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16-18P

Expert-verifiedFound in: Page 443

Book edition
7th

Author(s)
Douglas C. Giancoli

Pages
978 pages

ISBN
978-0321625922

**(III) Two charges, \( - {\bf{Q}}\) and \( - {\bf{3Q}}\) are a distance l apart. These two charges are free to move but do not because there is a third (fixed) charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium?**

The magnitude of the third charge is \(0.40Q\) at a distance \(0.37l\) from the \( - Q\) charge in between the two charges.

**The force between two point charges relies on the magnitude of both the charges and the separation between them. **

The expression for the force between two point charges is given as:

\(F = k\frac{{{Q_1}{Q_2}}}{{{r^2}}}\) … (i)

Here, *k* is the Coulomb’s constant, \({Q_1},\;{Q_2}\) are the charges and *r* is the separation between them.

**In an equilibrium state, the net force on each charge must be zero. **

The given two charges are \( - Q\) and \( - 3Q\).

The distance between the charges is \(l\).

Let there is *q* charge at a distance of *x* from the charge \( - Q\) in between the two charges.

The whole system is in equilibrium, then the force on each charge is zero.

For the equilibrium of the \( - Q\) charge, you can get,

\(\begin{aligned}{c}k\frac{{qQ}}{{{x^2}}} = k\frac{{3Q \times Q}}{{{l^2}}}\\\frac{q}{{{x^2}}} = \frac{{3Q}}{{{l^2}}}\\q = 3Q\frac{{{x^2}}}{{{l^2}}}\end{aligned}\) … (i)

Now, for the equilibrium of the charge *q* you get,

\(\begin{aligned}{c}k\frac{{qQ}}{{{x^2}}} = k\frac{{q \times 3Q}}{{{{\left( {l - x} \right)}^2}}}\\3{x^2} = {\left( {l - x} \right)^2}\\x = \frac{l}{{\sqrt 3 + 1}}\\x = 0.37l\end{aligned}\)

From equation (i), the magnitude of the third charge is,

\(\begin{aligned}{c}q = 3Q\frac{{{{\left( {\frac{l}{{\sqrt 3 + 1}}} \right)}^2}}}{{{l^2}}}\\ = 3Q \times \frac{{{l^2}}}{{{{\left( {\sqrt 3 + 1} \right)}^2}}} \times \frac{1}{{{l^2}}}\\ = \frac{3}{{{{\left( {\sqrt 3 + 1} \right)}^2}}}Q\\ = 0.40Q\end{aligned}\)

Thus, the magnitude of the third charge is \(0.40Q\) at a distance \(0.37l\) from the \( - Q\) charge in between the two charges.

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