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Physics / Chapter 24: Electric Potential / Section 24-10

Electric Potential Energy of a System of Point Charges — Question 1

Section 24-10 · Physics pages 3972–3973

Question 1 An equilateral assembly has negative total energy despite one repulsive pair Charges +q at A, +q at B and -2q at C occupy an equilateral triangle of side a, with q,a>0. Initially all charges are infinitely separated; ignore self-energies. A B C a Schematic; labels, not drawing lengths, specify the data. (a) Find the final interaction energy by counting each pair once. (b) Find the three quasistatic assembly works when A is placed first, B second and C last, then show the total for the reverse order.
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Question 1 - Solution An equilateral assembly has negative total energy despite one repulsive pair A B C a (a) Method and calculation U = k a[q2 −2q2 −2q2] = −3kq2 a . WA = 0, WB = kq2/a, WC = −4kq2/a. (b) Interpretation and further calculation WA + WB + WC = −3kq2/a. WC′ = 0, WB′ = −2kq2/a, WA′ = −kq2/a, again giving the same final energy. Physical check The work of an individual assembly step depends on the charges already present, but the total depends only on the final configuration.

Question 1

An equilateral assembly has negative total energy despite one repulsive pair

Charges +q at A, +q at B and -2q at C occupy an equilateral triangle of side a, with q,a>0. Initially all charges are infinitely separated; ignore self-energies.

Diagram shown in the original SVG worksheet.

  1. Find the final interaction energy by counting each pair once.

  2. Find the three quasistatic assembly works when A is placed first, B second and C last, then show the total for the reverse order.

Question 1 - Solution

An equilateral assembly has negative total energy despite one repulsive pair

Diagram shown in the original SVG worksheet.

(a) Method and calculation

U=ka[q2−2q2−2q2]=−3kq2a.U=\frac{k}{a}[q^2-2q^2-2q^2]=-\frac{3kq^2}{a}.WA=0,WB=kq2/a,WC=−4kq2/a.W_A=0,\quad W_B=kq^2/a,\quad W_C=-4kq^2/a.

(b) Interpretation and further calculation

WA+WB+WC=−3kq2/a.W_A+W_B+W_C=-3kq^2/a.WC′=0,WB′=−2kq2/a,WA′=−kq2/a,W_C\prime=0,\quad W_B\prime=-2kq^2/a,\quad W_A\prime=-kq^2/a,again giving the same final energy.

Physical check

The work of an individual assembly step depends on the charges already present, but the total depends only on the final configuration.

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