Download 1 Lesson 3 (1) Electric Field Defined A charge distribution is any

Lesson 3 (1) Electric Field Defined A charge distribution is any configuration of electrically charged objects. The distribution is discrete if it consists of a number of point charges. It is continuous if the charge is smeared out on a line, a surface, or over a volume. If a point charge q0
is ! placed at a point P in the vicinity of a charge distribution, it will experience a force F due to the distributed charges. From Coulomb’s law, this force is proportional to q0 . The quantity defined by !
! F
E = q0
is therefore independent of q0 , and is a property of the charge distribution and the point P only. It s called the electric field at the point P, and is a vector function of space. The point charge q0 is known as a test charge. The electric field can be considered as the force on a test charge of positive one Coulomb, and the unit of N/C. (2) Electric Field of Point Charges The simplest charge distribution is a single point charge q . If a test charge q0 is placed at a point P that is a distance r from q , the electrostatic force on q0 is !
qq
F = k 02 r̂ r
where r̂ is the unit vector directed from q toward the point P. Therefore the electric field produced by q at P is !
q
E = k 2 r̂ r
If the point charge q is placed at the origin of the x-­‐y plane, and if the point P is on the x-­‐axis with coordinate x , the electric field there always points away from the origin if q is positive and toward the origin if q is negative. It therefor has only x
component. This component is given by kq
E x = sign(x) 2 x
1 where sign(x) = +1 if x > 0 and sign(x) = !1 if x < 0 . The function E x ( x ) is sketched below: The electric field due to multiple point charges is obtained from vector addition of the electric field due to the individual point charges. Example: On the x-­‐axis, a point charge 3q is at the origin and another point charge !q is at the point x = a . Sketch the function E x ( x ) for !" < x < +" . Solution: The sketch is based on the following observations: 1. Near the point charges, the graph should be similar to those of a single point charge 2. E x cannot be zero between the point charges. But it should be zero at a point to the right of the negative charge. (smaller charge and shorter distance cancels the effect of larger charge at longer distance) 3. For x very large and positive or negative, (i.e., far away from the two point charges), the charge combination behaves like a single point charge of +q 2 (3) Electric Field Lines Electric field lines are curves drawn with an arrow so that at any point of the curve, the direction of the tangent is the direction of the electric field at that point, and is therefore also the direction of the force when a positive test charge is placed there. They are not paths of charged particles. They have the following properties: 1. Two field lines cannot intersect each other. (If they do, the direction of the electric field cannot be determined at the point of intersection) 2. They come out of positive charge and end on negative charge. (the charges can be at infinity) 3. The number of lines originating from or ending on a point charge is proportional to the strength of the charge. 4. Region of space with high density of lines has strong electric field. Examples: 3 Note the number of lines on each point charge in the last two examples. Note also the configuration near the point where the electric field vanishes in these examples. (4) Electric Dipole An electric dipole is a pair of point charges of opposite signs but equal magnitude. On the x-­‐y plane, consider a point charge q on the x-­‐axis with x = a and another point charge !q at x = !a . At a point P on the x-­‐axis, with coordinate x , the electric field has only x-­‐component. If P lies to the right of q , we have Ex =
kq
( x ! a)
2
!
kq
( x + a)
2
where the two terms are contributions from q and !q respectively. If P is far away from the two point charges, so that x >> a , we can use the approximation n ( n !1) 2
n
! +!
when ! << 1 (1+ ! ) = 1+ n! +
2
and write '2
$
1
1
1 ! a$
1!
a
=
=
1+
=
1'
2
+!
#
&
#
&
2
2
2
2
%
x
( x + a) x 2 (1+ a x ) x " x % x "
1
( x ' a)
2
=
x 2 (1' a x )
so that 4kqa 2kp
Ex ! 3 = 3
x
x
'2
1
2
=
$
1 ! a$
1!
a
1' & = 2 #1+ 2 +!& 2 #
%
x " x%
x "
x
x " # 4 after defining the electric dipole moment p to be the product of the charge on one member of the dipole and the distance between the two point charges: p = q(2a) Thus instead of falling off as 1 x 2 valid for a single point charge, the electric field of a dipole falls off more rapidly as 1 x 3 as x ! " . Next consider the point P to lie on the y-­‐axis with y-­‐coordinate y . The contribution to the !
electric fields from is the vector EL and that !
from q is ER as shown. ! !
!
The net field E = EL + ER has only x
!
component as the y components of EL and !
ER cancel out. Further, ELx = ERx by symmetry. Since the distance between P and the point charge !q is y 2 + a 2 , we have kq
kq
a
2kqa
E x = 2ELx = !2 2 2 cos! = !2 2 2
=!
2
2
2
2 32
y +a
y +a y +a
y
+
a
(
)
If P is far from the charges so that y >> a , we can neglect a 2 in comparison with y 2
and arrive at the approximation kp
E x ! " 3 y
demonstrating dependence on the dipole moment and the (1/distance )3 fall off at large distances. 5