Question

io charges q, 2q,-4q, and-2q (q is positive) occupy the four corners of a square of sides 2L, centered at the origin of a coor- dinate system (Fig. 5-1). (a) What is the net force on charge q due to the other charges? (b) What is the force on a new charge Q placed at the origin? 5 24 -O, L)

Answer 1

A point charge $q_1$ is located at point $\vec{r}_1$. Another point charge $q_2$ is located at point $\vec{r}_2$.

Electric force on $q_2$ due to $q_1$ is  $\vec{F}_{21}=\frac{1}{4\pi\epsilon_0}\frac{q_2q_1(\vec{r}_2-\vec{r}_1)}{|\vec{r}_2-\vec{r}_1|^3}$

Charges and their locations are

$q_1=+q$ located at $\vec{r}_1=L(\hat{i}+\hat{j})$

$q_2=-2q$ located at $\vec{r}_2=L(-\hat{i}+\hat{j})$

$q_3=-4q$ located at $\vec{r}_3=L(-\hat{i}-\hat{j})$

$q_4=+2q$ located at $\vec{r}_4=L(\hat{i}-\hat{j})$

(a)

Net force on charge $q_1=+q$ due to other charges

$\vec{F}_{1,net}=\vec{F}_{12}+\vec{F}_{13}+\vec{F}_{14}$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2(\vec{r}_1-\vec{r}_2)}{|\vec{r}_1-\vec{r}_2|^3}+\frac{1}{4\pi\epsilon_0}\frac{q_1q_3(\vec{r}_1-\vec{r}_3)}{|\vec{r}_1-\vec{r}_3|^3}+\frac{1}{4\pi\epsilon_0}\frac{q_1q_4(\vec{r}_1-\vec{r}_4)}{|\vec{r}_1-\vec{r}_4|^3}$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{q_1q_2(\vec{r}_1-\vec{r}_2)}{|\vec{r}_1-\vec{r}_2|^3}+\frac{q_1q_3(\vec{r}_1-\vec{r}_3)}{|\vec{r}_1-\vec{r}_3|^3}+\frac{q_1q_4(\vec{r}_1-\vec{r}_4)}{|\vec{r}_1-\vec{r}_4|^3} \right ]$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{q_1q_2(L(\hat{i}+\hat{j})-L(-\hat{i}+\hat{j}))}{|L(\hat{i}+\hat{j})-L(-\hat{i}+\hat{j})|^3}+\frac{q_1q_3(L(\hat{i}+\hat{j})-L(-\hat{i}-\hat{j}))}{|L(\hat{i}+\hat{j})-L(-\hat{i}-\hat{j})|^3}+\frac{q_1q_4(L(\hat{i}+\hat{j})-L(\hat{i}-\hat{j}))}{|L(\hat{i}+\hat{j})-L(\hat{i}-\hat{j})|^3} \right ]$$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{q_1q_2(2L\hat{i})}{|2L\hat{i}|^3}+\frac{q_1q_3(2L(\hat{i}+\hat{j}))}{|2L(\hat{i}+\hat{j})|^3}+\frac{q_1q_4(2L\hat{j})}{|2L\hat{j}|^3} \right ]$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{q_1q_2\hat{i}}{4L^2}+\frac{q_1q_3(2L(\hat{i}+\hat{j}))}{(2\sqrt{2}L)^3}+\frac{q_1q_4\hat{j}}{4L^2} \right ]$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{q_1q_2\hat{i}}{4L^2}+\frac{q_1q_3(\hat{i}+\hat{j})}{8\sqrt{2}L^2}+\frac{q_1q_4\hat{j}}{4L^2} \right ]$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{-2q^2\hat{i}}{4L^2}+\frac{-4q^2(\hat{i}+\hat{j})}{8\sqrt{2}L^2}+\frac{2q^2\hat{j}}{4L^2} \right ]$

$\vec{F}_{1,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{-q^2\hat{i}}{2L^2}+\frac{-q^2(\hat{i}+\hat{j})}{2\sqrt{2}L^2}+\frac{q^2\hat{j}}{2L^2} \right ]$

