Chapter 4: Magnetostatics

1- Introduction 

Magnetostatics is the branch of physics that studies magnetic fields in systems where the currents are steady (not changing with time). It is a subfield of electromagnetism, focusing on the behavior and properties of magnetic fields produced by constant currents.

Magnets have been known since ancient times as magnetite, a stone found in proximity to the city of Magnesia (Turkey). It is from this stone that the current name comes of magnetic field.
The Chinese were the first to use the properties of magnets, more than 1000 years ago, to make compasses. They consisted of a magnetite needle placed on top of straw floating on water contained in a graduated container.

2- Magnetostatic force (Lorentz and Laplace) 

In magnetostatics, the forces experienced by charged particles and current-carrying conductors in a magnetic field are described by the Lorentz and Laplace forces.

Lorentz Force

The Lorentz force describes the force on a charged particle moving through a magnetic field. It is given by:

                                       F=q(E+v×B) 

where:

• F is the force on the particle.
• q is the electric charge of the particle.
• E is the electric field.
• v: is the velocity of the particle.
• B is the magnetic field.

• In magnetostatics, we usually focus on the magnetic component of the Lorentz force, since the electric field component can be considered separately in electrostatics. Thus, the magnetic part of the Lorentz force is:

                                            F=q(v×B)

  This force is perpendicular to both the velocity of the particle and the magnetic field, causing the particle to move in a circular or helical path in the presence of a uniform magnetic field.

Laplace Force (Force on a Current-Carrying Conductor)

The Laplace force describes the force on a current-carrying conductor in a magnetic field. It is given by:

                                          dF=I(dl×B)

where:

• dF is the differential force on a small segment of the conductor.
• I: is the current through the conductor.
• d: is a differential length vector of the conductor.
• B:is the magnetic field.

For a finite length of conductor, the total force is obtained by integrating along the length of the conductor:

                                                                  F=I∫(dl×B)

This force is also perpendicular to both the direction of the current and the magnetic field.

Mathematical Examples

1. Lorentz Force Example:                                                                                                                                 A proton (charge q=+1.6×10^−19C) moving with velocity v= 10^6 m/s in the x-direction enters a magnetic field B= 0.1T in the z-direction.                                                                                                    The force on the proton is:   F=q(v×B)=(1.6×10^−19 C)(10^6 m/s i^×0.1 T k^)     F=1.6×10−14Nj^     ​The proton experiences a force of 1.6×10^−14 N in the y-direction.

2. Laplace Force Example:

   • A wire carrying a current I=5I is placed in a magnetic field $0.2$ T perpendicular to the wire, over a length $0.1$ m.                                                                                                                       The force on the wire is                                                                                                                                                     $$\mathbf{F}=ILB=(5\text{A})(0.1\text{m})(0.2\text{T})$$       $$\mathbf{F}=0.1\text{N The wire experiences a force of 0.1 N perpendicular to both the current direction and the magnetic field. }$$                                                                                         Understanding the Lorentz and Laplace forces is crucial for designing and analyzing systems that involve the interaction of electric currents and magnetic fields, forming the basis for many modern technological applications.

3- megnetic fields

Magnetic Fields and Magnetic Flux:

A magnetic field B: is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

Magnetic flux ( Φ) : through a surface is the integral of the magnetic field over that surface.

4- Biot's law and Sawark

The Biot-Savart Law is a fundamental principle in magnetostatics that describes how steady electric currents produce magnetic fields. It is analogous to Coulomb’s law in electrostatics but applies to the magnetic field generated by currents.

It provides a mathematical model to calculate the magnetic field ($\mathbf{B}$) at a point in space due to a small segment of current-carrying wire. The law is named after French physicists Jean-Baptiste Biot and Félix Savart, who discovered it in the early 19th century.

Significance

The Biot-Savart Law is essential for understanding the magnetic fields generated by currents in various geometries. It forms the basis for more complex calculations in electromagnetism and is foundational in the design of electrical devices such as motors, transformers, and inductors.