Electric Potential And Circuit

Electric Potential:

An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration.

It is a scalar quantity.

The difference in electric potential between two point is known as potential difference/voltage.

In Electrostatics:

The electric potential at a point "r" in static electric field E is given by:

$V_\mathbf{E} = - \int_C \mathbf{E} \cdot \mathrm{d} \boldsymbol{\ell} \,$

where C is an arbitary constant.

In this case, the electric field is conservative and determined by the gradient of the potential.

$\mathbf{E} = - \mathbf{\nabla} V_\mathbf{E}. \,$

By Gauss's law

$\mathbf{\nabla} \cdot \mathbf{E} = \mathbf{\nabla} \cdot \left (- \mathbf{\nabla} V_\mathbf{E} \right ) = -\nabla^2 V_\mathbf{E} = \rho / \varepsilon_0, \,$ (poisson's equation)

where,

- ρ is total charge density

-∇· is the divergence.

A test charge q has an electric potential energy U is given by:

$U_ \mathbf{E} = q\,V. \,$

Electric potential due to a point charge:

The electric potential arising from a point charge q at a point distance r from the charge is:

Generalization to electrodynamics:

When time-varying electric field is applied then, it is not possible to describe the electric field simply in a scalar potential V because the electric field is no longer conservative.

\textstyle\int_C \mathbf{E}\cdot \mathrm{d}\boldsymbol{\ell}

is path dependent because $\mathbf{\nabla} \times \mathbf{E} \neq \mathbf{0}$ (Faraday's law of induction)

Instead,

$\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}, \,$

where B is magnetic field.

$\mathbf{F} = \mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}$ is a conservative field by Faraday's law.

Therefore,

$\mathbf{E} = -\mathbf{\nabla}V - \frac{\partial\mathbf{A}}{\partial t}, \,$

where V is scalar potential defined by conservative field F

The SI unit of electric potential is Volt (in honor of Alessandro Volta)

Electric Circuit:

An electrical circuit is a path or line through which an electrical current flows. The path may be closed (joined at both ends), making it a loop. A closed circuit makes electrical current flow possible. It may also be an open circuit where the electron flow is cut short because the path is broken. An open circuit does not allow electrical current to flow.