INDUCTANCIA

Inductancia

� El inductor es un elemento de un circuito que guarda energía en el campo magnético que rodea a sus alambres portadores de corriente.

� Del mismo modo que un capacitor guarda dicha energía en el campo eléctrico formado entre sus placas cargadas.

� El inductor se caracteriza por su inductancia, la cual depende de la forma de dicho inductor.

Inductancia

dtdi

LL =εInductancia L:

InductanciaWe used a coil and the solenoid assumption to introduce the inductance. But the definition

holds for all types of inductance, including a straight wire. Any conductor has capacitance and inductance.

LLdI

dt

≡ − E

An inductor is usually made of a coil to make a large inductance (more loops = more flux). The circuit symbol is

The self-induced emf through this inductor under a changing current I is given by:

L

dIL

dt= −E

Unidades de la inductancia

� The SI unit for inductance is the henry (H)

� Named for Joseph Henry: � 1797 – 1878� American physicist� First director of the Smithsonian� Improved design of electromagnet� Constructed one of the first motors� Discovered self-inductance

AsV

1H1⋅=

Inductancia

dtdi

LVV ab −=−

ab VV >

ab VV <

Cálculo de la inductancia

iN

L BΦ=

Por ser ΦB proporcional a la corriente i, la razón de dichaecuación no depende de i y, por consiguiente, la inductancia (como la capacitancia) depende sólo de la forma del dispositivo.

NΦB conexiones de flujo

Cálculo de la inductancia de un solenoide

IWhen a current flows through a coil, there is magnetic field established. If we take the solenoid assumption for the coil:

E EL+

–

0B nIµ=When this magnetic field flux changes, it induces an emf, EL, called self-induction:

( ) ( )0 20

BL

d NAB d NA nId dI dIn V L

dt dt dt dt dt

µΦ µ= − = − = − = − ≡ −E

or: L

dIL

dt≡ −E

This defines the inductance L, which is a constant related only to the coil.

The self-induced emfεL is generated by (changing) current in the coil.According to Lenz’s Law, the emf generated inside this coil is always opposing

the change of the current which is delivered by the original emf ε.

For a solenoid: 20L n Vµ=

Wheren: # of turns per unit length.N: # of turns in length l.A: cross section areaV: Volume for length l.

La inductancia de un toroide

Recordemos…

Magnetic Field of a Toroid

� The toroid has N turns of wire

� Find the field at a point at distance r from the center of the toroid (loop 1)

� There is no field outside the coil (see loop 2)

2

2

o

o

d B πr µ N

µ NB

πr

⋅ = =

=

∫ B sr r

� ( ) I

I

La inductancia de un toroide

Inductores con materiales magnéticos

Recordemos…

Magnetización

0BBrr

mκ=

� La permeabilidad de la mayor parte de losmateriales comunes(excepto losferromagnéticos) tienevalores cercanos a 1.

� Con respecto a otrosmateriales que no son ferromagnéticos, la permeabilidad puededepender de propiedadescomo la temperatura y la densidad del material, perono del campo B0.

� Para los ferromagnéticos κmdepende del campo aplicado B0.

Put inductor L to use: the RL Circuit

� An RL circuit contains a resistor R and an inductor L.

� There are two cases as in a RC circuit (charging and discharging) but in an RLcircuit one changes current, not electric charge.

� Current increases:� When S2 is connected to

position a and when switch S1is closed (at time t = 0), the current through R and L begins to increase

� Current decreases: � When S2 is connected to

position b.

RL Circuit� Applying Kirchhoff’s loop rule to the

circuit in the clockwise direction gives

0d I

ε I R Ldt

− − =

( ) ( )τ− −= − ≡ −1 1Rt L tε εI e e

R R

Here because the current is increasing, the induced emf has a direction that should oppose this increase.

τ ≡ LR

� Solve for the current I, with initial condition that I(t=0) = 0, we find

� Where the time constant is defined as:

Constante de tiempoinductiva

RL Circuit� When switch S2 is moved to

position b, the original current disappears. The self-induced emfwill try to prevent that change, and this determines the emf direction (Lenz Law).

τ− −= ≡Rt L tε εI e e

R R

( )= = E0 RI t� Solve for the current I, with initial

condition that we find

0=+dtdI

LIR

Energy stored in an inductor

The increasing current I from the battery supplies power not only to the resistor, but also to the inductor. From Kirchhoff’s loop rule, we have

= + d Iε I R L

dtMultiply both sides with I:

= +2 d IεI I R LI

dtThis equation reads: powerbattery=powerR+powerL

So we have the rate of energy increase in the inductor as:

=LdU d ILI

dt dt

Solve for UL: = =∫2

0

12

I

LU LId I LI

Stored energy type and the Energy Density of a Magnetic Field

� Given UL = ½ L I2 and assume (for simplicity) a solenoid with L = µo n2 V

� Since V is the volume of the solenoid, the magnetic energy density, uB is

� This applies to any region in which a magnetic field exists (notjust the solenoid)

= =

2 221

2 2L oo o

B BU µ n V V

µ n µ

≡ =2

2L

Bo

U Bu

V µ

So the energy stored in the solenoid volume V is magnetic (B) energy.

And the energy density is proportional to B2.

RL and RC circuits comparison

Energy

Discharging

Charging

RCRL

−= Rt LεI e

R

( )−= −1 Rt LεI e

R

= 212LU LI

221

( )2 2C

QU C V

C∆= =

( )−

=t

RCεI t e

R

( )−

=t

RCQI t e

RC

Energy density

Electric fieldMagnetic field

=2

2Bo

Bu

µ= 21

2E ou ε E

Energy Storage Summary

� Inductor and capacitor store energy through different mechanisms� Charged capacitor

� Stores energy in the electric field

� When current flows through an inductor � Stores energy in the magnetic field

� A resistor does not store energy � Energy delivered is transformed into thermo energy

Oscilaciones electromagnéticas: cualitativas

Oscilaciones electromagnéticas: cualitativas

Oscilaciones electromagnéticas: cualitativas

� Analogía con el MAS� q↔x� i↔v� 1/C↔k� L↔m

mk

f == πω 2 LCf

12 == πω

Oscilaciones electromagnéticas: cualitativas

Oscilaciones electromagnéticas: cuantitativas

EB UUU +=

Cq

LiU2

2

21

21 +=

cteU = 0=dtdU

Oscilaciones electromagnéticas: cuantitativas

01

2

2

=+ qLCdt

qd02

2

=+ xmk

dtxd

)cos( φω += txx m

)cos( φω += tqq m

Oscilaciones electromagnéticas: cuantitativas

)(cos22

1 222

φω +== tC

qCq

U mE

)(21

21 2222 φωω +== tsenqLLiU mB

)(2

22

φω += tsenC

qU m

B

Sustituyendo ω: