Understanding Conservative Forces

A force is considered conservative if the work done by it in moving an object between two points is independent of the path taken. A key characteristic of conservative forces is that the work done over any closed path is zero. Mathematically, a force field $\vec{F}$ represented as $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ is conservative if its curl is equal to the zero vector, i.e., $\nabla \times \vec{F} = \vec{0}$.

The Curl Condition Explained

The curl of a vector field $\vec{F}$ in three dimensions is calculated using a determinant:

$ \nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} $

When this determinant is expanded, we get:

$ \nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \hat{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \hat{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \hat{k} $

For the force $\vec{F}$ to be conservative, the result of the curl operation must be the zero vector ($\vec{0}$). This implies that each component of the curl must be equal to zero:

• The $\hat{i}$ component must be zero: $\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0$
• The $\hat{j}$ component must be zero: $\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = 0$
• The $\hat{k}$ component must be zero: $\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0$

Analyzing the Options

Now, let's relate these conditions to the given options:

• Option A: $\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0$. This condition corresponds exactly to the $\hat{k}$ component of the curl being zero. Hence, it is a requirement for a conservative force.
• Option B: $\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0$. This condition corresponds exactly to the $\hat{i}$ component of the curl being zero. Hence, it is also a requirement for a conservative force.
• Option C: $\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = 0$. This condition corresponds exactly to the $\hat{j}$ component of the curl being zero. Hence, it is also a requirement for a conservative force.
• Option D: $\frac{\partial F_y} {\partial x} - \frac{\partial F_x} {\partial y} = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \ne 0$. This option states that the components of the curl are equal but non-zero. For a force to be conservative, all components of the curl *must* be zero. Therefore, this condition is incorrect for conservative forces.

Conclusion

For a force $\vec{F}$ to be conservative, its curl must be zero. This requires that all three partial derivative conditions represented by options A, B, and C must hold true simultaneously. Option D describes a scenario where the curl components are equal but not zero, which is not the condition for a conservative force.

Therefore, the correct relations are A, B, and C only.