Answer to Question #173178 in Electric Circuits for Shai

Question #173178

Consider three charges π‘ž1=1.5 πœ‡πΆ, π‘ž2Β =5πœ‡πΆ, and π‘ž3=-3 πœ‡πΆ located at (π‘₯1,𝑦1,𝑧1)=(1,βˆ’2,4) π‘š, (π‘₯2,𝑦2,𝑧2)=(2,-1, 3) π‘š, andΒ (π‘₯3,𝑦3,𝑧3)=(3, 3, -4) π‘š, respectively. Find the total net forceΒ F

βƒ—Β 

1

Fβ†’1Β on charge π‘ž1Β due to electric fieldsΒ E

βƒ—Β 

2

Eβ†’2Β (induce by π‘ž2)Β andΒ E

βƒ—Β 

3

Eβ†’3Β (induce by π‘ž3) . Note that by superposition of vectors:


1
Expert's answer
2021-03-21T11:31:05-0400

Given,

"q_1=1.5\\mu C"

"q_2=5\\mu C"

"q_3=-3\\mu C"

(π‘₯1,𝑦1,𝑧1)=(1,βˆ’2,4) π‘š,

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

(π‘₯2,𝑦2,𝑧2)=(2,-1, 3) π‘š,

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


AndΒ (π‘₯3,𝑦3,𝑧3)=(3, 3, -4) π‘š

"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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