$\vec{F}_{1,net}=\frac{q^2}{4\pi\epsilon_0L^2}\left[-\frac{\hat{i}}{2}-\frac{(\hat{i}+\hat{j})}{2\sqrt{2}}+\frac{\hat{j}}{2} \right ]$

$\vec{F}_{1,net}=\frac{q^2}{4\pi\epsilon_0L^2}\left[-\hat{i}\left(\frac{\sqrt{2}+1}{2\sqrt{2}} \right )+\hat{j}\left(\frac{\sqrt{2}-1}{2\sqrt{2}} \right ) \right ]$

Magnitude of the net force on charge $q_1=+q$ is

$|\vec{F}_{1,net}|=\frac{q^2}{4\pi\epsilon_0L^2}\left[\left(\frac{\sqrt{2}+1}{2\sqrt{2}} \right )^2+\left(\frac{\sqrt{2}-1}{2\sqrt{2}} \right )^2 \right ]$

$|\vec{F}_{1,net}|=\frac{3q^2}{16\pi\epsilon_0L^2}$

(b)

Net force on charge $Q$ located at $\vec{r}_c=0$ due to other charges

$\vec{F}_{c,net}=\vec{F}_{c1}+\vec{F}_{c2}+\vec{F}_{c3}+\vec{F}_{c4}$

$\vec{F}_{c,net}=\frac{1}{4\pi\epsilon_0}\left[\frac{Qq_1(\vec{r}_c-\vec{r}_1)}{|\vec{r}_c-\vec{r}_1|^3}+\frac{Qq_2(\vec{r}_c-\vec{r}_2)}{|\vec{r}_c-\vec{r}_2|^3}+\frac{Qq_3(\vec{r}_c-\vec{r}_3)}{|\vec{r}_c-\vec{r}_3|^3}+\frac{Qq_4(\vec{r}_c-\vec{r}_4)}{|\vec{r}_c-\vec{r}_4|^3} \right ]$

$\vec{F}_{c,net}=\frac{Q}{4\pi\epsilon_0}\left[\frac{q_1(0-L(\hat{i}+\hat{j}))}{|0-L(\hat{i}+\hat{j})|^3}+\frac{q_2(0-L(-\hat{i}+\hat{j}))}{|0-L(-\hat{i}+\hat{j})|^3}+\frac{q_3(0-L(-\hat{i}-\hat{j}))}{|0-L(-\hat{i}-\hat{j})|^3}+\frac{q_4(0-L(\hat{i}-\hat{j}))}{|0-L(\hat{i}-\hat{j})|^3} \right ]$

$\vec{F}_{c,net}=-\frac{Q}{4\pi\epsilon_0}\left[\frac{q_1L(\hat{i}+\hat{j})}{2\sqrt{2}L^3}+\frac{q_2L(-\hat{i}+\hat{j})}{2\sqrt{2}L^3}+\frac{q_3L(-\hat{i}-\hat{j})}{2\sqrt{2}L^3}+\frac{q_4L(\hat{i}-\hat{j})}{2\sqrt{2}L^3} \right ]$

$\vec{F}_{c,net}=-\frac{Q}{4\pi\epsilon_02\sqrt{2}L^2}\left[q_1(\hat{i}+\hat{j})+q_2(-\hat{i}+\hat{j})+q_3(-\hat{i}-\hat{j})+q_4(\hat{i}-\hat{j})\right ]$

$\vec{F}_{c,net}=-\frac{Q}{4\pi\epsilon_02\sqrt{2}L^2}\left[q(\hat{i}+\hat{j})-2q(-\hat{i}+\hat{j})-4q(-\hat{i}-\hat{j})+2q(\hat{i}-\hat{j})\right ]$

$\vec{F}_{c,net}=-\frac{Qq}{4\pi\epsilon_02\sqrt{2}L^2}(9\hat{i}+\hat{j})$

Magnitude of net force on charge $Q$ located at $\vec{r}_c=0$ due to other charges

$|\vec{F}_{c,net}|=\frac{Qq\sqrt{82}}{4\pi\epsilon_02\sqrt{2}L^2}$