Consider three charges π1=1.5 ππΆ, π2Β =5ππΆ, and π3=-3 ππΆ located at (π₯1,π¦1,π§1)=(1,β2,4) π, (π₯2,π¦2,π§2)=(2,-1, 3) π, andΒ (π₯3,π¦3,π§3)=(3, 3, -4) π, respectively. Find the total net forceΒ F
βΒ
1
Fβ1Β on charge π1Β due to electric fieldsΒ E
βΒ
2
Eβ2Β (induce by π2)Β andΒ E
βΒ
3
Eβ3Β (induce by π3) . Note that by superposition of vectors:
Given,
"q_1=1.5\\mu C"
"q_2=5\\mu C"
"q_3=-3\\mu C"
(π₯1,π¦1,π§1)=(1,β2,4) π,
"r_1=\\hat{i}-2\\hat{j}+4\\hat{k}"
(π₯2,π¦2,π§2)=(2,-1, 3) π,
"r_2=2\\hat{i}-\\hat{j}+3\\hat{k}"
AndΒ (π₯3,π¦3,π§3)=(3, 3, -4) π
"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}"
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