Electric Charge and Fields: Principles and Laws
**Concept of Electric Charge**
Electric charge is a fundamental property of matter. In nature, there are only two types of charges: positive (+) and negative (-). Electric charge is a scalar quantity and is measured in Coulombs (C). Attractive forces exist between bodies with opposite charges, while repulsive forces exist between bodies with the same charge. Matter consists of electrons and protons. The magnitude of the charge on both particles is the same. Protons are located in the nucleus, and electrons are in the outer shell. It is easy to separate them with little energy. A body with a deficiency of electrons has a positive charge, and a body with an excess of electrons has a negative charge. Currently, there are no known subatomic particles with fractional charges, but theoretically, quarks could consist of bosons.
**Coulomb’s Law**
The quantization of electric forces is due to Coulomb, who determined this through a torsion balance. Coulomb’s law states that: the force of attraction or repulsion between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance that separates them. (F1) The electric constant is defined as a function of the permittivity of the medium. (F2). Where E is the permittivity of the medium, and E0 is the permittivity of vacuum. (F3). Coulomb’s law is somewhat analogous to Newton’s law of universal gravitation. The forces can be attractive or repulsive. Moreover, K depends on the medium in which the charges are located.
**Concept of Electric Field**
An electric field exists in a given region of space if a test electric charge introduced into that region experiences a force. The electric field intensity is defined as the force per unit of electric charge. (F4).
**Electrical Potential Energy**
The electric force is a conservative central force. As such, a scalar function called electric potential energy can be defined at each point. (F5). The work done by the electric force between two points A and B is equal to the negative of the change in potential energy between these two points:
(F6). The electric potential energy at a point can be defined as the work performed by the electric force to move a charge q from that point to infinity.
**Lines of Force or Field**
These are imaginary lines that trace the path of a positive charge set free within an electric field. These lines have the following properties:
- Lines emerge from positive charges (sources) and enter negative charges (sinks).
- The number of lines coming into or out of a charge is proportional to the magnitude of the charge.
- At each point in the field, the number of lines per unit area perpendicular to them is proportional to the field strength.
- Two lines of force can never intersect. At each point, the field has only one direction, so two lines cannot cross because that would imply two directions at the same point.
The field strength is the number of lines of force crossing a unit surface placed perpendicularly to the lines.
**Flux of Electric Field: Gauss’s Law**
The electric field flux is a measure of the number of lines of force. If the electric field is uniform, the electric field flux is the scalar product of the field and a vector whose magnitude is the surface area and whose direction is perpendicular to that surface. (F7) If the field is not uniform, the flux is defined by the integral (F8).
Gauss’s theorem can be used to calculate the field value of a distribution of charges. It is also used to determine the electric field of one of these distributions. The distributions of charge may be wires, plates, etc.
**Definition of Electric Potential**
Electric potential at a point is defined as the potential energy per unit of positive charge placed at that point. (F9). A charge creates an electric field around it, which is a vector quantity. We can also assume that the presence of a charge creates a scalar property around it called electric potential. (F10). The potential source is at infinity, where the potential is equal to 0. Therefore, the work necessary to transfer a unit charge from infinity to that point is equal to the potential at that point. (F11). Since an electron has a negative charge, the work necessary to move an electron between two points in an electric field with a potential difference of 1 V is called an electronvolt (eV).
**Equipotential Surfaces**
These are surfaces that have the same potential. The work done to move a charge along an equipotential surface is 0. The electric field vector is perpendicular at all points to an equipotential surface.
**Relationship Between Field and Potential**
(F12) This implies that if there is no variation of potential in a given direction, the component of the field in that direction is zero.
Knowing the value of the potential at each point, we can determine the magnitude of the electric field. The direction of the field is towards decreasing potential, as indicated by the negative sign in the equation. If we know the value of the field at each point, we can obtain the value of the potential by integration.
**Oersted’s Experiment**
Oersted was the first to demonstrate the relationship between electrical and magnetic phenomena. Oersted placed a compass near a wire through which an electric current was flowing. He observed that when current passed through the wire, the compass needle oriented itself perpendicular to the wire. When the current ceased, the needle returned to its original orientation. Reversing the direction of the current changed the orientation of the needle. This experiment demonstrated that electric charges in motion produce magnetic fields. He also conducted an experiment with Ampere, placing two parallel conductors through which currents flowed. They observed that forces occurred between the two conductors, which were attractive if the currents had the same direction and repulsive if they had opposite directions. These experiments prove that magnetic fields generated by electric currents or magnets can be traced to moving charges.
**Magnetic Field**
A magnet or a moving charge creates a magnetic field in space. This field is characterized by the vector B. To characterize the magnetic induction, we will use the effects produced on an electric charge. When we place a test electric charge at a point in space where a magnetic field exists, we observe:
- If the charge is at rest, no force acts on it.
- If the charge is in motion: a. There is a direction of the velocity at which no force acts on the charge. b. When the direction of the velocity is perpendicular to the direction in (a), the force on the charge is maximum. c. The force is perpendicular to the velocity, and its magnitude is proportional to the velocity. d. The force is proportional to the charge and changes direction when the sign of the charge changes.
The mathematical relationship is given by the formula: (F13) This formula tells us that the force is perpendicular to the plane formed by the velocity and the magnetic induction. It also tells us that the force is maximum when the velocity is perpendicular to the magnetic induction. To determine the direction and sense, the right-hand rule for the vector product is used. For an electric charge to experience a force, it only needs to be in an electric field. For it to experience a magnetic force, the charge must be in motion. Often, we want to indicate the direction and sense of the magnetic induction. For this, the following symbols are used:
- = This symbol indicates that the magnetic induction is perpendicular to the plane of the paper and the direction is outward.
- 0X = This symbol indicates that the magnetic induction is perpendicular to the plane of the paper and the direction is inward.
**Magnetic Field Lines**
These lines are used to define the direction of the magnetic field at a point in space, which is the same as the direction of the magnetic induction. These are not lines of force, as they do not indicate the direction of the force. Magnetic field lines are assumed to leave the north pole and enter the south pole.
**Action of a Magnetic Field on a Moving Charge**
The force acting on a moving charge is perpendicular to the velocity. Therefore, the acceleration produced is normal, so the magnitude of the velocity does not change, but its direction does. (F.14). If the magnetic field is uniform and the velocity is perpendicular to the magnetic induction, the charge describes a circular motion with a radius given by (F.15). If the magnetic field is uniform but the velocity is not perpendicular to the magnetic induction, the charge describes a helical motion. The work done by the magnetic force is always zero because the force is always perpendicular to the displacement of the charge.