### Electric Potential and Electric Field Calculations**(a)** The electric potential in a region is given by the equation:\[V(x, y) = \left(3.00 \, \frac{V}{m^2}\right) xy + \left(1.00 \, \frac{V}{m^5}\right) x^2 y^3\]**Question:** What is the electric field in this region?---**(b)** The electric potential in a region is given by the equation:\[V(x, y, z) = \left(1.00 \, \frac{V}{m^2}\right) xy + \left(3.00 \, \frac{V}{m^3}\right) z^2 y + \left(2.00 \, \frac{V}{m^5}\right) x^2 z^3\]**Question:** What is the electric field in this region?---### Explanation:In these scenarios, the electric field **E** can be derived from the electric potential **V**. The electric field is the negative gradient of the electric potential:- In two-dimensional space, \(E(x, y) = -\nabla V(x, y)\), where the gradient \(\nabla V\) includes partial derivatives with respect to each variable. - In three-dimensional space, \(E(x, y, z) = -\nabla V(x, y, z)\), where the gradient \(\nabla V\) includes partial derivatives with respect to \(x\), \(y\), and \(z\).To find the electric field components, calculate the partial derivatives of the potential function with respect to each of its variables and apply the negative sign.

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Answered: (a) The electric potential in a region…

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(a) The electric potential in a region is given by the equation: V(x, y) = (3.002)xy + (1.005) x²3 What is the electric field in this region? (b) The electric potential in a region is given by the equation V(x, y, z) = (1.00) xy +(3.00)+(200x²2³ What is the electric field in this region?