Ohm's Law and Reluctance

Ohm's Law

Electric circuits share many of the same characteristics as magnetic circuits. One of the most important and fundamental equations describing electric circuits is Ohm’s Law. Georg Ohm, a German physicist, published this famous equation in 1827, drawing significant influence from Fourier’s previous work in heat conduction.

Ohm’s Law states that the resistance of a conductor is constant and inversely proportional to the current running through the conductor and directly proportional to the voltage across it, or expressed mathematically as

$R={\frac {V}{I}}$

The SI units for resistance are the Ohm ($\Omega$), voltage is expressed in volts (V) and current is measured in amperes (A).

Expressed differently, the current density J (A/$m^{2}$) is proportional to the product of conductivity sigma (V/m) and the electric field E (siemens/meter, s/m), or

$\displaystyle J=\sigma E$

Reluctance

Now, consider a material of very high permeability with flux $\Phi$ running through it and separated by a very small gap of area A, where the flux flows through this gap as if it were free space. The flux density crossing this gap is B. Therefore

$\displaystyle \Phi =BA$

The magnetic field intensity can be approximated by $H={\frac {B}{\mu _{0}}}$

The reluctance of this gap is analogous to the resistance described by Ohm’s Law, where it is the ratio of the magnetomotive force to the flux, or

$\mathbb {R} ={\frac {F}{\phi }}$