Question

Consider the vector field F = ax (4y - c1z) + ay (4x + 2z) +az (2y +z) in rectangular coordinated system (x, y, z) with unit vectors ax, ay, and az, If the filed F is irrotational (conservative), then the constant c₁ (in integer) is _____

Answer 1

Explanation:

If a vector F is irrotational, then the curl of F will be zero i.e. \(\nabla\times \vec F = 0\)

The curl of a vector

\(\vec F = \overrightarrow {{F_x}} \widehat {\dot i} + \overrightarrow {{F_y}} \widehat {\dot J} + \overrightarrow {{F_z}} \hat k\)

\(\nabla = \frac{\partial }{{\partial x}}\widehat { i} + \frac{\partial }{{\partial y}}\widehat { j} + \frac{\partial }{{\partial z}}\hat k\)

\(\nabla \times \vec F = Curl\left( {\vec F} \right) \)

\(= \left| {\begin{array}{*{20}{c}} i&{j}&k\\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\ {{F_x}}&{{F_y}}&{{F_z}} \end{array}} \right|\;\)

Calculation:

F = a_x (4y – c₁z) + a_y (4x + 2z) + a_z (2y + z)

\(\vec \nabla \times \vec F = \left| {\begin{array}{*{20}{c}} {{a_x}}&{{a_y}}&{{a_z}}\\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\ {\left( {4y - {c_1}z} \right)}&{\left( {4x + 2z} \right)}&{\left( {2y + z} \right)} \end{array}} \right|\)

\(= {a_x}\left[ {\frac{\partial }{{\partial y}}\left( {2y + z} \right) - \frac{\partial }{{\partial z}}\left( {4x + 2z} \right)} \right] - {a_y}\left[ {\frac{\partial }{{\partial x}}\left( {2y + z} \right) - \frac{\partial }{{\partial z}}\left( {4y - {c_1}z} \right)} \right] + {a_z}\left[ {\frac{\partial }{{\partial x}}\left( {4x + 2z} \right) - \frac{\partial }{{\partial y}}\left( {4y - {c_1}z} \right)} \right]\)

= a_x( 2 – 2) – a_y (0 + c₁) + a_z (4 - 4) = 0

∴  c₁ = 0