10. [1pt] Calculate the magnitude of Q 3 . The magnitudes of the three charges a

Question

10. [1pt]
Calculate the magnitude of Q3. The magnitudes of the three charges are in the exact ratios of 1 to 2 to 3.

Answer:  Last Answer: 8.8 C
Incorrect, tries 1/6.

Hint: The ratio 1:2:3 does not imply that Q1=1:Q2=2:Q3=3 You have to find out which charge is the smallest/largest from the plot Pick a point on the +5kV contour (or even better on the +2kV contour). Measure the distance to the 3 charges (in m). Calculate the potential assuming charges of Q, 2Q and 3Q (watch the signs) Note that 2Q refers to Q1 and Q refers to Q2, etc. Set the potential equal to 5 kV and solve. As usual with point charges, the voltage at infinity = zero, so you can use V_tot = V_1 + V_2 +.. = kQ_1/r_1+kQ_2/r_2 +..

Explanation / Answer

To find the charge Q3 we can use the SUPERPOSITION PRINCIPLE : the electrostatic potential at any point in the diagram is the sum of the potentials from all 3 charges. the potential at distance r from a charge Q is Q/4??r, where ? is the permittivity of free space. we pick any convenient point X on an equipotential and measure the distances r1, r2,r3 of Q1, Q2, Q3 from X. then
(1/ 4??)(Q1/r1 + Q2/r2 + Q3/r3) = P
where P = potential at X.

Taking the distance from each charge to the nearest equipotential (+5 or -5 kV), Q3 has the largest magnitude (because Q/r is the same for each charge, so large distance r to the equipotential means Q must also be large) and is +ve. Q1 is comes in between and is -ve, and Q2 is smallest, also -ve.
so Q3 : Q1 :Q2 = +3 : -2 : -1. if Q2 = -q then Q1 = -2q and Q3 = +3q. substituting above,
q(-2/r1 -1/r2 + 3/r3) = 4??P.

for accuracy, choose X as far from each charge as possible. X cannot be on the P= 0 V equipotential, because the equation would then give us q=0 for all values of r. i choose X as the point where the +1kV equipotential meets the left lower edge.

Distances measured to X from Q1, Q2, Q3 are 62, 70, 35.5 mm respectively.

P= +1kV, ? = 8.85x10^-12 F/m, 4??P = 1.1x10^-7 C/m.
(-2/r1 -1/r2 + 3/r3) = (-2/0.062 -1/0.070 + 3/0.0355) = 37.96 m^-1.
q = 1.1x10^-7 / 37.96 = 2.92 x 10^-9 C.

so Q3 = 3x (+2.92nC) = +8.8 nC

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