1. Exam 2: Multiple choice, 90-max

2. Gauss’s Law Can derive one from another Gauss’s law is more universal:
works at relativistic speeds
Gauss’s law:
If we know the field distribution on closed surface
we can tell what is inside.
Knowing E can conclude what is inside
Knowing charges inside can conclude what is E

3. Gauss’s Law: Circuits
Can we have excess charge inside in steady state?
We already have established that in steady state for a uniform conductor, the E-field is the same magnitude and follows the wire. Thus, Gauss’s law gives no net flux and hence no net charge contained within the conductor. But current carriers are all negative. Where does the positive charge reside?

4. Gauss’s Law: Junction Between two Wires
i1=i2
n1Au1E1 = n2Au2E2
There is negative charge along the interface!
n2<n1
u2<u1
Take Gaussian surface just inside the conductor since we are not interested in surface charges. We’ve already shown that within each uniform conductor there is no charge.

5. Clicker rA rB rC Q1 Q2
A solid conducting sphere of radius rA has charge +Q1 uniformly distributed over its surface.
Concentric with the solid sphere is a conducting shell of with charge Q2 on its outer surface.
What is the charge Q on the inner surface of the outer shell?
+Q2
-Q2
+Q1
-Q1

6. Patterns of Magnetic Field in Space
Is there current passing
through these regions?
There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area.
Can we predict current pattern from knowing B pattern?
Ampere’s law

7. Quantifying the Magnetic Field Pattern
Curly character – introduce:
Long wire pattern
Similar to Gauss’s law (Q/0)
Will it work for any circular path of radius r ?

8. A Noncircular Path (home study)
Need to compare and
Long wire pattern

9. Currents Outside the Path (home study)
Need to compare and
Long wire pattern
for currents outside the path

10. Three Current-Carrying Wires
Ampere’s law
Similar to Gauss's law – only charges inside matter

11. Ampère’s Law All the currents in the universe contribute to B
but only ones inside the path result in nonzero path integral
Ampere’s law is almost equivalent to the Biot-Savart law:
but Ampere’s law is relativistically correct
Andre Marie Ampere , never attended school (farther educated him). Read Encyclopedia through starting from A etc.
First paper at age 13 – not accepted…
. Maxwell, writing about this Memoir in 1879, says:-
We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it.
Ampere, 1826: Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience

12. Inside the Path Ampere’s law Choose the closed path
Imagine surface (‘soap film’) over the path
3. Walk counterclockwise around the path adding up
Count upward currents as positive, inward going as negative
(right hand rule: 4 fingers along the path then thumb is positive current)

13. Ampere’s Law: A Long Thick Wire
Can B have an out of plane component?
Is it always parallel to the path?
for thick wire:
(the same as for thin wire)
Symmetry makes problem easy to solve using Ampere's law – counting currents is simple, and path integral is simple due to the symmetry
Would be hard to derive using Biot-Savart law

14. Clicker R r1
What is the magnetic field inside a wire (wire radius is R) at distance r1 from its center?
B=20r1I/(R2)
B=r1I/(20R2)
B=0r1I/2R2
B=0R2I/2r1R2 C

15. Ampere’s Law: find B inside a solenoid
Number of wires: (N/L)d
What is on sides?
B outside is very small
(solenoid)
Talk importance:
for getting uniform magnetic field
electromagnets
On sides: draw on board (symmetry) to show that B is perp to path on sides (wires to left and right total in only perp component)
:ength d disappears – and it must, B should not depend on some mathematical path
Uniform: same B no matter where is the path

16. Gauss’s Law for Magnetism
Dipoles:
Electric field: ‘+’ and ‘–’ charges can be separated
Magnetic field: no monopoles
Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field.
One cannot separate them  total magnetic ‘charge’ is zero. or
Gauss’s law for magnetism

17. Maxwell’s Equations Three equations: Gauss’s law for electricity
Gauss’s law for magnetism
Ampere’s law for magnetism
Is anything missing?
(incomplete)
‘Ampere’s law for electricity’
First two: integrals over a surface
Second two: integrals along a path
Incomplete: no time dependence

18. Predicting Possible Field Configuration
Is the following “curly” pattern of electric field possible? dl dl
is always parallel to
Cannot produce with any stationary arrangement of charges – but could be in principle, will talk in later chapters.
Move from A to A dl
This “curly” pattern of electric field is impossible to produce by arranging any number of stationary point charges!

19. Chapter 23
Faraday’s Law

20. Faraday’s Experimentation
Set wire (connected to ammeter) around a solenoid.
Can a current be observed in this wire?

21. Changing Magnetic Field
Solenoid: inside
outside
Constant current: there will be no forces
on charges outside (B=0, E=0)
What if current is not constant in time? Let B increase in time
What will happen if we put a loop of wire in?
Round trip is not zero!
Got curly electric field
E~dB/dt
E~1/r
Non-Coulomb ENC !

22. Two Ways to Produce Electric Field
1. Coulomb electric field: produced by charges
2. Non-Coulomb electric field:
using changing magnetic field
Field outside of solenoid
Same effect on charges:
What will happen if we put a loop of wire in?

23. Direction of the Curly Electric Field
Right hand rule:
Thumb in direction of
fingers: ENC
Exercise:
Magnetic field points down
from the ceiling and is increasing.
What is the direction of E?
“Current opposes change in B”

24. Driving Current by Changing B
ENC causes current to
run along the ring
What is emf and I?
Ring has resistance, R
What will happen if we put a loop of wire in?
It can make a current run in a wire just as though a battery was present!

