5.1 Electrostatic potential

Electrostatic potential energy is the work done to assemble charges against electrostatic forces. It's measured in joules and can be calculated for point charges or charges in electric fields. Understanding this concept is key to grasping electrostatic interactions.

Electric potential, measured in volts, is the potential energy per unit charge at a point in an electric field. It's a scalar quantity related to the electric field through the gradient operator. Calculating electric potential for various charge distributions is crucial in electrostatics.

Electrostatic potential energy

• Electrostatic potential energy represents the work done to assemble a system of charges against electrostatic forces
• Potential energy is a scalar quantity measured in joules (J)

Potential energy of point charges

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• Potential energy between two point charges $q_1$ and $q_2$ separated by a distance $r$ is given by $U = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}$
  • $\epsilon_0$ is the permittivity of free space
• Potential energy is positive for like charges (repulsive) and negative for opposite charges (attractive)
• Potential energy approaches zero as the separation distance approaches infinity

Potential energy in electric fields

• Potential energy of a charge $q$ in an electric field $\vec{E}$ is given by $U = q\phi$, where $\phi$ is the electric potential
• Work done to move a charge from one point to another in an electric field equals the change in potential energy
  • $W = -\Delta U = -q\Delta\phi$
• Potential energy is independent of the path taken between two points in an electric field

Electric potential

• Electric potential, denoted by $\phi$, is the potential energy per unit charge at a point in an electric field
  • Measured in volts (V), where 1 volt = 1 joule/coulomb

Electric potential vs electric field

• Electric potential is a scalar quantity, while electric field is a vector quantity
• Electric field represents the force per unit charge, while electric potential represents the potential energy per unit charge
• The negative gradient of electric potential gives the electric field: $\vec{E} = -\nabla\phi$

Electric potential due to point charges

• Electric potential due to a point charge $q$ at a distance $r$ is given by $\phi = \frac{1}{4\pi\epsilon_0}\frac{q}{r}$
• Principle of superposition allows for calculating the total electric potential due to multiple point charges
  • $\phi_{total} = \sum_{i}\frac{1}{4\pi\epsilon_0}\frac{q_i}{r_i}$

Equipotential surfaces

• An equipotential surface is a surface on which all points have the same electric potential
• Electric field lines are always perpendicular to equipotential surfaces
• No work is done when moving a charge along an equipotential surface

Calculating electric potential

• Various methods exist for calculating electric potential, depending on the charge distribution and symmetry of the system

Electric potential of charge distributions

• Electric potential due to a continuous charge distribution is given by $\phi = \frac{1}{4\pi\epsilon_0}\int\frac{dq}{r}$
  • $dq$ is the charge element and $r$ is the distance from the charge element to the point of interest
• Examples of charge distributions include line charges, surface charges, and volume charges

Deriving electric field from potential

• Electric field can be derived from electric potential using the gradient operator: $\vec{E} = -\nabla\phi$
  • In Cartesian coordinates, $\vec{E} = -\left(\frac{\partial\phi}{\partial x}\hat{x} + \frac{\partial\phi}{\partial y}\hat{y} + \frac{\partial\phi}{\partial z}\hat{z}\right)$
• Negative sign indicates that the electric field points in the direction of decreasing potential

Laplace's equation in electrostatics

• Laplace's equation, $\nabla^2\phi = 0$, describes the electric potential in a region with no charges
  • $\nabla^2$ is the Laplacian operator
• Poisson's equation, $\nabla^2\phi = -\frac{\rho}{\epsilon_0}$, describes the electric potential in a region with charge density $\rho$
• Solutions to Laplace's and Poisson's equations provide the electric potential for various boundary conditions

Electrostatic potential applications

• Electrostatic potential plays a crucial role in various applications, such as capacitors, electrostatic shielding, and dielectrics

Capacitance and electric potential energy

• Capacitance is the ability of a system to store electric charge and is defined as $C = \frac{Q}{V}$, where $Q$ is the charge and $V$ is the potential difference
• Electric potential energy stored in a capacitor is given by $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C}$
  • Examples of capacitors include parallel plate capacitors and spherical capacitors

Electrostatic shielding

• Electrostatic shielding involves using conducting surfaces to protect a region from external electric fields
• Electric potential inside a conducting shell is constant, regardless of the external electric field
  • Faraday cages are an example of electrostatic shielding

Dielectrics in capacitors

• Dielectrics are insulating materials that can be polarized in the presence of an electric field
• Inserting a dielectric between the plates of a capacitor increases its capacitance by a factor of $\kappa$, the dielectric constant
  • $C = \kappa C_0$, where $C_0$ is the capacitance without the dielectric
• Dielectrics reduce the electric field inside the capacitor and increase the maximum voltage that can be applied before breakdown occurs

Electrostatic potential problems

• Solving electrostatic potential problems often involves applying Gauss's law, the method of images, and boundary value problems

Gauss's law and electric potential

• Gauss's law relates the electric flux through a closed surface to the charge enclosed: $\oint\vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
• Applying Gauss's law to solve for the electric field and then integrating to find the electric potential is a common problem-solving technique
  • Symmetry (spherical, cylindrical, or planar) simplifies the application of Gauss's law

Method of images

• The method of images is a problem-solving technique that replaces certain boundary conditions with imaginary charges
• Imaginary charges are placed to satisfy the boundary conditions and simplify the calculation of electric potential
  • Examples include point charges near conducting planes or spheres

Boundary value problems in electrostatics

• Boundary value problems involve solving for the electric potential given the potential or its derivative on the boundaries of a region
• Common boundary conditions include Dirichlet (fixed potential) and Neumann (fixed derivative of potential) conditions
• Techniques for solving boundary value problems include separation of variables, Fourier series, and numerical methods