The electric potential V at any point (x,y,z), all in metres in space is given by V = 4 x^2 volt ….

Q: The electric potential V at any point (x , y , z) all in metres in space is given by V = 4 x2 volt. Then electric field at the point (1, 0 ,2) in volt/metre, is

(a) 8 along positive X-axis

(b) 16 along negative X-axis

(c) 16 along positive X-axis

(d) 8 along negative X-axis

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Ans: (d)
Sol: Given V = 4 x2

$\displaystyle E_x = -\frac{\partial V}{\partial x}= -8x $

$\displaystyle E_y = -\frac{\partial V}{\partial y}= 0 $

$\displaystyle E_z = -\frac{\partial V}{\partial z}= 0 $

$\displaystyle \vec{E} = E_x\hat{i} + E_y \hat{j}+ E_z \hat{k} $

$\displaystyle \vec{E} = -8x \hat{i} + 0 \hat{j}+ 0 \hat{k} $

At a point (1,0,2),

$\displaystyle \vec{E} = -8 \times 1 \hat{i} $

E = 8 V/m along -ve X- direction.