Question

If potential (in volts) in a region is expressed as V (x, y, z) = 6 xy-y +2 yz, the electric field (in N/C) at point (1,1,0) is

$negative left parenthesis 3 bold i with bold hat on top space plus space 5 bold j with bold hat on top space plus 3 bold k with bold hat on top right parenthesis$
$negative left parenthesis 6 bold i with bold hat on top space plus space 5 bold j with bold hat on top space plus 2 bold k with bold hat on top right parenthesis$
$negative left parenthesis 2 bold i with bold hat on top space plus space 3 bold j with bold hat on top space plus bold k with bold hat on top right parenthesis$
$negative left parenthesis 6 bold i with bold hat on top space plus space 9 bold j with bold hat on top space plus bold k with bold hat on top right parenthesis$

Solution

negative left parenthesis 6 bold i with bold hat on top space plus space 5 bold j with bold hat on top space plus 2 bold k with bold hat on top right parenthesis

Given, potential in a region, V = 6xy - y + 2yz
Electric field in a region
$straight E space equals space minus fraction numerator partial differential straight V over denominator partial differential straight x end fraction bold i with bold hat on top space minus fraction numerator partial differential straight V over denominator partial differential straight y end fraction bold j with bold hat on top space minus fraction numerator partial differential straight V over denominator partial differential straight z end fraction bold k with bold hat on top straight E space equals space minus 6 straight y bold i with bold hat on top space minus space left parenthesis 6 straight x minus 1 right parenthesis bold j with bold hat on top space minus 2 straight y bold k with bold hat on top At comma space left parenthesis 1 comma 1 comma space 0 right parenthesis comma space electric space field space can space be space expressed straight E space equals space minus left parenthesis 6 space straight x space 1 stack right parenthesis bold i with hat on top space minus space left parenthesis 6 space straight x space 1 space minus space 1 right parenthesis bold j with bold hat on top space minus space 2 space straight x space 1. space bold k with hat on top equals negative left parenthesis 6 space straight i with hat on top space plus space 5 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space straight N divided by straight C$

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Electric Charges And Fields

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

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