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If potential (in volts) in a region is expressed as V(x, y, z) = 6xy − y + 2yz, the electric field (in N/C) at point (1, 1, 0) is:

1. −(6i + 5j + 2k)
2. −(2i + 3j + k)
3. −(6i + 9j + k)
4. −(3i + 5j + 3k)

Correct answer: −(6i + 5j + 2k)

Solution

The electric field is the negative gradient of the potential, E = -∇V. Calculating partial derivatives of V(x, y, z) = 6xy − y + 2yz at (1, 1, 0): ∂V/∂x = 6y = 6, ∂V/∂y = 6x - 1 + 2z = 5, ∂V/∂z = 2y = 2. Thus, E = -(6i + 5j + 2k).

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