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The electric potential at a point (x, y, z) is given by V = –x²y–xz² + 4. The electric field E⃗ at that point is

1. E⃗ = î 2xy + ĵ (x² + y²) + k̂ (3xz – y²)
2. E⃗ = î 2x + ĵ xyz + k̂ z²
3. E⃗ = î (2xy – z²) + ĵ xy² + k̂ 3z² x
4. E⃗ = î (2xy + z²) + ĵ x² + k̂ 3xz²

Correct answer: E⃗ = î (2xy + z²) + ĵ x² + k̂ 3xz²

Solution

The electric field is the negative gradient of the potential, E⃗ = -∇V. Differentiating V = -x²y - xz² + 4 with respect to x, y, and z gives E⃗ = î(2xy + z²) + ĵ(x²) + k̂(3xz²).

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