Learning Goal: To practice Problem-Solving Strategy 20.1 Electric forces and Cou

Question

Learning Goal: To practice Problem-Solving Strategy 20.1 Electric forces and Coulomb's law. Two charged particles, with charges q_1 = q and q_2 = 4q, are located at a distance d = 2 cm apart on the x axis. A third charged particle, with charge q_3 = q, is placed on the x axis such that the magnitude of the force that charge 1 exerts on charge 3 is equal to the force that charge 2 exerts on charge 3. Now use the information and the insights that you have accumulated to construct the necessary mathematical expressions and to derive the solution . Find the position of charge 3 when q = 1.0 nC . Note that you are given the magnitude of q, but the sign of the charge q is not specified, q can be positive or negative, so that the three charges (q_1, q_2, q_3) are either all positive charges or all negative charges. Find the two possible values x_3,r and X_3 for the position of charge 3. Express your answers in centimeters, separated by a comma. This question will be shown after you complete previous question(s).

Explanation / Answer

givens:
q1 = q
q2 = 4q
q3 = q
d = 2.00 cm
xq1 = 0 cm
xq2 = 2.00 cm
q = 1.00 nC

next lets determine the forces
F1on3= K|q1||q3| / r2 = K|q||q| / (d1-3)2 = Kq2 / (d1-3)2
F2on3= K|q2||q3| / r2 = K|4q||q| / (d3-2)2 = K4q2 / (d3-2)2

since the forces are supposed to equal each other in each case, we're able to solve for the variables in both cases

F1on3= F2on3
Kq2 / x2=K4q2 / (2.00 cm - x)2
1/x2 = 4/ [4.00 cm2 - (4.00 cmx) + x2]
[4.00 cm2 - (4.00 cmx) + x2]/x2 = 4
4.00 cm2/x2 - (4.00 cmx)/x2 +x2/x2 - 4 = 0
4.00 cm2/x2 - (4.00 cm)/x + 1 - 4 = 0
4.00 cm2/x2 - (4.00 cm)/x -3 = 0
3=4.00 cm2/x2 - (4.00 cm)/x
3x2=4.00 cm2 -4.00 cmx
3x2 + 4.00 cmx - 4.00 cm2 = 0
plugging that into the quadratic equation we get
x = -2 & x = 2/3
x3,r, x3, = 0.667 cm,-2.00 cm

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