What Is Conservative Force Field?

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Conservative force field:

Let us consider a particle is acted upon by a force \( \vec{F} \), as a result the particle is displaced by a distance \( d\vec{r} \). Then the work done by the force on the particle is given by,

\( dW=\vec{F}\cdot{d\vec{r}} \)

Since the work done is the scalar product of two vectors force and displacement, so the work done is a scalar quantity.

Fig.1

Now the total work done in moving the particle from the point \( A \)to point \( B \) along the curve \( C \) by the force \( \vec{F} \) is given by,

\( W=\displaystyle{\int_c}\left(\vec{F}\cdot{d\vec{r}}\right) \)

\( or,\ W=\displaystyle{\int_A^B}\left(\vec{F}\cdot{d\vec{r}}\right) \)

Now if the position vector of the points \( A \) and \( B \) are \( \vec{r_1} \) and \( \vec{r_2} \) respectively, then the work done by the force \( \vec{F} \) will be,

\( W=\displaystyle{\int_{r_1}^{r_2}}\left(\vec{F}\cdot{d\vec{r}}\right) \)

Now let us consider the force \( \vec{F} \) acting on the particle is given by,

\( \vec{F}=- \vec{\nabla}{V} \),
where V is the potential energy.

Now the total work done can be written as,

\( W=\displaystyle{\int_A^B}\left(\vec{F}\cdot{d}\vec{r}\right) \)

\( or,\ W=\displaystyle{\int_A^B}\left(-\vec{\nabla}{V}\cdot{d}\vec{r}\right) \)

Again, \( \vec{\nabla}{V}\cdot{d}\vec{r} \)

\( =\left(\displaystyle{\frac{\delta{V}}{\delta{x}}\hat{i}+\frac{\delta{V}}{\delta{y}}\hat{j}+\frac{\delta{V}}{\delta{z}}\hat{k}}\right)\cdot\left(dx\hat{i}+dy\hat{j}+dz\hat{k}\right) \) \( =\displaystyle{\frac{\delta{v}}{\delta{x}}dx+\frac{\delta{v}}{\delta{y}}dy+\frac{\delta{v}}{\delta{z}}dz} \) \( =dV \)

Therefore the work done is,

\( W=-\displaystyle{\int_A^B}dV=V_A-V_B \)

From the above equation, we can say that the work done is independent of the path joining the two points: initial point A and final point B, but depends only on the potential function \( V \) at the initial position A and final position B.

  1. Therefore the force \( \vec{F} \) acting on the particle is said to conservative if the work done by the force in moving the particle from the initial position to the final position is independent of the path but only depends on the initial and the final position of the particle.
  2. A force field \( \vec{F} \) is said to be conservative if and only if there exist a scalar field V which is continuous and differentiable, in such a way that \( \vec{F}=-\vec{\nabla}V \).
  3. Again, \( curl\ \vec{F}=\vec{\nabla}\times\vec{F} \) Now putting the value of \( \vec{F}=-\vec{\nabla}V \), we get \( \vec{\nabla}\times\vec{F}=-\vec{\nabla}\times\vec{\nabla}V =0 \). Therefore, a force \( \vec{F} \) acting on a particle is said to be conservative force if and only if the curl of that force is zero.
  4. A force \( \vec{F} \) is said to be conservative force the closed path integral of the force is zero, i.e. \( \displaystyle{\oint_{C}\vec{F}\cdot{d\vec{r}}=0} \). Here the force field is continuously differentiable and \( C \) is the non-intersecting closed curve.

Examples of Conservative Forces:

  • Gravitational force between any two masses.
  • Coulomb force between any two stationary charges.
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