Question

For a force F to be conservative, the relations to be satisfied are:
A. \(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0\)
B. \(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0\)
C. \(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = 0\)
D. \(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \neq 0\)
Choose the correct answer from the options given below:

• A. A and B only
• B. A, B and C only
• C. B, C and D only
• D. A, B, C and D only

Hint:

A force is conservative if it can be written as the gradient of a scalar potential, \(\vec{F} = -\vec{\nabla}V\). The condition \(\vec{\nabla} \times \vec{F} = 0\) is equivalent to this, due to the identity \(\vec{\nabla} \times (\vec{\nabla}V) = 0\).

Answer 1

The Correct Option is B

Step 1: Define a conservative force. A force field \(\vec{F}\) is conservative if its curl is the zero vector: \(\vec{abla} \times \vec{F} = \vec{0}\).
Step 2: Express the curl of \(\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}\) using the determinant: \[\vec{abla} \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\hat{i} - \left(\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z}\right)\hat{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\hat{k}\]
Step 3: Equate the curl to zero and examine its components. For the curl to be zero, each component must be zero:
- \(\hat{i}\) component: \(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0\). This corresponds to statement B.
- \(\hat{j}\) component: \(-\left(\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z}\right) = 0 \implies \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = 0\). This is statement C.
- \(\hat{k}\) component: \(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0\). This is statement A.

Consequently, for a force to be conservative, all three conditions A, B, and C must hold. Statement D describes a non-conservative force.