Question

Two identical particles, each with charge q, are placed on the x axis, one at the origin and the other at x = 5 cm. A third particle, with charge −q, is placed on the x axis so the potential energy of the three-particle system is the same as the potential energy at infinite separation. Its x coordinate is:​

Answer 1

potential energy of first two bodies = kq*q/5 let the third particle is placed at (a,0) Therefore total potential energy would be kq*q/5 - kq*q/(a-5) -kq*q/a = 0...

Answer 2

Answer:

The x - coordinate of third charge is 13.1 cm or 1.9 cm.

Explanation:

• Electric Potential Energy is the energy possessed by a system of charges by virtue of their positions.
• The electric potential energy of a system of point charges may be defined as the amount of work done in assembling the charges at their location by bringing them in, from infinity.

$U = k\frac{q1q2}{r}$

• When two like charges are infinite distance apart, their potential energy is zero because no work is done in moving the charge at infinite distance from each other.

Given that :

• Two identical charge, q are placed at origin and x = 5.
• Potential Energy of three particle system is same as charges at infinite separation.

To find :

• Position of third charge, -q

Solution :

• Let, the charge q be at A(0,0), charge q at B(5,0) and charge -q at C(a,0).
• Energy potential energy of the whole system is

$U = k \frac{ (q)(q)}{5} + k\frac{(q)( - q) }{a} + k\frac{ (q)( - q) }{a - 5} \\ = k {q}^{2} ( \frac{1}{5} - \frac{1}{a} - \frac{1}{a - 5} )$

• This energy is equal to energy possed by charges at infinite separation i.e. zero.

$U= 0 \\ k {q}^{2} ( \frac{1}{5} - \frac{1}{a} - \frac{1}{a - 5} ) = 0 \\ \frac{a(a - 5) - 5(a - 5) - 5a}{5a(a - 5)} = 0 \\ {(a - 5)}^{2} - 5a = 0 \\ {a}^{2} + 25 - 10a - 5a = 0 \\ {a}^{2} + 25 - 15a = 0$

• Solving above equation, we get

${a}^{2} + 25 - 15a = 0 \\ a = \frac{15 + - \sqrt{ {15}^{2} - (4 \times 25) } }{2} \\ a = \frac{15 + - ( \sqrt{125)} }{2} \\ a = 13.1 \: or \: 1.9$

• Hence, charge -q must be placed at (13.1 , 0) or at (1.9 , 0).

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