Top Banner
Question
30

Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Jan 17, 2016

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Question

Page 2: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Question

Page 3: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Question

Page 4: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Question 2

-+-+

-+-++

++

---

A)

+++

---

B)

-+-+

-+-+C)

+-+-+-

+----

+++

D)

E

An electric field polarizes a metal block as shown below. Selectthe diagram that represents the final state of the metal.

Page 5: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Chapter 16

Electric Field of Distributed Charges

Page 6: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Distributed Charges

Page 7: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Length: LCharge: Q

What is the pattern of electric fieldaround the rod?Cylindrical symmetry

Uniformly Charged Thin Rod

Could the rod be a conductor and be uniformly charged?

Page 8: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

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 DE 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

Page 9: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Apply superposition principle:Divide rod into small sections Dy with charge DQ

Assumptions:Rod is so thin that we can ignore its thickness.

If Dy is very small – DQ can be considered as a point charge

Step 1: Divide Distribution into Pieces

∆𝐸

>

>

𝑦 0

>

,

𝑥

Page 10: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

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:

Step 2: E due to one Piece

>

>

𝑦 0

,

𝑥

𝑟∆𝐸

Page 11: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Magnitude of r:

Unit vector r:

Magnitude of E:

Step 2: E due to one Piece

>

>

𝑦 0

,

𝑥

𝑟∆𝐸

Page 12: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 2: E due to one Piece

>

>

𝑦 0

,

𝑥

𝑟

Vector ΔE:

Page 13: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 2: E due to one Piece

>

>

𝑦 0

,

𝑥

𝑟

DQ in terms of integration Dy:

Page 14: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Components of :

Step 2: E due to one Piece

>

>

,

𝑟

∆𝐸

∆𝐸𝑥

∆𝐸 𝑦

Page 15: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Simplified problem: find electric field at the location <x,0,0>

Step 3: Add up Contribution of all Pieces

∆𝐸1

∆𝐸2

𝑥 ∆𝐸❑

Page 16: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Integration: taking an infinite number of slices

definite integral

Step 3: Add up Contribution of all Pieces

Page 17: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Evaluating integral:

Cylindrical symmetry:replace xr

Step 3: Add up Contribution of all Pieces

Page 18: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

In vector form:

Step 4: Check the results:

Direction:

Units:

Special case r>>L:

E of Uniformly Charged Thin RodAt center plane

Page 19: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Very long rod: L>>r

Q/L – linear charge density

1/r dependence!

Special Case: A Very Long Rod

Page 20: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

At distance r from midpoint along a line perpendicular to the rod:

For very long rod:

Field at the ends:

Numerical calculation

E of Uniformly Charged Rod

Page 21: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

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 DE 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

Page 22: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Origin: center of the ringLocation of piece: described by q, where q = 0 is along the x axis.

Step 1: Cut up the charge distribution into small pieces

Step 2: Write E due to one piece

A Uniformly Charged Thin Ring

Page 23: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 2: Write DE due to one piece

A Uniformly Charged Thin Ring

Page 24: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 2: Write DE due to one piece

Components x and y:

A Uniformly Charged Thin Ring

Page 25: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 2: Write DE due to one piece

Component z:

A Uniformly Charged Thin Ring

Page 26: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 3: Add up the contributions of all the pieces

A Uniformly Charged Thin Ring

Page 27: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Step 4: Check the results

Direction

Units

Special cases:

Center of the ring (z=0): Ez=0

Far from the ring (z>>R):

A Uniformly Charged Thin Ring

Page 28: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Distance dependence:

Far from the ring (z>>R):

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

Ez~1/z2

A Uniformly Charged Thin Ring

Page 29: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Electric field at other locations: needs numerical calculation

A Uniformly Charged Thin Ring

Page 30: Question. Question 2 -+ -+ -+ -+ ++++++ ------ A) ++++++ ------ B) -+ -+ -+ -+ C) +- +- +- +- ------ ++++++ D) E An electric field polarizes a metal.

Section 16.5 – Study this!

A Uniformly Charged Disk