The electric potential in a region of space is V=(150 x^2-....

The electric potential in a region of space is $V=\left(150 x^{2}-\right.$ $\left.200 y^{2}\right) V,$ where $x$ and $y$ are in meters. What are the strength and direction of the electric field at $(x, y)=(2.0 \mathrm{m}, 2.0 \mathrm{m}) ?$ Give the direction as an angle cw or ccw (specify which) from the positive $x$ -axis.

Key Concepts

Electric Potential

Electric potential is a scalar field that represents the electric potential energy per unit charge at any point in space. It provides a measure of the work done by an electric force in moving a test charge from a reference point to a particular location in an electric field.

Electric Field as the Negative Gradient of the Potential

The electric field is defined as the negative gradient of the electric potential. This concept links spatial variations in potential to the electric force experienced by charges; steeper changes in potential imply stronger electric fields, and the negative sign indicates the field points in the direction of decreasing potential.

Gradient Operation

The gradient of a scalar field is a vector that consists of the partial derivatives of the field with respect to each coordinate. It represents the rate and direction of maximum increase of the field and, when negated, gives the direction and magnitude of the electric field in relation to changes in potential.

Directional Angles and Coordinate Systems

Directional angles are used to specify the orientation of vectors relative to a reference axis, typically the positive x-axis. The angle is expressed with respect to a defined direction, such as clockwise (cw) or counterclockwise (ccw), providing an unambiguous method for describing the direction of a vector in a coordinate system.

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