electric potential

Examples of electric potential in the following topics:

• Stokes' Theorem

  • Therefore, electric field can be written as a gradient of a scalar field:
  • which is equivalent to saying that work done by the electric field only depends on the initial and final point of the motion.
  • The scalar field $\varphi$ in the case of electromagnetism is called electric potential.

• Fundamental Theorem for Line Integrals

  • By placing $\varphi$ as potential, $\nabla$ is a conservative field.
  • Electric field lines emanating from a point where positive electric charge is suspended over a negatively charged infinite sheet.
  • Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.
  • Therefore, electric force is a conservative force.

• The Chain Rule

  • $U$ could be electric potential energy at a location $(x,y)$.
  • What we want to calculate is the rate of change of the potential energy $U$ as a function of time $t$.

• Applications of Multiple Integrals

  • The gravitational potential associated with a mass distribution given by a mass measure $dm$ on three-dimensional Euclidean space $R^3$ is:
  • If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:
  • In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total magnetic and electric fields.
  • In the following example, the electric field produced by a distribution of charges given by the volume charge density $\rho (\vec r)$ is obtained by a triple integral of a vector function:

• Surface Integrals of Vector Fields

  • This applies, for example, in the expression of the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material.
  • An electric field from a point charge ($Q$) is given as:
  • If the charge is located at the center of a sphere with a radius $R$, the surface integral of the electric field over the surface is calculated at the following:

• Cylindrical and Spherical Coordinates

  • Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis, such as water flow in a straight pipe with a round cross-section, heat distribution in a metal cylinder, electromagnetic fields produced by an electric current in a long, straight wire, and so on.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.

• The Divergence Theorem

  • The first equation of the Maxwell's equations is often written as $\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$ in a differential form, where $\rho$ is the electric density.

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • When the above equation holds, $\varphi$ is called a scalar potential for $\mathbf{v}$.

• Power Series

  • These power series arise primarily in real and complex analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the $Z$-transform).

• Line Integrals

  • The line integral finds the work done on an object moving through an electric or gravitational field, for example.