Magnetic Field Sources Lesson 29 by Owen Borville 12.30.2025
The magnetic field created by a current-carrying wire is found by the Biot-Savart law, a fundamental principle in electromagnetism that describes the magnetic field generated by a steady electric current, stating that an infinitesimal current produces a tiny magnetic field that is proportional to the current, the element's length, and the sine of the angle to the observation point, while inversely proportional to the square of the distance. These values are integrated mathematically along the entire path, following the right hand rule for direction. The current element Idl produces a magnetic field a distance r away. dB = μ0/4π * Idl*r/r^2 , where I is the current, dl is the vector length of the current element, r is the unit vector and distance from dl to P.
The strength of the magnetic field created by current in a long straight wire is B = μ0I/2πR where I is the current, R is the shortest distance to the wire, and the constant μ0 = 4π x 10^-7 T*m/s is the permeability of free space. The absolute magnetic permeability of a material (μ) = (1+ 𝜒)μ0 where 𝜒 is the magnetic susceptibility (how much a material becomes magnetized in an external magnetic field and μ0 is the permeability of free space.
The direction of the magnetic field created by a long straight wire is determined by the right-hand-rule 2 (RHR-2): Point the thumb of the right hand in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.
The magnetic force between two parallel currents I1 and I2 separated by a distance r has a magnitude per unit length F/l = μ0I1I2/2πr. The force is attractive if the currents are in the same direction, and repulsive in they are in the opposite directions.
The magnetic field strength at the center of a circular loop is B = μ0I/2R where R is the radius of the loop. RHR-2 gives the direction of the field about the loop.
Ampere's Law states that the magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field. ∫Bdl = μ0I Ampere's law can be used to determine the magnetic field from a thin wire or a thick wire by a geometrically convenient path of integration. The results are consistent with the Biot-Savart law.
The magnetic field strength inside a solenoid (coil of wire that creates a magnetic field when current flows through it) is B = μ0nI where n is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.
The magnetic field inside a toroid (coil of wire around a ring core) is B = μ0NI/2πr where N is the number of windings. The field inside a toroid is not uniform and varies with the distance as 1/r.
Materials are classified as paramagnetic, diamagnetic, or ferromagnetic depending on how they behave in an applied magnetic field.
Paramagnetic materials have partial alignment of their magnetic dipoles with an applied magnetic field. This is a positive magnetic susceptibility. Only a surface current remains, creating a solenoid-like magnetic field.
Diamagnetic materials exhibit induced dipoles opposite to an applied magnetic field and is a negative magnetic susceptibility.
Ferromagnetic materials have groups of dipoles, called domains, which align with the applied magnetic field. However, when the field is removed , the ferromagnetic material remains magnetized, unlike paramagnetic materials. This magnetization of the material versus the applied field effect is called hysteresis.
The magnetic field created by a current-carrying wire is found by the Biot-Savart law, a fundamental principle in electromagnetism that describes the magnetic field generated by a steady electric current, stating that an infinitesimal current produces a tiny magnetic field that is proportional to the current, the element's length, and the sine of the angle to the observation point, while inversely proportional to the square of the distance. These values are integrated mathematically along the entire path, following the right hand rule for direction. The current element Idl produces a magnetic field a distance r away. dB = μ0/4π * Idl*r/r^2 , where I is the current, dl is the vector length of the current element, r is the unit vector and distance from dl to P.
The strength of the magnetic field created by current in a long straight wire is B = μ0I/2πR where I is the current, R is the shortest distance to the wire, and the constant μ0 = 4π x 10^-7 T*m/s is the permeability of free space. The absolute magnetic permeability of a material (μ) = (1+ 𝜒)μ0 where 𝜒 is the magnetic susceptibility (how much a material becomes magnetized in an external magnetic field and μ0 is the permeability of free space.
The direction of the magnetic field created by a long straight wire is determined by the right-hand-rule 2 (RHR-2): Point the thumb of the right hand in the direction of the current, and the fingers curl in the direction of the magnetic field loops created by it.
The magnetic force between two parallel currents I1 and I2 separated by a distance r has a magnitude per unit length F/l = μ0I1I2/2πr. The force is attractive if the currents are in the same direction, and repulsive in they are in the opposite directions.
The magnetic field strength at the center of a circular loop is B = μ0I/2R where R is the radius of the loop. RHR-2 gives the direction of the field about the loop.
Ampere's Law states that the magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field. ∫Bdl = μ0I Ampere's law can be used to determine the magnetic field from a thin wire or a thick wire by a geometrically convenient path of integration. The results are consistent with the Biot-Savart law.
The magnetic field strength inside a solenoid (coil of wire that creates a magnetic field when current flows through it) is B = μ0nI where n is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.
The magnetic field inside a toroid (coil of wire around a ring core) is B = μ0NI/2πr where N is the number of windings. The field inside a toroid is not uniform and varies with the distance as 1/r.
Materials are classified as paramagnetic, diamagnetic, or ferromagnetic depending on how they behave in an applied magnetic field.
Paramagnetic materials have partial alignment of their magnetic dipoles with an applied magnetic field. This is a positive magnetic susceptibility. Only a surface current remains, creating a solenoid-like magnetic field.
Diamagnetic materials exhibit induced dipoles opposite to an applied magnetic field and is a negative magnetic susceptibility.
Ferromagnetic materials have groups of dipoles, called domains, which align with the applied magnetic field. However, when the field is removed , the ferromagnetic material remains magnetized, unlike paramagnetic materials. This magnetization of the material versus the applied field effect is called hysteresis.