Net Electrostatic Force

(lecture notes from February 6, 2004)

The diagram on the right shows three point charges in a straight line on the x axis. Our task is to calculate the net force acting on each of the three charges, due to the presence of the other two (assuming that all three charges are at rest). Besides doing calculations with Coulomb's Law, you are expected to make use of symmetry considerations to predict your results qualitatively and to check your work.

We will first calculate the net force acting on charge Q₃. To predict the qualitative answer, remember

• unlike charges attract, and like charges repel. Q₃ will be repelled by Q₁, but attracted by Q₂.
• the strength of electrostatic force is inversely proportional to the square of the distance between two charges. Q₁ is twice as far from Q₃ as is Q₂, and that will have the biggest effect on the relative strengths of the forces.

So even though Q₁ has twice the charge magnitude of Q₂, the fact that it is twice as far from Q₃ means that it would exert one-fourth as much force on Q₃ considering distance. Putting it all together, the rightward push exerted on Q₃ by Q₁ is half as strong as the leftward pull exerted on Q₃ by Q₂, so the net force on Q₃ points left.

Coulomb's Law is

F₁₂ = (kQ₁Q₂)/(r₁₂)²

where k = 8.99 x 10⁹ Nm²/C². The electrostatic force between any two charges Q₁ and Q₂ is really a pair of equal and opposite forces (Newton's third law), so we usually calculate just the magnitude of the force (ignoring the signs on the charges) and add vector information (direction) by inspection (to decide whether to add or subtract forces to get net force). The force that charge Q₁ exerts on charge Q₃ is therefore equal to

F_1on3 = (9x10⁹Nm²/C²)(4x10^-6C)(2x10^-6C)/(0.50m)² = 0.29 N to the right.

The force that charge Q₂ exerts on charge Q₃ is equal to

F_2on3 = (9x10⁹Nm²/C²)(2x10^-6C)(2x10^-6C)/(0.25m)² = 0.58 N to the left.

Note that we could have saved ourselves the trouble of doing the second calculation if we just used our qualitative predictions. The net force on Q₃ is therefore equal to

F_3net = |F_2on3 | - |F_1on3 | = 0.29 N to the left.

Now let's repeat the analysis for the net force on the middle charge, Q₂. The symmetry considerations will allow a qualitative prediction. Both surrounding charges are equal distance from the center charge, and both surrounding charges are positive and want to attract the negative center charge. Since Q₁ has twice the charge magnitude of Q₃, and since distances are the same, the leftward pull on Q₂ will be twice as strong as the rightward pull on Q₂. So the net force on Q₂ will point left.

The force that Q₃ exerts on Q₂ is equal to

F_3on2 = (9x10⁹Nm²/C²)(2x10^-6C)(2x10^-6C)/(0.25m)² = 0.58 N to the right.

Note that we could have saved ourselves the trouble of doing this calculation if we had remembered Newton's third law: F_3on2 is equal and opposite to F_2on3.

Now with our symmetry considerations, we see that

|F_1on2| = 2|F_3on2| = 1.16 N (to the left)

so therefore

F_2net = |F_1on2 | - |F_3on2 | = 0.58 N to the left.

I will leave it to you to calculate the net force on charge Q₁, and show that it is 0.87 N to the right.

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