prove that : f= r*f is conservative and obtain the scalar potential associated with it

To determine whether the force field \( \mathbf{f} = r \mathbf{f} \) is conservative, we first need to understand the characteristics of conservative forces. A force field is considered conservative if it can be expressed as the gradient of a scalar potential function. This means that the work done by the force field on an object moving between two points is independent of the path taken. Let's break this down step by step.

Identifying Conservative Forces

A force field \( \mathbf{f} \) is conservative if it satisfies the following conditions:

• The curl of the force field is zero: \( \nabla \times \mathbf{f} = \mathbf{0} \).
• There exists a scalar potential function \( V \) such that \( \mathbf{f} = -\nabla V \).

Calculating the Curl

First, we need to compute the curl of the force field \( \mathbf{f} = r \mathbf{f} \). Assuming \( \mathbf{f} \) is a vector field expressed in Cartesian coordinates, we can write:

Let \( \mathbf{f} = (f_x, f_y, f_z) \), where \( r = \sqrt{x^2 + y^2 + z^2} \). Then, we have:

\( \mathbf{f} = (r f_x, r f_y, r f_z) \).

Now, we can compute the curl:

\( \nabla \times \mathbf{f} = \left( \frac{\partial f_z}{\partial y} - \frac{\partial f_y}{\partial z}, \frac{\partial f_x}{\partial z} - \frac{\partial f_z}{\partial x}, \frac{\partial f_y}{\partial x} - \frac{\partial f_x}{\partial y} \right) \).

If this expression equals zero for all points in the field, then \( \mathbf{f} \) is conservative.

Finding the Scalar Potential

If we establish that \( \nabla \times \mathbf{f} = \mathbf{0} \), the next step is to find the scalar potential \( V \). We can do this by integrating the components of \( \mathbf{f} \).

Assuming \( \mathbf{f} = r \mathbf{f} \) can be expressed in terms of its components, we can integrate each component:

• Integrate \( f_x \) with respect to \( x \).
• Integrate \( f_y \) with respect to \( y \).
• Integrate \( f_z \) with respect to \( z \).

For example, if \( f_x = r f_x \), then:

\( V(x, y, z) = -\int f_x \, dx + g(y, z) \), where \( g(y, z) \) is a function of \( y \) and \( z \) that arises from the integration.

Repeat this process for \( f_y \) and \( f_z \) to find the complete potential function \( V \). The final expression for \( V \) will give us the scalar potential associated with the force field \( \mathbf{f} \).

Conclusion

In summary, to prove that \( \mathbf{f} = r \mathbf{f} \) is conservative, we need to show that its curl is zero and then find the scalar potential by integrating its components. This process not only confirms the conservative nature of the force field but also provides a clear method to derive the associated potential function.