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Chapter 22: Electric Field

PHY2049: Chapter 22

1

Concept Quiz

Î

For which situations can we put a charge q’ on the left side of the two charges so that it is in equilibrium?

(1) (a) only

(2) (a) and (b)

(3) (c) and (d)

(4) (b) and (d)

(5) (c) only

Should be near the smaller charge because if it is near the larger charge, there can be no total cancellation

*q*

′

*q*

′

*q*

′

*q*

′

PHY2049: Chapter 22

2

Concept Quiz

Î

Which situation has the largest force acting on q?

*q q q q*

All E fields are in the same direction in (3). Note that (4) has E = 0.

PHY2049: Chapter 22

3

Electric Field of Single Point Charge

*q*

>

0

G

*E*

=

*kq r*

ˆ

*r*

2

PHY2049: Chapter 22

*q*

<

0

G

*E*

=

*kq r*

ˆ

*r*

2

4

Example: Electric Field on Proton

Î

At surface of proton

q = e = 1.6 x 10

-19

r = 10

-15 m

C

*E*

=

*kq r*

2

=

(

×

9

10

)(

−

15

2

−

19

)

=

21

N/C

Î

E points radially outward for + charge

PHY2049: Chapter 22

5

E Field of Two Equal, Positive Point Charges

PHY2049: Chapter 22

6

E Field of Two Equal, Unlike Point Charges

PHY2049: Chapter 22

7

Field Between Two Charged Parallel Plates

Î

Assume plates are much larger than separation

E is approx. constant between plates

E is zero outside the plates

This is a capacitor!

Î

E points from + plate to – plate

Î

We will calculate E in Chap. 23

Gauss’ law

Proportional to surface charge density

PHY2049: Chapter 22

8

1. Rank magnitude of E at P

1

, P

2

, P .

Assume charges on rings are +q and +q

PHY2049: Chapter 22

9

Answer to Question #1

Î

P

1 has E = 0 (equidistant from ring A and B and they are same sign)

Î

P

3 has largest E (contributions from ring A and B)

Î

P

2 has no contribution from ring B because it is at the center, thus it is only affected by ring A.

Î

So the order (smallest E to largest E) is P

1

, P

2

, P

3

PHY2049: Chapter 22

10

2. Which point has largest E?

Assume charges on rings are +q and − q

PHY2049: Chapter 22

11

Answer to Question #2

Î

P

1 has largest E field since it is equidistant from ring A and

B and their E contributions add, rather than cancel, as in the first question.

PHY2049: Chapter 22

12

Calculate E of Dipole ( ⊥ axis)

Î

At point x, E x

= 0 and E y

< 0. Why?

**+Q**

*r r*

=

*x*

2

+

*d*

2

/ 4 sin

θ

=

*d*

/ 2

*r*

θ

*d*

x

*r*

**-Q**

*E y*

=

−

2

*kQ*

sin

θ

*r*

2

=

2

⎛

⎝

*x*

2

−

*kQ*

+

*d*

2

/ 4

⎞

⎠

*d*

/ 2

*x*

2

+

*d*

2

/ 4

=

(

*x*

2

+

−

*kQd d*

2

/ 4

)

3/ 2

*E y*

≈ −

*kp x*

3

*x d p*

=

*Qd*

(dipole moment)

13

PHY2049: Chapter 22

Calculate E of Dipole (along axis)

Î

At point x, E x

> 0 and E y

= 0. Why?

−

*Q*

*d*

+

*Q*

x

*x*

*E x*

=

(

*kQ*

/ 2

−

*kQ x*

+

*d*

/ 2

)

2

=

(

*x*

2

2

*kQxd*

−

*d*

2

/ 4

)

2

Show this yourself

(algebra)

*E x*

≈

2

*kp x*

3

*x d p*

=

*Qd*

(dipole moment)

PHY2049: Chapter 22

14

Finding E Field from Charge Distribution

Î

Perform integral over charge distribution

Each component must be calculated separately (vector addition)

*dE y*

=

*kdq*

(

*r*

2

θ

)

Î

General helpful rules

Use symmetry to see if any component must be zero

Use symmetry to see if any component is doubled, etc.

Express dq, r and trig functions in terms of “natural” variables defined by the problem

Then we can integrate!

PHY2049: Chapter 22

15

Find E at Center of Uniformly Charged Circle

+q on top half

− q on the bottom half

Symmetry:

E = 0: E x

E x y doubled: E y

(right) = − E x

(top) = +E y

(left)

(bottom)

λ

*dq*

=

=

π

*q r*

=

λ

*ds*

charge / unit length

=

*r*

= const

16

PHY2049: Chapter 22

Center of Uniformly Charged Circle

Î

Find E

Express dq, r, sin θ in terms of θ

Top, bottom give same contribution

*dE y*

= −

*kdq r*

2 sin

θ

= −

*k*

(

*r*

2

) sin

θ

θ

*dq*

*E y*

=

=

∫

π

0

−

*r*

2

−

2

*k*

λ

−

4

*r k*

λ

(

− cos

θ

)

π

0 sin

θ

*r*

λ

=

*q*

π

*r*

*E y*

= −

4

*kq*

π

*r*

2

*dq*

=

λ

*ds*

=

17

PHY2049: Chapter 22