Question

In Fig. 21-25, four particles form a square. The charges areq₁=q₂=Q, and q₂=q₃=q. (a) What is Q/qif the net electrostatic force on particles 1 and 4 is zero? (b) Is there any value of qthat makes the net electrostatic force on each of the four particles zero? Explain.

Short answer

1. The value of Q/q if the electrostatic force on particles 1 and 4 is zero is 2.83 or -2.83.
2. There is no value of that makes the net electrostatic force on each of the four particles zero because the condition of the ratio does not give any equilibrium structure of the condition.

Step-by-step solution

The given data

The values of charges forming a square are: $q_1(=q_4)=Q$and $q_2(=q_3)=q$

Understanding the concept of Coulomb’s law

Using the dot product of the charges, we can get the value of the force. Using this value and the condition of the net force on particles 1 and 4 to be zero, we can find the required ratio of the charges. Again, using the concept for the y-component of particle 2, we can observe the equilibrium structure of the conditions.

Formula:

The magnitude of the electrostatic force between any two particles,

$F_1=\frac{1}{\left(4\pi {\epsilon }_0\right)}\left(\frac{\left|q_1\right|\left|q_2\right|cos\theta }{r^2}\overbrace{i}+\frac{\left|q_1\right|\left|q_2\right|\sin \theta }{r^2}\overbrace{j}\right)$ (1)

a) Calculation of the ratio Q/q 

For ease of presentation (of the computations below), we assume Q > 0 and

q< 0 (although the final result does not depend on this particular choice).

The x-component of the force experienced by q₁ =Q using equation (1) is given as:

$$\begin{gathered} {F_1}_x=\frac{1}{4\pi {\epsilon }_0}\left(\frac{-\left(Q\right)\left(Q\right)}{(\surd 2a{)}^2}cos45°+\frac{\left(\right|q\left|\right)\left(Q\right)}{a^2}\right) \\ {F_1}_x=\frac{Q\left|q\right|}{4\pi {\epsilon }_0{a}^2}\left(\frac{-(Q/|q|}{(2\surd 2)}+1\right) \\ \frac{Q\left|q\right|}{4\pi {\epsilon }_0{a}^2}\left(-\frac{Q/\left|q\right|}{2\surd 2}+1\right)=0 \end{gathered}$$

Therefore,

role="math" localid="1662710310471" $$\begin{gathered} Q/\left|q\right|=2\sqrt{2} \\ =2.83 \end{gathered}$$

Or, it can be,

$$\begin{gathered} Q/\left|q\right|=-2\sqrt{2} \\ =-2.83 \end{gathered}$$

Thus, the value of the ratio is 2.83 and -2.83.

 Step 4: b) Checking if there is any value of q for which the net electrostatic force on each charge is zero

The y-component of the net force on q₁=Q using equation (1) is given as:

$\begin{array}{l}{F}_{2y}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\left(\frac{{\left|q\right|}^2}{({\sqrt{2a)}}^2}\sin {45}^{\circ }-\frac{\left|q\right|\left(Q\right)}{a^2}\right)\\ F_{2y}=\frac{|q{|}^2}{4{\mathrm{\pi \epsilon }}_0{a}^2}\left(\frac{1}{2\sqrt{2}}-\frac{\left(Q\right)}{\left|q\right|}\right)\end{array}$

$(ifweconsidertheconditionofnetzeroforceoneachcharge,then)Q/q=(-1)/(2\surd 2)$

The result is inconsistent with that obtained in part (a). Thus, we are unable to construct an equilibrium configuration with this geometry, where the only forces present are given by Eq. 21-1.

Hence, there is no value of q.