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

When studying electric fields, a key concept is understanding the electric field components. These components can be used to fully describe the direction and magnitude of the electric field at any point in space. In our exercise, the electric field components at a point are found by taking the negative gradient of the electric potential function, which involves partial differentiation with respect to each spatial coordinate.

To visualize this, consider the electric potential as a hill, where the slope at any point is determined by these components. The steeper the slope, the stronger the electric field at that point. In our example, after calculating, the electric field at the point (1m, 1m, 1m) has components \(E_x = -1\text{V/m}\), \(E_y = -12\text{V/m}\), and \(E_z = 3\text{V/m}\). This implies that the strongest electric field component is in the 'y' direction, with a value of -12 V/m.

Partial differentiation is an essential tool in multiple fields of science and engineering, especially when dealing with functions of several variables. It allows us to analyze the rate at which a function changes with respect to one variable while keeping other variables constant. In the context of electric fields, partial differentiation helps us find the field components by differentiating the electric potential with respect to each spatial coordinate.

In our exercise, we differentiate the potential function \(V=2xy-3zx+5y^2\) with respect to \(x\), \(y\), and \(z\) to find the respective electric field components. Remember, the minus sign in \(E = -abla V\) indicates that the electric field points in the direction of the greatest decrease of potential.

The concept of an electric field is central to the understanding of electromagnetic phenomena. An electric field surrounds electric charges and exerts force on other charges within the field. This field is a vector field, meaning it has both magnitude and direction at each point in space.

The electric field is defined as the force per unit charge. In our problem, the negative gradient of the potential (\(-abla V\)) gives us the electric field vector at a specific location. The components derived from this process tell us how the electric field behaves in relation to each axis in three-dimensional space. At the point \(x=1m, y=1m, z=1m\), we determined the electric field components which together give us the complete electric field vector at that point.