1. Inductance

2. Inductance Learning Objectives
Definition and Calculation of Self-Inductance
To obtain an expression for the Energy Stored by an Inductor
Definition and Calculation of Mutual-Inductance

3. Let’s start with: SELF-INDUCTANCE
The phenomenon of self-inductance was discovered by Joseph Henry in 1832 (Princeton University).
Joseph Henry

4. Self Inductance
When current in the circuit changes, the flux changes also, and a self-induced voltage appears in the circuit

5. L is the self-inductance of the coil.
I constant, e= 0
I increasing or decreasing
, e = Vab>0 b a
L is the self-inductance of the coil.

6. (a) Definition used to find L
Suppose a current I in a coil of N turns causes a flux B to thread each turn
The self-inductance L is defined by the equation

8. From Faraday’s Law of Induction
(b) Definition that describes the behaviour of an inductor in a circuit
From Faraday’s Law of Induction

9. Two equivalent definitions of L

10. This is called the henry (H)
SI unit for inductance
V s A-1
This is called the henry (H)
If a current changing by 1A/s is to generate 1V, the inductance is 1H.

11. Calculation of Self-Inductance

12. The Self-Inductance of a Solenoid
n turns per unit length, radius R
and the length of the solenoid is l

13. Set up a current I, and we have a B field
Total number of turns is N=nl
Flux through each turn

14. number of turns per unit length)
The inductance does not depend
on current or voltage, it is a property
of the coil. (length, width, and
number of turns per unit length)

15. Find the self-inductance of a solenoid of length 10 cm, area 5 cm2, and 100 turns.
n = 100/0.1 = 1000 turns/m
At what rate must the current in the solenoid change to induce a voltage of 20 V?
Answer: 3.18  105 A/s

16. The Self-Inductance of a Toroidb a b

17. The Self-Inductance of a Toroiddr h r b
Consider an elementary strip of area hdr

18. The Self-Inductance of a Toroid
Inductance – like capacitance – depends only on geometric factors

19. Unit of 0 is H m-1 0 = 4  10-7 H m-1
From the worked examples it can be seen that:
Unit of 0 is H m-1
0 = 4  10-7 H m-1
0 = 4  10-7 wb/Am

20. The Energy Stored by an Inductor
I increasing a b
(Faraday’s law in disguise) 
The energy dU supplied to the inductor during an infinitesimal time interval dt is:

21. The Energy Stored by an Inductor
The total energy U supplied while the current increases from zero to a final value I is
This energy is stored in the magnetic field

22. The energy stored in the magnetic field of an inductor is analogous to that in the electric field of a capacitor

23. Example: the energy stored in a solenoid
Energy per unit volume (magnetic energy density)

24. The equation is true for all magnetic field configurations
MAGNETIC ENERGY DENSITY IN A VACUUM
The equation is true for all magnetic field configurations
Compare with the energy density in an electric field

25. Review and Summary
The self-inductance L is defined by the equations

26. Review and Summary
An inductor with inductance L carrying current I has potential energy
This potential energy is associated with the magnetic field of the inductor. In a vacuum, the magnetic energy per unit volume is

27. How would the self-inductance of a solenoid be changed if
the same length of wire were wound onto a cylinder of the same diameter but twice the length?
twice as much wire were wound onto the same cylinder?
the same length of wire were wound onto a cylinder of the same length but twice the diameter?

28. (a) Since the diameter does not change, the number of turns and the area A remain constant. However, n2 is diminished by a factor of 4 and l is increased by a factor of 2. Thus L is reduced by a factor of 2.
(b) Using twice as much wire and making no other change, n2 and L are increased by a factor of 4.
(c) With twice the diameter, n2 is reduced by a factor of 4, but A is increased by the same factor; L is unchanged.

29. Mutual Inductance
A changing current in loop 1 causes a changing flux in loop 2 inducing a voltage

30. Mutual Inductance
(1) The mutual inductance M21 is defined by the equation:
(2) The mutual inductance M21 may also be defined by the equation:

31. Mutual Inductance
It can be proved that the same value is obtained for M if one considers the flux threading the first loop when a current flows through the second loop
(mutual inductance)
(mutually induced voltages)

32. A Metal Detector Sinusoidally varying current
Parallel to the magnetic field of Ct

33. Review and Summary
If two coils are near each other, a changing current in either coil can induce a voltage in the other. This mutual induction phenomenon is described by
where M (measured in henries) is the mutual inductance for the coil arrangement

34. Revision – Ampere’s Circuital Law
Where
is the line integral round a closed loop
and
Ienclosed is the current enclosed by the loop

35. The B Field Due to a Long Straight Wire

36. Next Installment
Magnetic Materials