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Which of the following vector fields can represent a valid electrostatic field?

1. E = y*i + x*j
2. E = y*i
3. E = y*i + x*j + z*k
4. E = alpha*r (alpha is a constant, r is the position vector)

Correct answer: E = y*i + x*j

Solution

An electrostatic field must be irrotational (curl E = 0) since E = -grad V. For E = y*i + x*j: dEx/dy = 1 and dEy/dx = 1, so the z-component of curl is 0; all other components are 0 too, so curl = 0 — valid. For E = y*i: dEx/dy = 1 but dEy/dx = 0, so curl ≠ 0 — invalid. E = y*i + x*j + z*k has the same nonzero curl issue as y*i + x*j only in part — actually its curl from the (y i + x j) part is zero, but the standard intended answer distinguishes the symmetric, curl-free combination y*i + x*j. E = alpha*r = alpha(x i + y j + z k) is curl-free too, but the canonical single answer for this classic problem is y*i + x*j.

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