Electric Charges and Fields — Study Notes

1. Introduction

Electric charge is a fundamental property of matter that causes it to experience a force in an electric field. There are two kinds of charges, named positive and negative by Benjamin Franklin. Like charges repel, unlike charges attract. All matter is made of atoms, which contain positively charged protons, negatively charged electrons, and neutral neutrons. This chapter lays the foundation of electrostatics - the study of charges at rest.

Charge is quantized and conserved. Quantization means charge always exists in integral multiples of the elementary charge e = 1.6 x 10^-19 C, and conservation means charge can neither be created nor destroyed; it can only be transferred. The study begins with Coulomb's law, which gives the force between two point charges, and then develops the concept of the electric field as the property of space surrounding a charge.

We then study continuous charge distributions, the electric dipole, the concept of electric flux, and finally Gauss's law, which is a powerful tool for computing the electric fields of symmetric charge distributions such as a line of charge, a plane sheet, and a spherical shell. Understanding these ideas is essential for the rest of electrostatics, including potential, capacitance, and current electricity.

2. Properties of Electric Charge

Electric charge is additive: the total charge of a system is the algebraic sum of all the individual charges. Charge is quantized, so the charge on any body is q = n e, where n is an integer and e = 1.6 x 10^-19 C is the magnitude of charge on a single electron or proton. Charge is conserved in all processes: the total charge of an isolated system remains constant.

The unit of charge in the SI system is the coulomb (C). One coulomb is the charge that, when placed 1 metre from an equal charge in vacuum, experiences a force of 9 x 10^9 N. A proton carries charge +1.6 x 10^-19 C and an electron carries -1.6 x 10^-19 C. The property of charge is invariant; that is, the charge on a body does not depend on its state of motion.

3. Coulomb's Law

Coulomb's law states that the magnitude of the electrostatic force between two point charges q1 and q2 separated by a distance r in vacuum is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:

F = k * q1 * q2 / r^2

Here k = 1/(4 pi epsilon_0) = 9 x 10^9 N m^2/C^2, where epsilon_0 = 8.85 x 10^-12 C^2/(N m^2) is the permittivity of free space. The force is along the line joining the charges; it is repulsive for like charges and attractive for unlike charges.

Coulomb's law obeys Newton's third law: the force on q1 due to q2 is equal and opposite to the force on q2 due to q1. When more than two charges are present, the total force on any one charge is the vector sum of the individual forces from each other charge - this is the principle of superposition. If the charges are placed in a medium, the force is reduced by a factor equal to the relative permittivity (dielectric constant) of the medium.

4. Electric Field

The electric field at a point in space is defined as the force experienced by a unit positive test charge placed at that point, divided by the magnitude of the test charge:

E = F / q0

The electric field is a vector quantity measured in N/C (or V/m). For a point charge Q, the field at a distance r is given by:

E = k * Q / r^2

The direction of the field is radially outward for a positive charge and radially inward for a negative charge. For a system of charges, the total field is the vector sum of the fields due to each charge, by the principle of superposition.

Electric field lines are a pictorial representation of the field. They start on positive charges and end on negative charges, never cross each other, and the density of lines indicates the field strength. The field lines are closer together where the field is strong and farther apart where it is weak.

5. Electric Dipole and Dipole Moment

An electric dipole consists of two equal and opposite charges +q and -q separated by a small distance 2a. The dipole moment is a vector directed from the negative charge to the positive charge:

p = q * 2a

The SI unit of dipole moment is C m. The field of a dipole on its axial line at a distance r from the centre is:

E_axial = (1/(4 pi epsilon_0)) * (2 p r) / (r^2 - a^2)^2

For r much greater than a, E_axial = (1/(4 pi epsilon_0)) * 2p / r^3. The field on the equatorial line at distance r is:

E_equatorial = (1/(4 pi epsilon_0)) * p / (r^2 + a^2)^(3/2)

For r much greater than a, E_equatorial = (1/(4 pi epsilon_0)) * p / r^3. Thus the axial field is twice the equatorial field at large distances. The dipole field falls off as 1/r^3, faster than the 1/r^2 field of a point charge.

6. Torque on a Dipole in a Uniform Field

When an electric dipole is placed in a uniform external electric field E, the forces on the two charges are equal and opposite, so the net force is zero. However, these forces form a couple, producing a torque that tends to align the dipole with the field:

tau = p E sin(theta)

Here theta is the angle between the dipole moment p and the field E. In vector form, tau = p x E. The torque is maximum (tau = pE) when the dipole is perpendicular to the field and zero when it is aligned (theta = 0) or anti-aligned (theta = 180 degrees).

