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Chapter 4: Q62P (page 717)

Two identical spheres with mass m are hung from silk threads of length L (Fig. P21.62). The spheres have the same charge, so q1 = q2 = q. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. Show that if the angle u is small, the equilibrium separation d between the spheres is d = (q2尝/2饾洃蔚0尘驳)1/3 (Hint: If饾泬is small, then tan饾泬鈮卻in饾泬.)

Short Answer

Expert verified

If the angle u is small, the equilibrium separation d between the spheres is d=2Lq24蟺蔚omg13.

Step by step solution

01

Component of forces

Since the charges are in equilibrium:

鈭�Fx=Tsin胃=14蟿蟿蔚oq2d鈭�Fy=Tcos胃=mg

02

Step 2:

From Step 1, we get

Tsin胃Tcos胃=14蟺蔚oq2d2mgtan胃=14蟺蔚oq2d2mgd2=q24蟺蔚o尘驳迟补苍胃

03

Step 3:

sin胃=d2L

Since胃<<1, use the approximation

tan胃鈮�sin胃=d2Ld2=2Lq24蟺蔚omgdd3=2Lq24蟺蔚omgd=2Lq24蟺蔚omg1/3

Hence, proved.

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