Notes from toppers on the topic of Field Due To Dipole And Continuous Charge Distributions

1. Electric Field Due to a Dipole:

• Definition and concept of a dipole:

  • A dipole consists of two equal and opposite charges separated by a small distance.
  • The dipole moment is a measure of the strength of the dipole and is defined as the product of the magnitude of one of the charges and the distance between the charges.

• Calculation of electric field due to a dipole at various points:

  • The electric field due to a dipole at a point on the dipole axis is given by:

$$|\overrightarrow{E}|=\frac{1}{4\pi {\epsilon }_0}\frac{2qs}{r^3}$$

• The electric field due to a dipole at a point perpendicular to the dipole axis is given by:

$$|\overrightarrow{E}|=\frac{1}{4\pi {\epsilon }_0}\frac{qs}{r^3}$$

• Electric potential due to a dipole:
  • The electric potential due to a dipole at a point is given by

$$\varphi =\frac{1}{4\pi {ϵ}_0}\frac{2qs}{r}\cos \theta$$

2. Field Due to a Continuous Charge Distribution:

• Linear charge distribution:
  • The electric field due to a thin, uniformly charged rod of length L and total charge Q is given by:

$$|\overrightarrow{E}|=\frac{1}{4\pi {\epsilon }_0}\frac{2Q}{L}\ln \left(\frac{x+L}{x}\right)$$

• The electric potential due to a thin, uniformly charged rod is given by:

$$V=\frac{1}{4\pi {\epsilon }_0}\frac{2Q}{L}\ln \left(\frac{x+L}{x}\right)$$

• Surface charge distribution:
  • The electric field due to a uniformly charged flat disk of radius R and surface charge density $\sigma$ is given by:

$$|\overrightarrow{E}|=\frac{1}{4\pi {\epsilon }_0}\frac{\sigma }{2}(1+\frac{z^2}{R^2})$$

• The electric potential due to a uniformly charged flat disk is given by:

$$V=\frac{1}{4\pi {ϵ}_0}\sigma (z+\sqrt{z^2+R^2})$$

• Volume charge distribution:
  • The electric field due to a uniformly charged sphere of radius R and total charge Q is given by:

$$|\overrightarrow{E}|=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}\text{ for }r>R$$

$$|\overrightarrow{E}|=\frac{1}{4\pi {\epsilon }_0}\frac{Qr}{R^3}\text{ for }r<R$$

• The electric potential due to a uniformly charged sphere is given by:

$$V=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}\text{ for }r>R$$

$$V=\frac{1}{4\pi {ϵ}_0}\frac{1}{2}\frac{Qr}{R^2}\text{ for }r<R$$

3. Gauss’s Law:

• Statement and mathematical form of Gauss’s law:
  • Gauss’s law states that the total electric flux through a closed surface is proportional to the enclosed charge.
  • Mathematically, Gauss’s law is expressed as:

$$\oint \overrightarrow{E}\cdot \hat{n}dA=\frac{Q_{enc}}{{\epsilon }_0}$$

• Where $\overrightarrow{E}$ is the electric field, $\hat{n}$ is the unit normal vector perpendicular to the surface, dA is the differential area element, $Q_{enc}$ is the total charge enclosed by the surface, and ${ϵ}_0$ is the permittivity of free space.

• Applications of Gauss’s law to calculate electric fields:

  • Gauss’s law can be used to calculate the electric field of a uniformly charged sphere, a conducting sphere, and a charged infinite plane.

• Use of Gauss’s law to determine the charge enclosed by a surface:

  • Gauss’s law can be used to determine the charge enclosed by a surface by calculating the total electric flux through the surface.

