Physics 2102 Physics 2102 Lecture 18Lecture 18

Ch30: Ch30:

Inductors & Inductance IIInductors & Inductance II

Physics 2102

Jonathan Dowling

Nikolai Tesla

dA

Faraday’s LawFaraday’s Law• A time varying magnetic

FLUX creates an induced EMF

• Definition of magnetic flux is similar to definition of electric flux

Bn

dt

dEMF BΦ

−=

B

S

B dAΦ = ⋅∫rr

• Take note of the MINUS sign!!• The induced EMF acts in such a way that it OPPOSES the change in magnetic flux (“Lenz’s Law”).

dA

Another formulation of Another formulation of Faraday’s LawFaraday’s Law

• We saw that a time varying magnetic FLUX creates an induced EMF in a wire, exhibited as a current.

• Recall that a current flows in a conductor because of the forces on charges produced by an electric field.

• Hence, a time varying magnetic flux must induce an ELECTRIC FIELD!

• But the electric field line would be closed!!?? What about electric potential difference V=∫E•ds?

Bn

dt

dsdE B

C

Φ−=⋅∫

rr

Another of Maxwell’s equations!To decide SIGN of flux, use right hand rule: curl fingers around loop C, thumb indicates direction for dA.

Example Example A long solenoid has a circular cross-section of radius R.The current through the solenoid is increasing at a steady rate di/dt. Compute the electric field as a function of the distance r from the axis of the solenoid.

R

First, let’s look at r < R:

2

20

(2 ) ( )

( )

dBE r r

dtdi

r ndt

π π

π μ

=

=

0

2

n diE r

dt

μ=

Next, let’s look at r > R:

dt

dBRrE )()2( 2ππ =

20

2

n di RE

dt r

μ=

magnetic field lines

electric field lines

dt

dsdE B

C

Φ−=⋅∫

rr

The electric current produces a magnetic field B=μ0ni, which changes with time, and produces an electric field.The magnetic flux through circular disks Φ=∫BdA is related to the circulation of the electric field on the circumference ∫Eds.

Example (continued)Example (continued)

r

E(r)

r = R

0

2

n diE r

dt

μ=

20

2

n di RE

dt r

μ=

magnetic field lines

electric field lines

SummarySummary

Two versions of Faradays’ law:– A varying magnetic flux produces an EMF:

– A varying magnetic flux produces an electric field:

dt

dEMF BΦ

−=

dt

dsdE B

C

Φ−=⋅∫

rr

Inductors are with respect to the magnetic field what capacitors are with respect to the electric field. They “pack a lot of field in a small region”. Also, the higher the current, the higher the magnetic field they produce.

Capacitance how much potential for a given charge: Q=CV

Inductance how much magnetic flux for a given current: Φ=Li

(Henry)H Ampere

mTesla][ :Units

2

≡⋅=LJoseph Henry (1799-1878)

Inductors: SolenoidsInductors: Solenoids

dt

diLEMF −=Using Faraday’s law:

““Self”-Inductance of a solenoidSelf”-Inductance of a solenoid• Solenoid of cross-sectional

area A, length l, total number of turns N, turns per unit length n

• Field inside solenoid = μ0 n i

• Field outside ~ 0

i

Al

NNAn

2

00 μμ ==L = “inductance”

LiniNANABB ===Φ 0μ

ExampleExample • The current in a 10 H inductor is

decreasing at a steady rate of 5 A/s.

• If the current is as shown at some instant in time, what is the magnitude and direction of the induced EMF?

• Magnitude = (10 H)(5 A/s) = 50 V • Current is decreasing• Induced emf must be in a direction

that OPPOSES this change.• So, induced emf must be in same

direction as current

(a) 50 V

(b) 50 V

i

dt

diLEMF −=