Question

Question 5: A point charge of -100 nC is surrounded by oil having a permittivity of 20 x 10-12 C2N 1m2. How much work would be done on a proton in transporting it from a radial distance of 0.50 m to 1.50 m? Additional challenges: Question 6: Two very small metal spheres carrying charges of 60 uC and -25 uC are located at coordinates (0,0) and (3 m, 0) respectively, in air. How much work would have to be done to bring a third sphere with a charge of -10 uC from very far away to the point (0, 4 m)? Question 7

Answer 1

Question 5)

Potential due to the charge $q=-100\,nC$ at distance $r_1=0.50\,m$ is

$V_1=\frac{q}{4\pi\epsilon r_1}$

Potential due to the charge $q=-100\,nC$ at distance $r_2=1.50\,m$ is

$V_2=\frac{q}{4\pi\epsilon r_2}$

$V_2-V_1=\frac{q}{4\pi\epsilon }\left ( \frac{1}{r_2}-\frac{1}{r_1} \right )$

Work done in moving a proton from $r_1=0.50\,m$ to $r_2=1.50\,m$ is $W=e(V_2-V_1)=\frac{qe}{4\pi\epsilon }\left ( \frac{1}{r_2}-\frac{1}{r_1} \right )$

$W=\frac{-100*10^{-9}*1.6*10^{-19}}{4\pi(20*10^{-12}) }\left ( \frac{1}{1.50}-\frac{1}{0.50} \right )$

$W=8.5*10^{-17}\,J$

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Question 6)

Charge $q_1=60\,\mu C$ located at $(x_1,y_1)=(0,0)\,m$

Charge $q_2=-25\,\mu C$ located at $(x_2,y_2)=(3,0)\,m$ ​​​​​​​

Third charge $q_3=-10\,\mu C$ located at $(x_3,y_3)=(0,4)\,m$

Work done on the third charge to bring it from infinity to point $(x_3,y_3)=(0,4)\,m$ is

$W=\frac{q_1q_3}{4\pi\epsilon_0 \sqrt{(x_3-x_1)^2+(y_3-y_1)^2}}+\frac{q_2q_3}{4\pi\epsilon_0 \sqrt{(x_3-x_2)^2+(y_3-y_2)^2}}$

$W=\frac{60*10^{-6}*-10*10^{-6}}{4\pi(8.85*10^{-12}) \sqrt{(0)^2+(4)^2}}+\frac{-25*10^{-6}*-10*10^{-6}}{4\pi(8.85*10^{-12}) \sqrt{(-3)^2+(4)^2}}$

$W=-1.3485+0.4495=-0.9\,J$