1

Three point charges lie at the vertices of an equilateral triangle as shown. All three charges have the same magnitude, but Charges #1 and #2 are positive (+q) and Charge #3 is negative (–q).

The net electric force that Charges #2 and #3 exert on Charge #1 is in

A. the +x-direction. B. the –x-direction.

C. the +y-direction. D. the –y-direction.

E. none of the above

CPS question

Charge #1

Charge #2

Charge #3

+q

+q

–qx

y

Three point charges lie at the vertices of an equilateral triangle as shown. All three charges have the same magnitude, but Charge #1 is positive (+q) and Charges #2 and #3 are negative (–q).

The net electric force that Charges #2 and #3 exert on Charge #1 is in

A. the +x-direction. B. the –x-direction.

C. the +y-direction. D. the –y-direction.

E. none of the above

CPS question

Charge #1

Charge #2

Charge #3

+q

–q

–qx

y

2

Electric field

0q

FE =

We can define the “electric field” as the electric force per unit of charge

EqF 0=

The force exerted on an arbitrary point charge by a field is:

q0 is called a “test charge”

Remember: the force on a charged body is exerted by a field created by other charges

Electric field (rigorous definition)

000

limq

FE

q →=

The presence of the “test charge” may affect the distribution of charge on the body creating the field (e.g. metal). So the fieldbefore and after may be different we probe it with the charge. We want to avoid this situation by taking the “limit”:

In practice, we take the charge distribution to be fixed

3

Electric fields may be mapped by force on a test charge

If one measured the force on a test charge at all points relative to another charge or

charges, an electric field may be mapped.

Electric field of a point charge

rr

qE

rr

qqF

ˆ4

1

ˆ4

1

20

20

0

πε

πε

=⇒

=

4

Electric field of a point charge (cont.)

rr

qE ˆ

4

12

0πε=

The field points away from the positive charge, and

toward the negative charge

Electron trajectory in a uniform electric field

m

eE

m

Fa y

y

−==

5

Electron trajectory in a uniform electric field

2

0

20

0

2

1

2

1t

m

eEytatvy

tvx

oyvyy

x

−=→+=

=

=

2

2

1x

mv

eEy

ox

−=⇒

Electric field calculations

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=

+++=

...ˆˆˆ4

1

...

323

322

2

212

1

1

0

321

rr

qr

r

qr

r

q

EEEE

πε

rrrr

6

Two point charges and a point P lie at the vertices of an equilateral triangle as shown. Both point charges have the same magnitude qbut opposite signs. There is nothing at point P.

The net electric field that Charges #1 and #2 produce at point P is in

CPS question Charge #1

Charge #2

–q

+qx

y

P

A. the +x-direction. B. the –x-direction.

C. the +y-direction. D. the –y-direction.

E. none of the above

Two point charges and a point P lie at the vertices of an equilateral triangle as shown. Both point charges have the same negative charge (–q). There is nothing at point P.

The net electric field that Charges #1 and #2 produce at point P is in

CPS question Charge #1

Charge #2

–q

–qx

y

P

A. the +x-direction. B. the –x-direction.

C. the +y-direction. D. the –y-direction.

E. none of the above

7

Field of an electric dipole

Charge density

∫=→= dxQdx

dQ λλ

∫∫=→=.Surf

dsQds

dQ σσ

∫∫∫=→=Volume

dvQdv

dQ ρρ

Linear distribution:

Surface (2D) distribution:

Volume (3D) distribution:

8

Field of a ring of charge

Field of a line of charge

9

Field of a disk of charge

Field of two parallel charged sheets

10

Electric field lines

Imaginary lines or curves drawn through a region of space such that their tangent at any point are in the

direction of the electric field vector at that point

Electric field lines (cont.)

•Higher density of lines → stronger field

•Arrows define the direction of the field

•Field lines never intersect (the field is uniquely defined at each point)

11

Electric dipolesPair of charges with equal magnitude and opposite

sign (q,-q), separated by a small distance d

Force and torque on electric dipole