25. Effect of the Ring Geometry
1. Change radius r2 by a factor of 2.
emf does not depend on radius of the ring!
Double radius -> half the electric field
2. One can easily show that emf will be the same for any circuit
surrounding the solenoid

26. Round-Trip Not Encircling the Solenoid=0 =0
for any path which does not
enclose the solenoid!
The non-coulomb electric field will polarize the wire, but would not drive a current around the loop.
Since integral does not depend on radial distance (due to cancellation of 1/r dependence of E_NC and r dependence of dl ( =r*d(theta) ), emf = 0.

27. Exercise
Is there current in these circuits?

28. Quantitative Relationship Between B and EMF
Can observe experimentally:
I=emf/R
ENC~emf
ENC~dB/dt
ENC~ cross-section of a solenoid
Increase current -> current runs clockwise out if the + terminal of the ammeter
emf does not depend on loop geometry!
Our demo set is not good enough to detect this emf, need more sensitive ammeter etc

29. Magnetic Flux magnetic flux mag on the area encircled by the circuit
(wire loop)
Magnetic flux on a small area A:
Magi. flux is calc the same way as electric flux
Definition of magnetic flux:
This area does not enclose a volume!

30. Faraday’s Law
Michael Faraday
( )
Faraday’s law cannot be derived from the other fundamental principles we have studied
Formal version of Faraday’s law:
Unlike motional emf – Faraday cannot be derived
. A British physicist and chemist, he is best known for his discoveries of electro-magnetic rotation, electro-magnetic induction and the dynamo.
Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. He demonstrated this principle of induction in Faraday expressed the electric current induced in the wire in terms of the number of lines of force that are cut by the wire. The principle of induction was a landmark in applied science, for it made possible the dynamo, or generator, which produces electricity by mechanical means
Sign: given by right hand rule
“Current opposes change in B”

31. Various ways of making changing magnetic field
Change current through the solenoid
Time varying B can be
produced by moving coil:
by moving magnet:
move coil – get changing B and curly E
How to create a time-varying magnetic field?
by rotating magnet:
(or coil)

32. Clicker
Solenoid has circumference 10 cm. The current I1 is increasing causing increasing B1 as in the figure.
Solenoid is surrounded by a wire with finite resistance.
When length of the wire is changed from 30 to 20 cm what will happen to detected current?
It will decrease
Increase
Does not change
ammeter + - B

33. A Circuit Surrounding a Solenoid
Example:
B1 changes from 0.1 to 0.7 T in 0.2 seconds; area=3 cm2.
Application of Faraday’s law
Circuit acts as battery.
What is the ammeter reading? (resistance of ammeter+wire is 0.5)

34. Clicker: A Circuit Not Surrounding a Solenoid
If we increase current through solenoid what will be ammeter reading?
positive current
negative current
zero
zero

35. The EMF for a Coil With Multiple Loops
Each loop is subject to similar
magnetic field  emf of loops
in series:
Series connection of many loops with the same emf

36. Exercise
1. A bar magnet is moved toward a coil. What is the ammeter reading (+/-)?
2. The bar magnet is moved away from the coil.
What will ammeter read?
S side in = B –field increasing and points to the left, to opposes the change current should run CW – positive current into (+) side of voltmeter
move coil – get changing B and curly E
Positive
Negative

37. Two Ways to Produce Electric Field
1. Coulomb electric field: produced by charges
2. Non-Coulomb electric field:
using changing magnetic field
Field outside of solenoid
Same effect on charges:
What will happen if we put a loop of wire in?

38. Direction of the Curly Electric Field
Right hand rule:
Thumb in direction of
fingers: ENC
Exercise:
Magnetic field points down
from the ceiling and is increasing.
What is the direction of E?
“Current opposes change in B”

39. Driving Current by Changing B
ENC causes current to
run along the ring
What is emf and I?
Ring has resistance, R
What will happen if we put a loop of wire in?
It can make a current run in a wire just as though a battery was present!

40. Effect of the Ring Geometry
1. Change radius r2 by a factor of 2.
emf does not depend on radius of the ring!
Double radius -> half the electric field
2. One can easily show that emf will be the same for any circuit
surrounding the solenoid

41. Round-Trip Not Encircling the Solenoid=0 =0
for any path which does not
enclose the solenoid!
The non-coulomb electric field will polarize the wire, but would not drive a current around the loop.
Since integral does not depend on radial distance (due to cancellation of 1/r dependence of E_NC and r dependence of dl ( =r*d(theta) ), emf = 0.

42. Exercise
Is there current in these circuits?

43. Quantitative Relationship Between B and EMF
Can observe experimentally:
I=emf/R
ENC~emf
ENC~dB/dt
ENC~ cross-section of a solenoid
Increase current -> current runs clockwise out if the + terminal of the ammeter
emf does not depend on loop geometry!
Our demo set is not good enough to detect this emf, need more sensitive ammeter etc

44. Magnetic Flux magnetic flux mag on the area encircled by the circuit
(wire loop)
Magnetic flux on a small area A:
Magi. flux is calc the same way as electric flux
Definition of magnetic flux:
This area does not enclose a volume!

45. Faraday’s Law
Michael Faraday
( )
Faraday’s law cannot be derived from the other fundamental principles we have studied
Formal version of Faraday’s law:
Unlike motional emf – Faraday cannot be derived
. A British physicist and chemist, he is best known for his discoveries of electro-magnetic rotation, electro-magnetic induction and the dynamo.
Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. He demonstrated this principle of induction in Faraday expressed the electric current induced in the wire in terms of the number of lines of force that are cut by the wire. The principle of induction was a landmark in applied science, for it made possible the dynamo, or generator, which produces electricity by mechanical means
Sign: given by right hand rule