A charge, q1, of +17 micro-coulombs is located at x = 0, y = 5.3 cm, a charge q2 of -42 micro-coulombs is located at x = 0, y = -4.0 cm, an unknown charge of q3, is located at x =- 4.9 cm, y = 0 cm, and a charge, q4, -8 micro-coulombs is located at x = +18.2 cm, y = 0 cm. The direction of the total electric force on q4 is 135.0 degrees from the +x axis, counter-clockwise. What is the unknown charge, q3, in micro-coulombs? If negative, include a negative sign in your answer.
Using Pythagorean theorem
r14 = sqrt(r12 + r42) = sqrt((0.053)2 + (0.182)2) = 0.1896 m
r24 = sqrt(r22 + r42) = sqrt((0.04)2 + (0.182)2) = 0.1863 m
also
1 = tan-1(r1/r4) = tan-1(0.053/0.182) = 16.24 deg
2 = tan-1(r2/r4) = tan-1(0.04/0.182) = 12.4 deg
r34 = r3 + r4 = 0.049 + 0.182 = 0.231 m
F1 = force by charge q1 on q4 = k q1 q4/r142 = (9 x 109) (17 x 10-6) (8 x 10-6)/(0.1896)2 = 34.05 N
= - (F1 Cos1) i + (F1 Sin1) j = - (34.05 Cos16.24) i + (34.05 Sin16.24) j = - 32.7 i + 9.5 j
F2 = force by charge q2 on q4 = k q2 q4/r242 = (9 x 109) (42 x 10-6) (8 x 10-6)/(0.1863)2 = 87.13 N
= (F2 Cos2) i + (F2 Sin2) j = (87.13 Cos12.4) i + (87.13 Sin12.4) j = 85.1 i + 18.7 j
F = magnitude of net force
= F Cos135 i + F Sin135 j = - (0.71) F i + (0.71) F j
F3 = magnitude of force by charge q3 on q4
= F3 i
net force is given as
= + +
- (0.71) F i + (0.71) F j = (- 32.7 i + 9.5 j ) + ( 85.1 i + 18.7 j) + (F3 i)
comparing the coefficients of "j" on both side
(0.71) F = 9.5 + 18.7
F = 39.1 N
comparing the coefficients of "i" on both side
- 0.71 F = - 32.7 + 85.1 + F3
- 0.71 (39.1) = - 32.7 + 85.1 + F3
F3 = - 80.2 N
since the force F3 is negative , hence force by charge q3 comes out to be towards negative x-axis. hence charge q3 is positive
F3 = k q3 q4/r342
80.2 = (9 x 109) q3 (8 x 10-6)/(0.231)2
q3 = 59.4 x 10-6 C
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