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In a certain region of space, the potential is given by $V = k [2 x^{2} - y^{2} + z^{2}]$ . The electric field at the point $(1, 1, 1)$ has magnitude :

A

$k 6$

B

$2 k 6$

C

$2 k 3$

D

$4 k 3$

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Solution

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Correct option is B)

Given, $V = k [2 x^{2} - y^{2} + z^{2}]$

Electric field , $E = - (\frac{d V}{d x} \hat{i} + \frac{d V}{d y} \hat{j} + \frac{d V}{d z} \hat{k})$

or, $E = - k (4 x \hat{i} - 2 y \hat{j} + 2 z \hat{k})$

or, $E_{(1, 1, 1)} = - k (4 \hat{i} - 2 \hat{j} + 2 \hat{k})$

Magnitude of electric field $= ∣ E_{(1, 1, 1)} ∣ = k^{2} (16 + 4 + 4) = k 24 = 2 k 6$

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