Potential energy:
When an object is placed in some field, then the body possesses some amount of energy by virtue of its position or configuration, this energy is known as potential energy.
A force field is conservative if and only if there exists a continuous differentiable scalar field \( V \) in such a way that \( \vec{F}=-\vec{\nabla}V \) or equivalently \( \vec{\nabla}\times\vec{F}=0 \). Here the scalar fiend \( V \) is called the potential energy.
The total work done in moving a particle by the force acting on the particle from initial point A to final point B is given by,
\( \displaystyle{W=\int_{A}^{B}\vec{F}\cdot{d\vec{r}}} \)
\( or,\ W=\displaystyle{\int_{A}^{B}-\vec{\nabla}V\cdot{d}\vec{r}} \)
\( or,\ W=\displaystyle{\int_{A}^{B}-dV} \)
\( or,\ W=-V|_{A}^{B}=V(A)-V(B) \)
This work done will appear as the potential energy of the particle. Thus the difference in potential energies between two positions A and B of the particle is equal to the work done by the external force in moving the particle from point A to point B.
The potential energy of a particle at a position \( \vec{r} \) is defined as the amount of work done by an applied force in moving the particle from infinity to that position.
\( \displaystyle{V(r)=-\int_{\infty}^r(\vec{F}\cdot{d}\vec{r})} \)
Types of potential energies:
- Gravitational Potential Energy: When two objects interact by the gravitational force of attraction, then the energy associated with the state of separation between these two objects is known as gravitational potential energy. Let us consider two objects of masses \( m_1 \) and \( m_2 \) are seperated by a distance \( r \), then the gravitational potential energy is given by, \( \displaystyle{V=-G\frac{m_1\ m_2}{r}} \), where, \( G \) is the Gravitational constant.
- Electronic Potential Energy: When there is an interaction between two charged particles due to the electric force, then the energy associated with the state of separation between these two charged particles, is known as electronic potential energy. Let us consider two charged particles of charges \( q_1 \) and \( q_2 \) are separated by a distance r, so the potential energy is given by, \( \displaystyle{V=\frac{1}{4\pi\epsilon_0}\frac{q_1\ q_2}{r}} \) ,
- Elastic Potential Energy: When an elastic object (for example spring) is compressed or elongated, then the energy associated with the state of compression or elongation of that object is known as elastic potential energy. Let us consider a spring of force constat \( k \) is elongated along the distance \( x \), then the elastic potential energy is given by, \( \displaystyle{V=\frac{1}{2}kx^2} \).