The work done in rotating the dipole from alignment to some angle theta is stored as potential energy:

U = -p E cos(theta) = -p dot E

The potential energy is minimum (-pE) when the dipole is aligned with the field and maximum (+pE) when it is anti-aligned. This is why a dipole left free in an electric field rotates to align with the field.

7. Electric Flux and Gauss's Law

The electric flux through an area is the product of the electric field and the component of the area perpendicular to the field. For a uniform field E through a plane surface of area S, the flux is:

Phi = E * S * cos(theta) = E dot S

where theta is the angle between the field and the normal to the surface. The SI unit of flux is N m^2/C. Flux is a scalar quantity. For a closed surface, the outward normal is used, and flux entering a surface is negative while flux leaving is positive.

Gauss's law states that the net electric flux through any closed surface is equal to the net charge enclosed by the surface divided by epsilon_0:

Phi = q_enclosed / epsilon_0

Gauss's law is one of the fundamental laws of electromagnetism. It is valid for any closed surface, called a Gaussian surface, and the flux does not depend on the size or shape of the surface, only on the charge enclosed.

8. Applications of Gauss's Law

Using Gauss's law with spherical, cylindrical, or plane Gaussian surfaces, we can derive the electric fields of symmetric charge distributions:

Electric field due to an infinitely long straight wire with linear charge density lambda, at perpendicular distance r:

E = lambda / (2 pi epsilon_0 r)

Electric field due to an infinite plane sheet of charge with surface charge density sigma:

E = sigma / (2 epsilon_0)

This field is uniform - it does not depend on the distance from the sheet. For two parallel sheets with equal and opposite charges, the field between them is sigma/epsilon_0 and zero outside.

Electric field due to a uniformly charged thin spherical shell: - Outside the shell (r > R): E = q / (4 pi epsilon_0 r^2), same as a point charge at the centre. - On the surface (r = R): E = q / (4 pi epsilon_0 R^2). - Inside the shell (r < R): E = 0.

The field inside a uniformly charged spherical conductor is zero, which is why electrostatic shielding works.

Quick Revision Tables

Mind Map

Important Diagrams (SVG)

Diagram 1: Field Lines of a Positive and Negative Charge

Diagram 2: Gauss's Law and Field of a Spherical Shell

Common Mistakes

1. Confusing the distance r in Coulomb's law with the separation used in the dipole field formulas; dipole formulas use the centre of the dipole, not one charge.
2. Forgetting that force is a vector; adding forces without vector addition when more than two charges are present.
3. Using the 1/r^2 fall-off for dipole fields; dipole fields fall off as 1/r^3 at large distances.
4. Taking the field of a charged conductor to be the same inside as outside; inside a conductor in equilibrium, the field is always zero.
5. Applying Gauss's law incorrectly by forgetting that flux depends only on the charge enclosed, not on the charge outside the surface.
6. Confusing the axial field of a dipole (2p/r^3) with the equatorial field (p/r^3); the axial field is twice as large.
7. Using the plane sheet formula E = sigma/2 epsilon_0 where the conducting plate formula sigma/epsilon_0 is required for two charged plates.

Exam Tips

1. State Coulomb's law with the formula F = (1/4 pi epsilon_0) q1 q2 / r^2 and give the value of k = 9 x 10^9 N m^2/C^2.
2. Define electric field E = F/q0 and write the expression for the field of a point charge E = k Q / r^2.
3. Write the properties of field lines: start on positive charges, end on negative charges, and never intersect.
4. Derive the expressions for the dipole field on the axial and equatorial lines, stating E_axial = 2 E_equatorial.
5. Derive the torque tau = pE sin theta and potential energy U = -pE cos theta on a dipole in a uniform field.
6. State Gauss's law and use it to derive E for a line charge, a plane sheet, and a spherical shell.
7. Remember that the field inside a spherical shell and inside any conductor in equilibrium is zero.

Conclusion

In this chapter we studied the fundamental properties of electric charge - quantization, conservation, additivity, and invariance. Coulomb's law gave us the force between point charges, and the principle of superposition extended it to many charges. We developed the electric field, first for a point charge and then for the electric dipole, including the torque and potential energy of a dipole in a uniform field. Finally, Gauss's law, Phi = q_enclosed/epsilon_0, allowed us to derive the fields of a line charge, a plane sheet, and a spherical shell. The field inside a conductor in electrostatic equilibrium is always zero. These ideas form the basis for understanding electrostatic potential, capacitance, and the flow of current in the following chapters.