Inductance and InductorsSelf InductanceExamplesInductance of SolenoidInductance of a ToroidMutual InductanceEnergy Stored in an InductorEnergy Density of a Magnetic FieldPHY2061 Enriched Physics 2 Lecture Notes InductanceInductance and InductorsDisclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.Self InductanceAn inductor is a circuit element that stores magnetic field. If the magnetic field is changing, i.e. the current is changing, it will have an induced EMF across it with a magnitude proportional to the rate of change of current:diLdte =The proportionality constant L is called the inductance of the device. It is a property of the device (geometry, windings) and does not depend on the current. Inductance is measured in units of “henrys”, where 1 henry = 1 volt-second/ampere. For circuit analysis, it is enough to know just the inductance of the device and not the specific geometry.As per Lenz’s Law, the sign of the EMF is determined such that it opposes the change in the magnetic flux through the device. When going from point “a” to point “b” on each end of the device, the EMF is given by: b adiV V V Ldte =D = - =-So it increases in going from one end of the device to the other if the current is decreasing(and vice versa).Note that by Faraday’s Law of Induction:B N is the number of windings, is the flux through one windingso BBBd diN Ldt dtLi NNLieF=- =- F= FF=D. Acosta Page 1 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes InductanceExamplesInductance of SolenoidFor a solenoid, 0B nim= where n is the number of turns per unit length Nn =l. ( ) ( ) ( ) ( ) ( )02020BdiLdtNLin BA n ni AL n Ai iL n Vemmm=-F=� = = ==l llwhere V is the volume of the solenoid.Inductance of a ToroidFor a toroid, 02NiBrmp=The magnetic field is perpendicular to the radius vector from the center of the circular toroid, thus it is parallel to the area vector through any radial slice across the toroid.0020 where h is the thickness of the toroid2ln2ln2babaBrBrrBrbBabBad BdABhdrNidrhrNi rhrN h rNLi rmpmpmpF = � =F =F =� �F =� �� �� �F= =� �� �� ���B AMutual Inductance(to be filled in)D. Acosta Page 2 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes InductanceEnergy Stored in an InductorRecall that the EMF is defined by dWdqe =, the work done per unit charge by a source of EMF.Power is dW dW dqP idt dq dte= = =, and this is the power supplied by the source of EMF to maintain a current i. This work applied by the EMF source is the negative of the work done by the magnetic field in setting up the EMF, and so this power directly changes the magnetic potential energy of the setup:W U=D, sodUPdt=For a circuit containing only an inductor, this EMF is given by diLdte =, so 212di dUP Lidt dtdU LidiU Li diU Li= =� =��� ==�This is the magnetic potential energy stored in an inductor. Contrast this to the electric potential energy stored in a capacitor: ( )2212 2qU C VC= D =Energy Density of a Magnetic FieldTake the specific case of a solenoid:2022 200, so12 2L n VBU n Vi Vmmm== =The energy density (energy per unit volume) can be defined as:202U BuV m= =D. Acosta Page 3 1/14/2019PHY2061 Enriched Physics 2 Lecture Notes InductanceThis is a general equation beyond just a solenoid and represents the energy density stored in a magnetic field. When combined with the electric field energy density, we have:220012 2Bu Eem= +Let’s compute the total stored energy in the solenoid used for the CMS experiment:B = 4T, r =3m, l = 13 m( )( )26 372 2 394 T6.4 10 J/m2 4 10 Tm/AThe volume is 3 13 368 m2.3 10 J (2.3 GJ!)uV rUpp p-= = �= = � � =� = �lThis is the most stored energy of any constructed magnet. Note that 1 ton TNT equivalentis 94.2 10 J�, so the energy stored in this magnet is equivalent to ½ ton TNT! One must be very carefully how this energy is dissipated… D. Acosta Page 4
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