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: Recall the definition of a conservative force. A force field \(\vec{F}\) is conservative if its curl is equal to the zero vector. \[ \vec{\nabla} \times \vec{F} = \vec{0} \]
Step 2: Write out the components of the curl. The curl of \(\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}\) is given by the determinant: \[ \vec{\nabla} \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}
F_x & F_y & F_z \end{vmatrix} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\hat{i} - \left(\frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z}\right)\hat{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\hat{k} \]
Step 3: Set the curl to zero and analyze the components. For the curl to be zero, each of its vector components must be zero. - \(\hat{i}\) component: \(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0\). This is 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. Therefore, for a force to be conservative, all three relations A, B, and C must be satisfied. Statement D describes a non-conservative force.