Determining the Electric Field in a Given Potential at a Specific Point

Understanding the relationship between electric potential and electric fields is fundamental in classical electrodynamics. This article will guide you through the process of determining the electric field E at a specific point in space when the potential V is given by a specific function. We'll cover step-by-step instructions on how to compute the components of the electric field and evaluate them at a given point.

Understanding the Basics: Electric Field and Potential

The electric field E and the electric potential V are two key concepts in electrostatics. The electric field is a vector field that describes the force per unit charge exerted on a test charge at each point in space, while the electric potential is a scalar field that represents the potential energy per unit charge at each point in space.

Determining the Electric Field from Potential

The link between the electric field and the electric potential is defined by the gradient of the potential. The electric field E can be derived from the electric potential V using the formula:

E -?V

Where ? is the gradient operator.

Step-by-Step Process of Calculating the Electric Field

Let's consider the potential V given by:

V 125y2 - 3z2

We need to find the components of the electric field at the point (x, y, z) (-1, 2, 3).

Step 1: Computing the Gradient of the Potential

The gradient of a scalar field in three dimensions is given by:

?V ( ?V/?x, ?V/?y, ?V/?z )

We need to compute each partial derivative of V.

Partial Derivative with Respect to x

?V/?x 2

Partial Derivative with Respect to y

?V/?y 20y

Partial Derivative with Respect to z

?V/?z -6z

Therefore, the gradient of the potential V is:

?V (2, 20y, -6z)

Step 2: Evaluating the Gradient at the Point (-1, 2, 3)

Now, let's evaluate the partial derivatives at the point (x, y, z) (-1, 2, 3).

?V/?x 2

?V/?y 20(2) 40

?V/?z -6(3) -18

Thus, the gradient of the potential at this point is:

?V (2, 40, -18)

Step 3: Calculating the Electric Field Components

The electric field E can be found by taking the negative gradient of the potential:

E -?V (-2, -40, 18)

The components of the electric field at the point (-1, 2, 3) are:

E_x -2 N/C, E_y -40 N/C, E_z 18 N/C

Conclusion

The three components of the electric field E at the point (-1, 2, 3) are:

E -2 N/C, -40 N/C, 18 N/C

General Process for Calculating Electric Field from Potential

The general process for determining the electric field E from a given potential V involves the following steps:

Step 1: Calculate the Electric Field

Given the electric potential V(x, y, z), the electric field E can be calculated using:

E -?V

This can be broken down into the partial derivatives:

E_x -?V/?x, E_y -?V/?y, E_z -?V/?z

Step 2: Evaluate at the Given Point

Substitute the coordinates of the point (x, y, z) into the expressions for E_x, E_y, and E_z to find the components at that specific location.

Conclusion

In conclusion, determining the electric field from a given potential involves calculating the gradient of the potential and evaluating it at the desired point. This process is crucial in understanding the behavior of electric fields in various applications, from theoretical physics to engineering.