Electric Charges and Forces
25-27
Solve: (a) In the figure, the distances are '1 = '3 = d(1 cm)2 +(3 cm)* = 3.162 cm and the angle is 8 = tan-](l/ 3) = 18.43". Using the equation for the field of a point charge, ~(q,( E = E =-= 1
3
'12
(9.0 x IO9 N m2 / C2)(1.0 x lo4 C)' = 9000N/C (0.03162 m)'
We now use the angle 8 to find the components of the field vectors:
-
E, = E, cos& - E, sin($
,$ = E3cos& + E3sin6 = (85401 + 28403 ) N / C
)
= (85401 - 28403 N / C
E2 is easier since it has only an x-component. Its magnitude is ~lq,l
E2=-r,'
-
(9.0 x lo9 N m2 / C2)(1.0 x
C)
(0.0300 m)'
= 10,000N / C
E, = E,;
(b) The electric field is defined in terms of an electric force acting on charge q:
= lO,OOOi^N / C
E = F/q. Since forces obey a
fi
principle of superposition ( = + F; + ...) it follows that the electric field due to several charges also obeys a principle of superposition. - - . . . (c) The net electric field at a point 3 cm to the right of q2is = E, + E2 + E3 = 27,100; N / C. The y-components of
E, and E, cancel, giving a net field pointing along the x-axis.
25.67. Model: Model the proton and the electron as point charges. Solve: (a) The force that an electric field E exerts on a charge q is = qE . A proton has q = e. Thus,
Fpromn =e(200i^ + 4ooj^) N / C where we used e = 1.60 x C. (b) The charge on an electron is q = -e. Thus,
Fe,-,,
=
= (3.201
+ 6 . 4 O j ^ ) ~ l O " ~N
-Fpown= (-3.201 - 6.403) x lo-''
N
(c) From Newton's second law,
(d) The electron experiences a force of the same magnitude but it has a different mass. Thus,
The forces may be the same, but the electron has a much larger acceleration due to its much smaller mass. Assess: The two forces in parts (a) and (b) are equal in magnitude but opposite in direction.
25.68. Model: The charged ball attached to the string is a point charge. Visualize:
-
w
The ball is in static equilibrium in the external electric field when the string makes an angle 8 = 20" with the vertical. The three forces acting on the charged ball are the electric force due to the field, the weight of the ball, and the tension force.
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