Question

Three charges +Q1, +Q2, and q are placed on a straight line such that q is somewhere in between +Q1 and +Q2 . If this system of charges is in equilibrium, what should be the magnitude and sign of charge q?​

Answer 1

Given:

Three charges +Q1, +Q2, and q are placed on a straight line such that q is somewhere in between +Q1 and +Q2 .

To find:

Magnitude and sign of q ?

Solution:

Let's consider that distance between Q1 and Q2 is "d" and q is placed at a distance "x" from Q2.

Now, if we consider the equilibrium of Q2, we can say that Q2 will be:

• Repelled by Q1 , because like charges repel each other.

• So, in order to gain equilibrium , q has to be negative such that it can attract Q2 and nullify the force exerted by Q1.

$\therefore \: \dfrac{1}{4\pi \epsilon_{0}} \bigg \{ \dfrac{(Q_{2})q}{ {x}^{2} } \bigg \} = \dfrac{1}{4\pi \epsilon_{0}} \bigg \{ \dfrac{(Q_{2})(Q_{1})}{ {d}^{2} } \bigg \}$

$\implies \: \dfrac{(Q_{2})q}{ {x}^{2} } = \dfrac{(Q_{2})(Q_{1})}{ {d}^{2} }$

$\implies \: \dfrac{q}{ {x}^{2} } = \dfrac{(Q_{1})}{ {d}^{2} }$

$\implies \: q= \dfrac{(Q_{1}) {x}^{2} }{ {d}^{2} }$

So, charge q will be:

$\boxed{ \bold{ \: q= - \dfrac{(Q_{1}) {x}^{2} }{ {d}^{2} } }}$