Evaluate The Work Done Between Point 1 And Point 2 For The Conservative Field F. F= (y+z)i + Xj + Xk
Understanding how to evaluate work done by a vector field between two points is a fundamental concept in vector calculus, particularly in the study of conservative fields. In this article, we will thoroughly analyze the work done by the vector field F = (y + z)i + xj + xk as it moves from Point 1 to Point 2. We will explore the properties of the field, verify whether it is conservative, and then compute the work done accordingly.
Understanding the Vector Field F
Before diving into the calculations, it is essential to understand the given vector field:
F = (y + z)i + xj + xk
This means that at any point (x, y, z) in space, the vector field assigns a vector:
- In the x-direction: (y + z)
- In the y-direction: x
- In the z-direction: x
This field combines variables in a way that suggests the potential for a scalar potential function, which would classify it as a conservative field.
Verifying Whether F is a Conservative Field
A key step in evaluating the work done by a vector field between two points is to determine if the field is conservative. A conservative field F is one where there exists a scalar potential function ϕ(x, y, z) such that:
F = ∇ϕ = (∂ϕ/∂x)i + (∂ϕ/∂y)j + (∂ϕ/∂z)k
If F is conservative, the work done moving from one point to another is path-independent and can be computed as the difference in potential function values at those points.
Step 1: Check Curl of F
For a vector field to be conservative (in simply connected regions), its curl must be zero:
∇ × F = 0
Calculate the curl:
\[
\nabla \times F = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
y + z & x & x
\end{vmatrix}
\]
Compute each component:
- i-component:
\[
\frac{\partial}{\partial y} (x) - \frac{\partial}{\partial z} (x) = 0 - 0 = 0
\]
- j-component:
\[
\frac{\partial}{\partial z} (y + z) - \frac{\partial}{\partial x} (x) = 1 - 1 = 0
\]
- k-component:
\[
\frac{\partial}{\partial x} (x) - \frac{\partial}{\partial y} (y + z) = 1 - 1 = 0
\]
Since all components are zero, ∇ × F = 0, confirming that F is a conservative field in a simply connected region.
Conclusion: The vector field F is conservative. Therefore, there exists a potential function ϕ(x, y, z) such that F = ∇ϕ.
Finding the Scalar Potential Function ϕ(x, y, z)
Given:
\[
\nabla \varphi = \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right) = (y + z, x, x)
\]
We now solve for ϕ:
Step 1: Integrate with respect to x:
\[
\frac{\partial \varphi}{\partial x} = y + z
\]
Integrate:
\[
\varphi(x, y, z) = x(y + z) + C(y, z)
\]
where C(y, z) is an arbitrary function of y and z.
Step 2: Find partial derivatives with respect to y and z to determine C(y, z):
- From the ∂ϕ/∂y component:
\[
\frac{\partial \varphi}{\partial y} = x + \frac{\partial C}{\partial y}
\]
But from the vector field:
\[
\frac{\partial \varphi}{\partial y} = x
\]
Set equal:
\[
x + \frac{\partial C}{\partial y} = x \Rightarrow \frac{\partial C}{\partial y} = 0
\]
Thus, C(y, z) does not depend on y; it is a function of z only:
\[
C(y, z) = C(z)
\]
- From the ∂ϕ/∂z component:
\[
\frac{\partial \varphi}{\partial z} = x + \frac{d C(z)}{d z}
\]
From the vector field:
\[
\frac{\partial \varphi}{\partial z} = x
\]
Set equal:
\[
x + \frac{d C(z)}{d z} = x \Rightarrow \frac{d C(z)}{d z} = 0
\]
Therefore, C(z) is a constant; without loss of generality, we can set it to zero.
Final potential function:
\[
\boxed{
\varphi(x, y, z) = x(y + z)
}
\]
---
Calculating the Work Done Between Two Points
Since F is conservative, the work done W moving from Point 1 (x₁, y₁, z₁) to Point 2 (x₂, y₂, z₂) is:
\[
W = \varphi(x2, y2, z2) - \varphi(x1, y1, z1)
\]
Step 1: Identify Points
Suppose the points are:
- Point 1: \((x1, y1, z_1)\)
- Point 2: \((x2, y2, z_2)\)
If specific points are given, substitute their coordinates into the potential function to find the work.
Step 2: Compute the difference
\[
W = x2(y2 + z2) - x1(y1 + z1)
\]
This expression gives the total work done by the field F in moving an object from Point 1 to Point 2.
---
Practical Example: Calculating Work with Specific Points
Suppose:
- Point 1: \((1, 2, 3)\)
- Point 2: \((4, 5, 6)\)
Calculate:
\[
W = 4(5 + 6) - 1(2 + 3) = 4 \times 11 - 1 \times 5 = 44 - 5 = 39
\]
Thus, the work done by the field F in moving from \((1, 2, 3)\) to \((4, 5, 6)\) is 39 units.
---
Summary and Final Thoughts
Evaluating the work done by a vector field between two points is straightforward once the field is identified as conservative. In our case, the key steps involved:
- Verifying the conservative nature of F via curl calculation.
- Finding the potential function ϕ(x, y, z).
- Applying the potential difference formula to compute work.
Since F is conservative, the work is path-independent and depends solely on the initial and final points. This property simplifies many calculations in physics and engineering, especially in fields like electrostatics, gravity, and fluid flow.
Whether dealing with theoretical problems or practical applications, understanding the process of evaluating work in conservative vector fields enhances problem-solving efficiency and deepens comprehension of the underlying physics.
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Additional Tips for Students and Practitioners:
- Always verify if a vector field is conservative before attempting to compute work; use curl as a quick test.
- When finding the potential function, integrate component-wise carefully, and determine any integration functions by matching partial derivatives.
- Remember that in non-conservative fields, work depends on the path taken, requiring line integral calculations.
By mastering these steps and concepts, you can confidently evaluate work in a variety of vector fields, streamlining your analysis of physical systems.