4. Applications of Electrostatics:

• Electrostatic potential energy
  • The electrostatic potential energy of a system of point charges is given by

$$U_e=\frac{1}{4\pi {\epsilon }_0}\sum _{i=1}^N\sum _{j=i+1}^N\frac{q_i{q}_j}{r_{ij}}$$

• Capacitance and capacitors

  • Capacitance is the ability of a system to store electrical charge.
  • The capacitance of a capacitor is given by $$C=\frac{Q}{V}$$

• Dielectrics and polarization

  • Dielectric materials are non-conducting materials that can be polarized when placed in an electric field.
  • Polarization is the process by which the charges within a dielectric material are separated when subjected to an electric field.

• Energy stored in a capacitor

• The energy stored in a capacitor is given by $$U_e=\frac{1}{2}QV=\frac{1}{2}C{V}^2=\frac{Q^2}{2C}$$

• Electrostatic machines

• Electrostatic machines are devices that use electrostatic principles to generate high voltages.

• Examples of electrostatic machines include the Van de Graaff generator and the Wimshurst machine.

5. Electrostatic Potential:

• Definition and concept of electrostatic potential:

  • Electrostatic potential at a point is the amount of electrical potential energy per unit charge at that point due to the presence of charges in the vicinity.

• Calculation of electrostatic potential due to point charges, dipoles, and continuous charge distributions:

  • The electrostatic potential due to a point charge Q at a distance r is given by

$$\varphi =\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r}$$

• The electrostatic potential due to a dipole at a distance r and an angle $\theta$ with the dipole moment $\overrightarrow{p}$ is given by

$$\varphi =\frac{1}{4\pi {\epsilon }_0}\frac{1}{r^2}(\overrightarrow{p}\cdot \hat{r})$$

• The electrostatic potential due to a continuous charge distribution can be calculated using integration.

• Equipotential surfaces and their properties:

  • Equipotential surfaces are surfaces where the electrostatic potential is constant.
  • Equipotential surfaces are always perpendicular to the electric field lines.
  • No work is done in moving a charge along an equipotential surface.

• Relation between electric field and electrostatic potential:

  • The electric field is the negative gradient of the electrostatic potential, i.e.,

$$\overrightarrow{E}=-\mathrm{\nabla }\varphi$$

6. Boundary Conditions:

• Boundary conditions for electric field and electric potential at interfaces:
  • The tangential component of the electric field is continuous across a boundary between two dielectrics.
  • The normal component of the electric displacement field is continuous across a boundary between two dielectrics.
  • The electric potential is continuous across a boundary between two dielectrics.

7. Method of Images:

• Principle and applications of the method of images:
  • The method of images is a technique used to solve electrostatics problems involving conducting surfaces.
  • The method of images involves placing imaginary charges in such a way that the boundary conditions are satisfied.

8. Laplace’s Equation:

• Definition and properties of Laplace’s equation:
  • Laplace’s equation is a second-order partial differential equation that is satisfied by the electrostatic potential in a region free of charges.

$${\mathrm{\nabla }}^2\varphi =0$$

• Solutions to Laplace’s equation in different coordinate systems:
  • Laplace’s equation can be solved in different coordinate systems using separation of variables.

9. Multipole Expansion:

• Concept of multipole expansion:

  • Multipole expansion is a technique used to represent the electrostatic potential of a charge distribution as a sum of terms that depend on the distance from a reference point.

• Multipole moments (dipole moment, quadrupole moment, etc.):

  • Multipole moments are coefficients in the multipole expansion of the electrostatic potential.
  • The dipole moment is the first multipole moment and represents the overall polarity of the charge distribution.
  • The quadrupole moment is the second multipole moment and represents the deviation of the charge distribution from a perfect sphere.

10. Electrostatic Energy:

• Energy of a system of point charges $$U_e=\frac{1}{4\pi {\epsilon }_0}\sum _{i=1}^N\sum _{j=i+1}^N\frac{q_i{q}_j}{r_{ij}}$$
• Electrostatic potential energy $$U=q_{0}V({\overrightarrow{r}}_0)$$
• Work done in moving a charge in an electric field $$W=q_0[\phi