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Magnetic Intensity

If potential ...

Question

If potential $V = x^{2} + y^{2} + z^{2}$ , then the magnetic of intensity of electric field at $(1, 1, 1)$ is

6 N / C

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\sqrt{2} N / C

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2 \sqrt{3} N / C

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\frac{2}{\sqrt{3}} N / C

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Solution

The correct option is A $2 \sqrt{3} N / C$
Given: $V = x^{2} + y^{2} + z^{2}$ , (x,y,z)=(1,2,3)
Solution: As we know that the relation between E,V & r is given by;
$E = - \frac{d V}{d r}$
$E_{x} = - \frac{d V}{d x} = \frac{d (x^{2})}{d x} = 2 x = 2$ (x=1)
$E_{y} = - \frac{d V}{d y} = \frac{d (y^{2})}{d y} = 2 y = 2$ (y=1)
$E_{z} = - \frac{d V}{d z} = \frac{d (z^{2})}{d z} = 2 z = 2$ (z=1)
$E = \sqrt{E_{x}^{2} + E_{y}^{2} + E_{z}^{2}} = \sqrt{2 + 2 + 2} = 2 \sqrt{3}$
Hence the correct option is C

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