6.1 Scalar potential

Scalar potential is a fundamental concept in electromagnetism, describing the potential energy per unit charge at each point in an electric field. It's crucial for understanding electric fields, as the negative gradient of scalar potential gives the electric field vector.

This topic connects electric fields, potential energy, and work done by electric forces. It introduces important equations like Laplace's and Poisson's, which are essential for solving complex electromagnetic problems and analyzing charge distributions in various systems.

Definition of scalar potential

• Scalar potential, denoted as $\phi$, is a scalar field that describes the potential at each point in an electric field
• Defined as the work per unit charge required to move a positive test charge from infinity to a specific point in the electric field
• Measured in units of volts (V), where 1 volt = 1 joule/coulomb

Relationship between electric field and scalar potential

• The electric field $\vec{E}$ is the negative gradient of the scalar potential $\phi$: $\vec{E} = -\nabla \phi$
• This relationship allows the electric field to be calculated from the scalar potential and vice versa
• The negative sign indicates that the electric field points in the direction of decreasing potential
• In regions where the electric field is stronger, the scalar potential changes more rapidly

Calculation of scalar potential from electric field

Line integral method

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• The scalar potential difference $\Delta \phi$ between two points can be calculated by integrating the electric field along a path connecting the points: $\Delta \phi = -\int \vec{E} \cdot d\vec{l}$
• The line integral is independent of the path chosen, as long as the endpoints remain the same
• This method is useful when the electric field is known and the potential difference between two points is needed

Gradient operator

• The gradient operator $\nabla$ is used to calculate the electric field from the scalar potential: $\vec{E} = -\nabla \phi$
• In Cartesian coordinates, the gradient is given by $\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}$
• The gradient points in the direction of the greatest rate of increase of the scalar potential
• This method is useful when the scalar potential is known, and the electric field needs to be calculated

Electric potential energy

Definition and formula

• Electric potential energy $U$ is the energy stored in a system due to the configuration of charges in an electric field
• The potential energy of a charge $q$ at a point with scalar potential $\phi$ is given by $U = q\phi$
• The potential energy is measured in joules (J)

Relationship to work and scalar potential

• The work $W$ done by the electric field on a charge $q$ moving from a point with potential $\phi_1$ to a point with potential $\phi_2$ is equal to the negative change in potential energy: $W = -\Delta U = -q(\phi_2 - \phi_1)$
• The work done by the electric field is independent of the path taken, as it depends only on the potential difference between the initial and final points
• The scalar potential is the potential energy per unit charge: $\phi = U/q$

Equipotential surfaces

Definition and properties

• An equipotential surface is a surface on which all points have the same scalar potential
• The electric field is always perpendicular to the equipotential surface at every point
• No work is done by the electric field when a charge moves along an equipotential surface

Relationship to electric field lines

• Electric field lines are always perpendicular to equipotential surfaces
• The density of electric field lines is proportional to the magnitude of the electric field
• In regions where equipotential surfaces are closely spaced, the electric field is stronger, and the field lines are more dense

Boundary conditions for scalar potential

Conductor surfaces

• The electric field inside a conductor is zero at electrostatic equilibrium
• The scalar potential is constant throughout the conductor and on its surface
• The electric field just outside the conductor surface is perpendicular to the surface

Dielectric interfaces

• At the interface between two dielectric materials with permittivities $\epsilon_1$ and $\epsilon_2$, the normal component of the electric field satisfies: $\epsilon_1 E_{1n} = \epsilon_2 E_{2n}$
• The tangential component of the electric field is continuous across the interface: $E_{1t} = E_{2t}$
• The scalar potential is continuous across the dielectric interface

Laplace's equation for scalar potential

Derivation in free space

• In a region with no charges (free space), the electric field satisfies $\nabla \cdot \vec{E} = 0$ (Gauss's law)
• Combining this with the relationship between electric field and scalar potential $(\vec{E} = -\nabla \phi)$ leads to Laplace's equation: $\nabla^2 \phi = 0$
• Laplace's equation states that the sum of the second partial derivatives of the scalar potential in all directions is zero

