Question

Let a uniform surface charge density of 5 nC/m2 be present at the z = 0 plane, a uniform line charge density of 8 nC/m be located at x = 0, z = 4, and a point charge of 2pC be present at P(2, 0, 0). If V 0 at M(O, 0, 5), find Vat N(1, 2, 3)

Answer 1

For a point charge, the potential function is given by

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$V_{p}=\frac{q}{4\pi \epsilon _{o}r}$

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For an infinite line charged, the potential function is

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$V_{l}=-\frac{\lambda \ln \rho }{2\pi \epsilon _{o}}+C_{l}$

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For the sheet, the potential function is

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$V_{s}=-\frac{\sigma z }{2 \epsilon _{o}}+C_{s}$

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Thus, the total potential function can be expressed as

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$V_{T}=V_{p}+V_{l}+V_{s}$

$\Rightarrow \: \: \: V_{T}=\frac{q}{4\pi \epsilon _{o}r}-\frac{\lambda \ln \rho }{2\pi \epsilon _{o}}+C_{l}-\frac{\sigma z }{2 \epsilon _{o}}+C_{s}$

$\Rightarrow \: \: \: V_{T}=\frac{q}{4\pi \epsilon _{o}r}-\frac{\lambda \ln \rho }{2\pi \epsilon _{o}}-\frac{\sigma z }{2 \epsilon _{o}}+C$

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We will express variables r and $\rho$ in cartesians coordinates,

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$r =\sqrt{(x-2)^{2}+y^{2}+z^{2}}$

$\rho =\sqrt{x^{2}+(z-4)^{2}}$

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thus,

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$V_{T}(x,y,z)=\frac{q}{4\pi \epsilon _{o}\sqrt{(x-2)^{2}+y^{2}+z^{2}}}-\frac{\lambda \ln \sqrt{x^{2}+(z-4)^{2}} }{2\pi \epsilon _{o}}-\frac{\sigma z }{2 \epsilon _{o}}+C$

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Evaluating the total potential function at M(0,0,5), we can obtain the unkown constant C

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$V_{T}(0,0,5)=0=\frac{q}{4\pi \epsilon _{o}\sqrt{(0-2)^{2}+0+5^{2}}}-\frac{\lambda \ln \sqrt{0+(5-4)^{2}} }{2\pi \epsilon _{o}}-\frac{5\sigma }{2 \epsilon _{o}}+C$

$\Rightarrow \: \: \: 0=\frac{q}{4\sqrt{29}\pi \epsilon _{o}}-\frac{5\sigma }{2 \epsilon _{o}}+C$

$\Rightarrow \: \: \: C=\frac{5\sigma }{2 \epsilon _{o}}-\frac{q}{4\sqrt{29}\pi \epsilon _{o}}$

$\Rightarrow \: \: \: C=\frac{5(5\times 10^{-9}C/m^{2}) }{2 (8.85\times 10^{-12}\, C^{2}/N\cdot m^{2})}-\frac{2\times 10^{-6}C}{4\sqrt{29}\pi (8.85\times 10^{-12}\, C^{2}/N\cdot m^{2})}$

$\Rightarrow \: \: \: C=-1.93\times 10^{3}\, V$

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Thus, the total potential function is

$\:$

$V_{T}(x,y,z)=\frac{q}{4\pi \epsilon _{o}\sqrt{(x-2)^{2}+y^{2}+z^{2}}}-\frac{\lambda \ln \sqrt{x^{2}+(z-4)^{2}} }{2\pi \epsilon _{o}}-\frac{\sigma z }{2 \epsilon _{o}}-1.93\times 10^{3}\, V$

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Evaluating at point N(1,2,3), we obtain

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${\color{Blue} V_{T}(1,2,3)=1.98\times 10^{3} \, V}$