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The electrostatic potential at any point (x, y, z) is given by V = 4x² (in SI units). Which statement(s) is/are correct?

1. the y- and z-components of the electric field at any point are zero
2. the x-component of the electric field at a point is given by (-8x) in the x-direction
3. the x-component of the electric field at the point (1, 0, 2) is -8 in the x-direction
4. the y- and z-components of the electric field are non-zero constants in magnitude.

Correct answer: the y- and z-components of the electric field at any point are zero

Solution

This is a multiple-correct question. E = -grad V. Ex = -dV/dx = -8x; Ey = -dV/dy = 0; Ez = -dV/dz = 0. So options about Ey = Ez = 0 (correct), Ex = -8x (correct), and Ex at (1,0,2) = -8 (correct) are all true; the option claiming non-zero y,z components is false. The canonical key here is the statement that y- and z-components are zero.

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