The electric potential in a region is given by V=2 x y-3 z...

The electric potential in a region is given by $V=2 x y-3 z x+5 y^{2}$, with $V$ in volts and the coordinates in meters. Find (a) the potential and (b) the components of the electric field at the point $x=3 \mathrm{~m}$, $y=3 \mathrm{~m}, z=1 \mathrm{~m}$.

Key Concepts

Gradient Operator

The gradient operator is a mathematical tool used to determine the rate and direction of the fastest increase of a scalar field. When applied to the electric potential, the gradient gives a vector field that, when negated, produces the electric field. It is computed by taking the partial derivatives of the potential with respect to the spatial coordinates.

Partial Derivative Evaluation

Partial derivative evaluation involves differentiating a multivariable function with respect to one variable while holding the others constant. In the context of electric fields, computing the partial derivatives of the potential function with respect to x, y, and z allows for the determination of the corresponding components of the electric field at any point in space.

Electric Potential

Electric potential is a scalar field that represents the work done per unit charge in bringing a charge from a common reference point (often taken at infinity) to a specific point in space. It is measured in volts and is crucial for understanding energy distributions in an electric field.

Electric Field

The electric field is a vector field that defines the force exerted per unit charge on a test charge placed in the field. Its direction indicates the force direction, and its magnitude indicates the strength of the force. The electric field is inherently linked to the electric potential through the negative gradient, implying that it points towards regions of decreasing potential.

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