Question

Consider three charges 𝑞₁=1.5 𝜇𝐶, 𝑞₂ =5𝜇𝐶, and 𝑞₃=-3 𝜇𝐶 located at (𝑥₁,𝑦₁,𝑧₁)=(1,−2,4) 𝑚, (𝑥₂,𝑦₂,𝑧₂)=(2,-1, 3) 𝑚, and (𝑥₃,𝑦₃,𝑧₃)=(3, 3, -4) 𝑚, respectively. Find the total net force F

⃗ 

1

F→1 on charge 𝑞₁ due to electric fields E

⃗ 

2

E→2 (induce by 𝑞₂) and E

⃗ 

3

E→3 (induce by 𝑞₃) . Note that by superposition of vectors:

Answer 1

Given,

$q_1=1.5\mu C$

$q_2=5\mu C$

$q_3=-3\mu C$

(𝑥₁,𝑦₁,𝑧₁)=(1,−2,4) 𝑚,

$r_1=\hat{i}-2\hat{j}+4\hat{k}$

(𝑥₂,𝑦₂,𝑧₂)=(2,-1, 3) 𝑚,

$r_2=2\hat{i}-\hat{j}+3\hat{k}$

And (𝑥₃,𝑦₃,𝑧₃)=(3, 3, -4) 𝑚

$r_3=3\hat{i}+3\hat{j}-4\hat{k}$

$d_{12}=\sqrt{(2-1)^2+(-1+2)^2+(3-4)^2}=\sqrt{1+1+1}=\sqrt{3}$

$\overrightarrow{d_{12}}=\hat{i}+\hat{j}-\hat{k}$

$d_{13}=\sqrt{(3-1)^2+(3+2)^2+(-4-4)^2}=\sqrt{4+25+64}=\sqrt{93}$

$\overrightarrow{d_{13}}=2\hat{i}+5\hat{j}-8\hat{k}$

$d_{23}=\sqrt{1+4^2+8^2}=\sqrt{81}=9$

$\overrightarrow{d_{23}}=\hat{i}+4\hat{j}-8\hat{k}$

Hence, the required force $F_{12}=\frac{9\times 10^{9}\times 1.5\times 10^{-6}\times 5\times 10^{-6}}{3}\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$ N

$=7.5\sqrt{3}\times 10^{-3}(\hat{i}+\hat{j}-\hat{k})N$

$F_{13}=-\frac{9\times 10^{9}\times 1.5\times 3\times 10^{-12}}{93}\frac{2\hat{i}+5\hat{j}-8\hat{k}}{\sqrt{93}}$

$=-0.044\times 10^{-3}(2\hat{i}+5\hat{j}-8\hat{k})$

$F_{net}=F_{12}+F_{13}=7.5\sqrt{3}\times 10^{-3}(\hat{i}+\hat{j}-\hat{k})-0.044\times 10^{-3}(2\hat{i}+5\hat{j}-8\hat{k})$

$=12.9\hat{i}+12.77\hat{j}-12.63\hat{k}$

$F_{23}=\frac{9\times 15\times 10{-3}}{81}\frac{\hat{i}+4\hat{j}-8\hat{k}}{9}$