location <x,0,0>
Show students how the y-component integrad, being odd in y, must give zero upon integration.

23. Step 3: Add up Contribution of all Pieces
Integration: taking an infinite number of slices
 definite integral
dy – infinitesimal increment along y axis

24. Step 3: Add up Contribution of all Pieces
Evaluating integral:
Cylindrical symmetry:
replace xr

25. E of Uniformly Charged Thin Rod
At center plane
In vector form:
Check the results: 
Direction: 
Units: 
Special case r>>L: 
Compare with numerical calculation:
L=1 m, r=0.05 m

26. Special Case: A Very Long Rod
Very long rod: L>>r
For a very long (infinite) rod, it does not make sense to keep Q and L separately. Let both Q and L approach infinity while the ratio, Q/L remains constant.
Q/L – linear charge density
1/r dependence!

27. E of Uniformly Charged Rod
At distance r from midpoint along a line perpendicular to the rod:
For very long rod:
Field at the ends:
Numerical calculation

28. General Procedure for Calculating Electric Field of Distributed Charges
Cut the charge distribution into pieces for which the field is known
Write an expression for the electric field due to one piece
(i) Choose origin
(ii) Write an expression for E and its components
Add up the contributions of all the pieces
(i) Try to integrate symbolically
(ii) If impossible – integrate numerically
Check the results:
(i) Direction
(ii) Units
(iii) Special cases