Suppose an electric potential is described by V(x, y, z) = −(5x2 + y + z)𝑉(𝑥, 𝑦, 𝑧) = −(5𝑥2 + 𝑦 + 𝑧) in volts. Which of the following expressions describes the associated electric field, in units of volts per meter?(a)  E⃗ =5x^+2y^+2z^𝐸→=5𝑥^+2𝑦^+2𝑧^ (b)  E⃗ =10xx^𝐸→=10𝑥𝑥^ (c)  E⃗ =5xx^+2y^𝐸→=5𝑥𝑥^+2𝑦^ (d)  E⃗ =10xx^+y^+z^𝐸→=10𝑥𝑥^+𝑦^+𝑧^ (e)  E⃗ =0𝐸→=0 The correct response is

The electric field E is related to the potential V by the equation E = -∇V, where ∇ is the gradient operator. This means that the electric field is equal to the negative of the gradient of the potential.

The gradient of a scalar function in three dimensions, like V(x, y, z), is given by the vector (∂V/∂x, ∂V/∂y, ∂V/∂z).

Given V(x, y, z) = -(5x^2 + y + z), we can compute the partial derivatives:

∂V/∂x = -10x, ∂V/∂y = -1, ∂V/∂z = -1.

So, the gradient of V is (-10x, -1, -1).

Therefore, the electric field E = -∇V is given by the negative of this gradient, which is (10x, 1, 1).

So, the correct answer is (d) E = 10x x^ + y^ + z^.