Question

There are two fields of forces which are represented by $\vec F = ay\hat i$ and $\vec F = ax\hat i + by\hat j$ where $a$ and $b$ are constants. Find out whether these forces are conservative or not.

Answer 1

In vector algebra, when a force is represented in the form of $\vec F = P\hat i + Q\hat j$ it means the components of force vector in X and Y directions are $P$ and $Q$ respectively. A conservative force is one whose work done on a body is independent of the path taken.

Complete step by step answer:
We know that, if a force is conservative then its work done must be independent of path and hence, if a force field $\vec F$ is conservative its curl is always zero which means $\vec \nabla \times \vec F = 0$.Let us check the first given force field $\vec F = ay\hat i$ as its $Y$ and $Z$ components are zero.

Checking the $\vec \nabla \times \vec F = 0$ for force $\vec F = ay\hat i$ we get,
\vec \nabla \times \vec F = \left( {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\\ {\dfrac{\partial }{{\partial x}}}&{\dfrac{\partial }{{\partial y}}}&{\dfrac{\partial }{{\partial z}}} \\\ {ay}&0&0 \end{array}} \right)
Finding determinant of above matrix we get,
$\vec \nabla \times \vec F = 0 + ( - a\hat k)$
$\therefore \vec \nabla \times \vec F = - a\hat k$
Hence, the value of $\vec \nabla \times \vec F \ne 0$ for the given force $\vec F = ay\hat i$.Hence, force $\vec F = ay\hat i$ is not a conservative force.

Similarly, Let us check for the force $\vec F = ax\hat i + by\hat j$
We have,
\vec \nabla \times \vec F = \left( {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\\ {\dfrac{\partial }{{\partial x}}}&{\dfrac{\partial }{{\partial y}}}&{\dfrac{\partial }{{\partial z}}} \\\ {ax}&{by}&0 \end{array}} \right)
Again, finding determinant of above matrix, we have
$\vec \nabla \times \vec F = 0 + 0 + 0$
$\therefore \vec \nabla \times \vec F = 0$
Hence, the value of $\vec \nabla \times \vec F = 0$ for the given force $\vec F = ax\hat i + by\hat j$. Hence, the force $\vec F = ax\hat i + by\hat j$ is a conservative force.

So, force $\vec F = ax\hat i + by\hat j$ is a conservative force while force $\vec F = ay\hat i$ is not a conservative force.

Note: Remember the operator $\vec \nabla$ is known as del operator and its defined as $\dfrac{\partial }{{\partial x}}\hat i + \dfrac{\partial }{{\partial y}}\hat j + \dfrac{\partial }{{\partial z}}\hat k$ and this del operator is widely used in differential calculus in order to find the divergence and curl of a given field the term $\vec \nabla \times \vec F$ for any vector field $\vec F$ is known as curls of the field and if this value is zero it’s said to be vector field is irrotational.