Question

The electric potential at a point (x, y, Z) is given by

V = - x²y -xz³ + 4

The electric field E at that point is

$bold E with bold rightwards arrow on top space equals bold i with bold hat on top space left parenthesis 2 space xy space plus space straight z cubed right parenthesis space plus space bold j with hat on top space straight x squared space plus space bold k with bold hat on top space 3 xz squared$
$bold E with bold rightwards arrow on top space equals bold i with bold hat on top space 2 space xy space plus space bold j with hat on top space left parenthesis straight x squared space plus straight y squared space right parenthesis space plus space bold k with bold hat on top space left parenthesis 3 xz space minus straight y squared right parenthesis$
$bold E with bold rightwards arrow on top space equals bold i with bold hat on top space left parenthesis straight z cubed right parenthesis space plus space bold j with hat on top space left parenthesis xyz space right parenthesis space plus space bold k with bold hat on top space straight z squared$
$bold E with bold rightwards arrow on top space equals bold i with bold hat on top left parenthesis 2 space xy space minus space straight z cubed right parenthesis space plus space bold j with hat on top space left parenthesis xy squared space right parenthesis space plus space bold k with bold hat on top space 3 straight z squared straight x$

Solution

bold E with bold rightwards arrow on top space equals bold i with bold hat on top space left parenthesis 2 space xy space plus space straight z cubed right parenthesis space plus space bold j with hat on top space straight x squared space plus space bold k with bold hat on top space 3 xz squared

The electric field at a point is equal to the negative gradient of the electrostatic potential at that point.
Potential gradient relates to electric field according to the following relation E = -dV/dr
$straight E with rightwards arrow on top space equals space minus space fraction numerator partial differential straight V over denominator partial differential straight r end fraction space equals open square brackets fraction numerator partial differential straight V over denominator partial differential straight x end fraction bold i with hat on top space minus fraction numerator partial differential straight V over denominator partial differential straight y end fraction space bold j with hat on top space minus fraction numerator partial differential straight V over denominator partial differential straight x end fraction space bold k with hat on top close square brackets equals negative left square bracket space straight i with hat on top space left parenthesis space 2 xy space plus space straight z cubed right parenthesis space plus space bold J with hat on top straight x squared space plus space bold k with hat on top space 3 xz squared right square bracket$

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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?

Figure shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?

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