1. ELEC 3105 Basic EM and Power Engineering
Faraday’s Law
Lenz’s Law
Displacement Current

2. Introduction: So far we have
Faraday’s Law
Introduction: So far we have
These equations are OK for static fields, i.e. those fields independent of time. When fields vary as a function of time the curl equations acquire an additional term.
gets a
gets a

3. ELEC 3105 Basic EM and Power Engineering
Faraday’s Law

4. Faraday’s Law
Consider the following experiment. Pull a wire loop through a region of non-uniform magnetic field
Wire loop, path encloses a surface S
area gained in time
area lost in time
Magnetic field vector points into the page

5. Faraday’s Law
Consider the following experiment. Pull a wire loop through a region of non-uniform magnetic field
Charge in wire
Charge in wire
Magnetic field vector points into the page

6. Faraday’s Law
𝑊 𝑞 =𝑉
Early definition of potential
Consider the work done on +1C test charge moved around the loop; this is the “emf” electromotive force.
Examine
Expression
Charge in wire
Charge in wire

7. Faraday’s Law
Consider the work done on +1C test charge moved around the loop; this is the “emf” electromotive force.
Now:
Flux change at right side of loop
Flux change at left side of loop
Note that B and da are in opposite directions

8. Faraday’s Law
This is a general result. Even if we hold the loop stationary and change B, the emf is still given by the negative rate of change of the flux.
N S
Move magnet

9. The induced emf always opposes the change in
Faraday’s Law
Further generalization is possible. Consider moving loop in time varying magnetic field.
The induced emf always opposes the change in
flux
MOTIONAL emf
TRANSFORMER emf
N S
Move magnet
Move loop

10. ELEC 3105 Basic EM and Power Engineering
Lenz’s Law

11. The induced emf always opposes the change in flux
Faraday’s Law / Lenz’s Law
The induced emf always opposes the change in flux
motional emf
Move loop towards magnet
B increases in loop
Flux increases in loop
Current induced through emf
Current produces magnetic field in loop: 2nd postulate
This magnetic field in opposite in direction to magnetic field of magnet I
N S
MOVE LOOP

12. The induced emf always opposes the change in flux
Faraday’s Law / Lenz’s Law
The induced emf always opposes the change in flux
motional emf
Move loop away from magnet
B decreases in loop
Flux decreases in loop
Current induced
Current produces magnetic field in loop
This magnetic field in same direction to magnetic field of magnet I
N S
MOVE LOOP

13. The induced emf always opposes the change in flux
Faraday’s Law / Lenz’s Law
The induced emf always opposes the change in flux
transformer emf
Move magnet towards loop
B increases in loop
Flux increases in loop
Current induced
Current produces magnetic field in loop
This magnetic field in opposite direction to magnetic field of magnet I
N S
MOVE MAGNET

14. The induced emf always opposes the change in flux
Faraday’s Law / Lenz’s Law
The induced emf always opposes the change in flux
transformer emf
Move magnet away from loop
B decreases in loop
Flux decreases in loop
Current induced
Current produces magnetic field in loop
This magnetic field is in same direction to magnetic field of magnet I
N S
MOVE MAGNET

15. Faraday’s Law N S MOVE MAGNET
Suppose loop is stationary so we have only transformer emf.
Equivalent battery to drive current around loop
N S
MOVE MAGNET

16. Faraday’s Law in derivative form: Valid for all points in space

17. The induced emf always opposes the change in
Note on: Faraday’s Law / Lenz’s Law
The induced emf always opposes the change in
flux
WE KNOW THAT
𝐵 𝑡 ∝𝐼 𝑡
𝐵 𝑡 = 𝐾 ∗𝐼 𝑡
constant
Inductance: relates the induced emf to the time rate of change of the current
𝑣 𝑡 =−𝐿 𝜕𝐼 𝑡 𝜕𝑡

18. Note on: Faraday’s Law / Lenz’s Law
Rotating current loop in constant magnetic field 𝜔 O 𝜃 𝐵 X
𝐵 ∙ 𝑑𝑎 =𝐵 𝑑𝑎 𝑐𝑜𝑠 𝜔𝑡
Sinusoidal voltage change
Amplitude of voltage depends on B, A and 
POWER GENERATOR

19. ? Note on: Faraday’s Law / Lenz’s Law
Moving rod in constant magnetic field 𝐵 X X X X X X X X L X X X X - +
Which end of the rod is positive and which end is negative? ? + -

20. ELEC 3105 Basic EM and Power Engineering
Displacement Current

21. NOW it might make sense, if
Figure Application of Ampere’s law to a circuit with an air-filled capacitor and time-varying current.
NOW it might make sense, if
i = dD/dt Sp
Taken from ELEC 3909

22. Displacement Current For surface A chosen
So far we have the following expression for Ampere’s law.
WIRE
Certainly there is no problem in evaluating the integral. The path shown encloses an area A through which the wire cuts.
We are in fact in the process of charging the capacitor through the current I of the wire.
Capacitor
For surface A chosen

23. Displacement Current For path A’ chosen Capacitor
So far we have the following expression for Ampere’s law.
WIRE
Certainly there is a problem in evaluating the integral. The path shown encloses an area A’ through which the wire does not cut. Yet the integral value evaluated here for the surface that does not contain the wire A’ and the surface of the previous slide A which does contain the wire must be the same since the surfaces A’ and A are arbitrary.
We are in fact in the process of charging the capacitor through the current I of the wire.
Capacitor
For path A’ chosen
Both integrals should give the same result??????

24. Displacement Current We have a problem Capacitor Capacitor WIRE WIRE
For different surfaces

25. Solution to the problem
Displacement Current
Solution to the problem
Between the capacitor plates we have a changing electric flux density. This changing electric flux density is equivalent to a current density.

26. Solution to the problem
Displacement Current
Solution to the problem
Ampere’s law in integral form
Apply Stoke’s theorem
Ampere’s law in derivative form

27. Ampere’s Law in derivative form: Valid for all points in space
Displacement Current