Magnetism, Flux and Inductance

Name	Formula	Related Formulae	Variables/Constants
Lorentz force	$\vec{F}=q\left(\vec{E}+\vec{v}\times \vec{B}\right)$	$\vec{F}=I\vec{l}\times \vec{B}(\text{current-carrying wire})$	$\vec{F}:force$ $q:charge$ $\vec{E}:electricfield$ $\vec{v}:velocity$ $\vec{B}:magneticfluxdensity$ $I:current$ $\vec{l}:wirelengthvector$
Biot-Savart law	$d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{l}\times \hat{r}}{r^2}$	$B=\frac{{\mu }_{0}I}{2\pi r}(\text{long straight wire})$	$d\vec{B}:fieldfromacurrentelement$ ${\mu }_0:permeabilityoffreespace$ $I:current$ $d\vec{l}:currentelement$ $\hat{r}:unitvectortothefieldpoint$ $r:distance$
Ampere’s law	$\oint \vec{B}\cdot d\vec{l}={\mu }_0{I}_{enc}$	$B={\mu }_{0}nI(\text{long solenoid})$	$\vec{B}:magneticfluxdensity$ $d\vec{l}:pathelement$ $I_{enc}:enclosedcurrent$ $n:turnsperunitlength$
Magnetic flux	$\Phi =\int \vec{B}\cdot d\vec{A}$	$\Phi =BA\cos \theta (\text{uniform field}),1\mathrm{Wb}=1\mathrm{T}{\mathrm{m}}^2$	$\Phi :magneticflux(weber)$ $\vec{B}:magneticfluxdensity(tesla)$ $d\vec{A}:areaelement$ $\theta :anglebetweenBandthesurfacenormal$
Faraday’s law	$\epsilon =-N\frac{d\Phi }{dt}$	$\oint \vec{E}\cdot d\vec{l}=-\frac{d\Phi }{dt},\epsilon =Blv(\text{moving rod})$	$\epsilon :inducedemf$ $N:numberofturns$ $\Phi :fluxthroughoneturn$ $t:time$ $B:field$ $l:rodlength$ $v:rodvelocity$
Flux linkage and inductance	$\lambda =N\Phi =LI$	$v=L\frac{di}{dt},1\mathrm{H}=1Wb/A$	$\lambda :fluxlinkage$ $N:numberofturns$ $\Phi :fluxperturn$ $L:inductance(henry)$ $I,i:current$ $v:voltageacrosstheinductor$
Inductance of a solenoid	$L=\frac{\mu N^{2}A}{l}$	$L=\frac{N^2}{\mathcal{R}}$	$\mu :permeabilityofthecore$ $N:numberofturns$ $A:cross-sectionalarea$ $l:length$ $\mathcal{R}:reluctanceofthemagneticpath$
Mutual inductance	$M=\frac{N_2{\Phi }_{21}}{I_1}$	$v_2=M\frac{d{i}_1}{dt},M=k\sqrt{L_1{L}_2}$	$M:mutualinductance$ $N_2:turnsofcoil2$ $\begin{array}{r}{\Phi }_{21}:fluxthroughcoil2duetocurrent{I}_1\\ incoil1\end{array}$ $k:couplingcoefficient(0to1)$ $L_1,L_2:self-inductances$
Energy stored in an inductor	$W=\frac{1}{2}L{I}^2$	$w_m=\frac{B^2}{2\mu }$	$W:storedenergy$ $L:inductance$ $I:current$ $w_m:magneticenergydensity$ $B:fluxdensity$ $\mu :permeability$
RL circuit	$i\left(t\right)=\frac{V_s}{R}\left(1-e^{-t/\tau }\right),\tau =\frac{L}{R}$	${\omega }_0=\frac{1}{\sqrt{LC}}(\text{LC circuit})$	$i:inductorcurrent$ $V_s:sourcevoltage$ $R:resistance$ $\tau :timeconstant$ $L:inductance$ ${\omega }_0:naturalangularfrequency$ $C:capacitance$
Maxwell’s equations	$\nabla \cdot \vec{D}={\rho }_v,\nabla \cdot \vec{B}=0,\nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t},\nabla \times \vec{H}=\vec{J}+\frac{\partial \vec{D}}{\partial t}$	$c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$	$\vec{D}:electricdisplacement$ ${\rho }_v:chargedensity$ $\vec{B}:magneticfluxdensity$ $\vec{E}:electricfield$ $\vec{H}:magneticfieldstrength$ $\vec{J}:currentdensity$ $c:speedoflight$

Important Questions

Ques: What is the quickest way to remember the inductance-flux relation?

