A charge, q1, of +16 micro-coulombs is located at x = 0, y = 8.8 cm, a charge q2 of -41 micro-coulombs is located at x = 0, y = -3.9 cm, an unknown charge of q3, is located at x =- 3.6 cm, y = 0 cm, and a charge, q4, -8 micro-coulombs is located at x = +13.1 cm, y = 0 cm. The direction of the total electric force on q4 is 128.2 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.
Let us assume that the charge at 3 is +ve therefore force due to
all charges are shown in the figure
Force due to the charge 1 on charge 4
F41 =
(kq1*q4)/r12
Now putting the values
F41 =
(9*109*16*10-6*8*10-6)
/(0.158)2
F41 = 46.146 N
Now due to the charge 2
F42 =
(kq2*q4)/r22
Now putting the values
F42 =
(9*109*41*10-6*8*10-6)
/(0.137)2
F42 =157.28 N
Now due to charge 3
Let us assume that q3 = X uC
F43 =
(kq3*q4)/r32
Now putting the values
F43 =
(9*109*X*10-6*8*10-6)
/(0.167)2
F43 = 2.581*XN
Now in horizontal direction
FX = -F43 - F41Cos33.89 +
F42Cos16.58
FX = -F43 -38.306 + 150.74 =
112.435 - F43 ------------(1)
Now in the Y direction
FY = F41Sin33.89 + F42Sin16.58 =
70.611 N ---------(2)
Now the direction is given
tan(theta) = FY/FX
tan128.2 = 70.611 /(112.435 - F43 )
112.435 - F43 = 70.611/tan128.2
112.435 - F43 = -55.565
F43 = 168 N
Now we know that
F43 = 2.581*X
2.581*X= 168
X= 65.09
hence the charge at 3 will be q3 = 65.09
*10-6 C = 65.09 uC
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