Electrostatics deals with the study of electric charges at rest. One of the key concepts in this domain is electric potential, which is closely linked with the idea of potential energy and the work done by electric forces.
When two point charges interact, they possess potential energy due to their positions in the electric field. This energy is called electrostatic potential energy. It is defined as the energy required to bring a charge from infinity to a given point in the presence of another charge.
The work done in moving a charge in an electric field is stored as potential energy. Mathematically, this is given by:
\( W = Potential Energy = U \)
For two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \), the electrostatic potential energy is:
\( U = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{r} \)
Electric potential at a point is defined as the work done per unit positive test charge in bringing the charge from infinity to that point, without acceleration.
Mathematically:
\( V = \frac{U}{q} = \frac{Electric \,Potential\,Energy}{Charge} \)
The electric potential created by a point charge \( q \) at a distance \( r \) is expressed as:
\( V = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q}{r} \)
Here,
Electric field and electric potential are closely related. The electric field is the negative gradient of electric potential:
\[ \vec{E} = -\nabla V \]
In one dimension:
\[ E = -\frac{dV}{dr} \]
Electric potential is a scalar quantity, so the total potential due to multiple charges is the algebraic sum of the potentials due to individual charges:
\( V_{total} = \sum_i \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_i}{r_i} \)
Understanding these relationships helps in solving problems involving work done by electric fields, potential differences, and energy conservation in electrostatics.
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