Pair it with $Q=CV$. The flux linkage of a coil, its flux per turn multiplied by the number of turns, equals inductance times current: $\lambda =N\Phi =LI$. Read L as flux linkage per ampere, just as C is charge per volt. Three checks make it stick. Units: a henry is a weber per ampere. Derivative: differentiating gives $d\lambda /dt=Ldi/dt$, and Faraday’s law says $d\lambda /dt$ is the induced voltage, so $v=Ldi/dt$. Duality: swapping C for L, V for I and Q for $\lambda$ turns every capacitor formula into an inductor formula, for example $\frac{1}{2}C{V}^2$ into $\frac{1}{2}L{I}^2$ and $i=Cdv/dt$ into $v=Ldi/dt$. Keeping this pairing in mind also makes it easy to recall the RL time constant $L/R$ next to the RC time constant $RC$. Reference: W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. (McGraw-Hill, 2012); W. H. Hayt Jr. and J. A. Buck, Engineering Electromagnetics, 8th ed. (McGraw-Hill, 2012).

Ques: Why is there a minus sign in Faraday’s law?

Because the induced emf opposes the change of flux that produces it, which is Lenz’s law. If the flux through a loop increases, the induced current flows so that its own magnetic field points against the increase; if the flux decreases, the induced current supports it. Without the minus sign a small change would induce a current that reinforced itself, creating energy from nothing. In circuit analysis the sign is absorbed into the passive sign convention, so the voltage across an inductor is written $v=+Ldi/dt$, measured as a drop in the direction of the current. The same law explains transformers, generators and the voltage spike when an inductor’s current is interrupted. Reference: D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, 2013); D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. (Wiley, 2014).

Ques: Why does inductance scale with the square of the number of turns?

Because the turns count twice. More turns produce a stronger field, $B=\mu nI$ inside a solenoid, so the flux through each turn grows with N; and that flux links every one of the N turns, so the flux linkage grows with N again. The result is $L=\mu N^{2}A/l$, or in magnetic-circuit form $L=N^2/\mathcal{R}$. Doubling the turns on the same core therefore quadruples the inductance. For example, 100 turns on a 1 cm^2 former 1 cm long in air give about 126 $\mu$H (computed), and a core with relative permeability ${\mu }_r$ multiplies this by ${\mu }_r$ until the core saturates. Spreading the same turns over a longer former lowers the inductance in proportion to the length. Reference: W. H. Hayt Jr. and J. A. Buck, Engineering Electromagnetics, 8th ed. (McGraw-Hill, 2012); S. J. Chapman, Electric Machinery Fundamentals, 5th ed. (McGraw-Hill, 2012).

Ques: Why is the permeability of free space written as 4 pi times 10^-7?

Because the ampere was historically defined through the force between parallel wires, which made ${\mu }_0=4\pi \times 10^{-7}$ H/m exact by definition, and textbooks use this value. Since the 2019 redefinition of the SI units, which fixed the elementary charge instead, ${\mu }_0$ is a measured quantity whose value differs from $4\pi \times 10^{-7}$ by less than one part in a billion (background knowledge), so the textbook value remains correct for every practical calculation. The speed of light is still exact, and $c=1/\sqrt{{\mu }_0{ϵ}_0}$ links ${\mu }_0$ to the permittivity of free space, $8.854\times 10^{-12}$ F/m. With ${\mu }_0$ known, the field of a long wire carrying 1 A is 20 $\mu$T at 1 cm (computed), comparable to the Earth’s field of roughly 50 $\mu$T. Reference: D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, 2013); W. H. Hayt Jr. and J. A. Buck, Engineering Electromagnetics, 8th ed. (McGraw-Hill, 2012).

Ques: Where is the energy of an inductor stored, and why do switched inductors spark?

In the magnetic field. The stored energy is $\frac{1}{2}L{I}^2$, and its density in the field is $B^2/(2\mu )$. Because $v=Ldi/dt$, an inductor’s current cannot change instantly; opening a switch tries to drop the current to zero almost at once, so the voltage rises until something conducts, often an arc across the contacts. With 10 $\mu$H, interrupting 1 A in 1 $\mu$s already needs 10 V, and in 10 ns 1000 V (computed). This is why relay coils have flyback diodes, why switching converters use snubbers, and why boost converters and ignition coils can generate high voltages from low supplies. The same energy is what an inductor hands back to the circuit when its current falls. Reference: C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. (McGraw-Hill Education, 2017); D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. (Wiley, 2014).

References

1. W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. New York: McGraw-Hill, 2012.
2. W. H. Hayt Jr. and J. A. Buck, Engineering Electromagnetics, 8th ed. New York: McGraw-Hill, 2012.
3. D. J. Griffiths, Introduction to Electrodynamics, 4th ed. Boston: Pearson, 2013.
4. D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. Hoboken, NJ: Wiley, 2014.
5. S. J. Chapman, Electric Machinery Fundamentals, 5th ed. New York: McGraw-Hill, 2012.
6. C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. New York: McGraw-Hill Education, 2017.