P22_012.PDF

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Chapter 22 - 22.12
12. (a) The distance between q 1 and q 2 is
r 12 = ( x 2
x 1 ) 2 +( y 2
y 1 ) 2 = (
0 . 020
0 . 035) 2 +(0 . 015
0 . 005) 2 =0 . 0559 m .
The magnitude of the force exerted by q 1 on q 2 is
F 21 = k |
q 1 q 2 |
r 12
= 8 . 99
×
10 9 3 . 0
×
10 6 4 . 0
×
10 6
=34 . 5N .
0 . 0559 2
The vector F 21 is directed towards q 1 and makes an angle θ with the + x axis, where
θ =tan 1 y 2
y 1
=tan 1 1 . 5
0 . 5
=
10 . 3 .
x 2
x 1
2 . 0
3 . 5
(b) Let the third charge be located at ( x 3 ,y 3 ), a distance r from q 2 .Wenotethat q 1 , q 2 and q 3 must
be colinear; otherwise, an equilibrium position for any one of them would be impossible to find.
Furthermore, we cannot place q 3 on the same side of q 2 wherewealsofind q 1 , since in that region
both forces (exerted on q 2 by q 3 and q 1 ) would be in the same direction (since q 2 is attracted
to both of them). Thus, in terms of the angle found in part (a), we have x 3 = x 2
r cos θ and
r sin θ (which means y 3 >y 2 since θ is negative). The magnitude of force exerted on q 2
by q 3 is F 23 = k
|
q 2 q 3 |
/r 2 , which must equal that of the force exerted on it by q 1 (found in part (a)).
Therefore,
r = r 12 q 3
k |
q 2 q 3 |
r 2
= k |
q 1 q 2 |
r 12
=
q 1 =0 . 0645 cm .
Consequently, x 3 = x 2
r cos θ =
2 . 0cm
(6 . 45 cm) cos(
10 . 3 )=
8 . 4cmand y 3 = y 2
r sin θ =
1 . 5cm
(6 . 45 cm) sin(
10 . 3 )=2 . 7cm .
y 3 = y 2
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