Solutions in different coordinate systems

• In Cartesian coordinates, Laplace's equation is given by $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$
• In cylindrical coordinates $(r, \theta, z)$, Laplace's equation takes the form $\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \phi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$
• In spherical coordinates $(r, \theta, \phi)$, Laplace's equation is given by $\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \phi}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial \phi}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 \phi}{\partial \phi^2} = 0$

Poisson's equation for scalar potential

Derivation with source charges

• In the presence of a charge density $\rho$, Gauss's law becomes $\nabla \cdot \vec{E} = \rho/\epsilon_0$
• Combining this with the relationship between electric field and scalar potential leads to Poisson's equation: $\nabla^2 \phi = -\rho/\epsilon_0$
• Poisson's equation relates the scalar potential to the charge density distribution

Green's function method for solutions

• The Green's function $G(\vec{r}, \vec{r}')$ is a solution to Poisson's equation for a point charge located at $\vec{r}'$
• The scalar potential can be expressed as a superposition of Green's functions weighted by the charge density: $\phi(\vec{r}) = \frac{1}{4\pi\epsilon_0}\int G(\vec{r}, \vec{r}')\rho(\vec{r}')d^3r'$
• In free space, the Green's function is given by $G(\vec{r}, \vec{r}') = \frac{1}{|\vec{r} - \vec{r}'|}$

Multipole expansion of scalar potential

Monopole, dipole, and quadrupole terms

• The scalar potential can be expanded in a series of terms with increasing complexity, known as the multipole expansion
• The monopole term represents the potential due to a single point charge and varies as $1/r$
• The dipole term represents the potential due to a pair of equal and opposite charges, and varies as $1/r^2$
• The quadrupole term represents the potential due to a configuration of four charges arranged in a square, and varies as $1/r^3$

Far-field approximations

• In the far-field region, where the distance from the charge distribution is much larger than the size of the distribution, the higher-order terms in the multipole expansion become negligible
• The monopole term dominates the far-field potential for a net charge, while the dipole term dominates for a neutral charge distribution
• Far-field approximations simplify the calculation of the scalar potential and electric field for complex charge distributions

Scalar potential in electrostatic systems

Capacitors and capacitance

• A capacitor is a device that stores electric potential energy in an electric field between two conducting plates
• The capacitance $C$ is the ratio of the charge $Q$ stored on the plates to the potential difference $\Delta V$ between them: $C = Q/\Delta V$
• The capacitance depends on the geometry of the plates and the dielectric material between them

Charge distributions and Coulomb's law

• The scalar potential due to a discrete charge distribution can be calculated using Coulomb's law: $\phi(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{|\vec{r} - \vec{r}_i|}$
• For a continuous charge distribution with density $\rho(\vec{r}')$, the scalar potential is given by $\phi(\vec{r}) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|}d^3r'$
• Coulomb's law and the superposition principle allow the calculation of the scalar potential for complex charge distributions

Scalar potential in time-varying fields

Electrodynamic potentials

• In time-varying fields, the scalar potential $\phi$ and the vector potential $\vec{A}$ are used to describe the electric and magnetic fields
• The electric field is given by $\vec{E} = -\nabla \phi - \frac{\partial \vec{A}}{\partial t}$, which includes a term for the time-varying magnetic field
• The magnetic field is related to the vector potential by $\vec{B} = \nabla \times \vec{A}$

Retarded potentials and Liénard-Wiechert potentials

• In time-varying fields, the potentials at a point $\vec{r}$ and time $t$ depend on the charge distribution at an earlier time $t' = t - |\vec{r} - \vec{r}'|/c$, known as the retarded time
• The retarded scalar potential is given by $\phi(\vec{r}, t) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho(\vec{r}', t')}{|\vec{r} - \vec{r}'|}d^3r'$
• The Liénard-Wiechert potentials describe the scalar and vector potentials due to a moving point charge, taking into account the retarded time and relativistic effects