The story so far

1
The story so far
dB
r
dI
Magnetic field generated by current element
Biot-Savart
I
Amperes law
closed path
surface bounded by path
2
Exam 2 results

• Grade cutoffs
• A 86
• AB 79
• B 66
• BC 58
• C 37
• D 23

Ave69
3
Amperes law

• Sum up component of B around path
• Equals current through surface.

Component of B along path
I
closed path

• Amperes law

surface bounded by path
4
Amperes law in electrostatics
Work done by E-field
So is work per unit charge to bring charge back
to where it started.
This is zero.
5
Gauss law in electrostatics

• Electric flux through surface ? charge enclosed

What about magnetic flux?
6
Magnetic flux

• Magnetic flux is defined exactly as electric
  flux
• (Component of B ? surface) x (Area element)

zero flux
Maximum flux
SI unit of magnetic flux is the Weber ( 1 T-m2 )
7
Magnetic flux

• What is that magnetic flux through this surface?

1. Positive
2. Negative
3. Zero

8
Gauss law in magnetostatics

• Net magnetic flux through any closed surface is
  always zero

Compare to Gauss law for electric field
No magnetic charge, so right-hand side0
for mag. Basic magnetic element is the dipole
9
Comparison with electrostatics
Gauss law
Amperes law
Electrostatics
Magnetostatics
10
Time-dependent fields

• Up to this point, have discussed only magnetic
  and electric fields constant in time.
• E-fields arise from charges
• B-fields arise from moving charges (currents)

Faradays discovery

• Another source of electric field
• Time-varying magnetic field creates electric field

11
Measuring the induced field

• A changing magnetic flux produces an EMF around
  the closed path.
• How to measure this?
• Use a real loop of wire for the closed path.

The EMF corresponds to a current flow
12
Current but no battery?

• Electric currents require a battery (EMF)
• Faraday Time-varying magnetic field creates EMF

Faradays law
EMF around loop - rate of change of mag. flux
13
Faradays law
EMF around loop
Magnetic flux through surface bounded by path
EMF no longer zero around closed loop
14
Quick quiz

• Which of these conducting loops will have
  currents flowing in them?

I(t) increases
Constant I
Constant v
Constant v
Constant I
Constant I
15
Faradays law

• Faradays law
• Time-varying B-field creates E-field
• Conductor E-field creates electric current

• Biot-Savart law
• Electric current creates magnetic field

• Result
• Another magnetic field created

16
Lenzs law

• Induced current produces a magnetic field.
• Interacts with bar magnet just as another bar
  magnet
• Lenzs law
• Induced current generates a magnetic field that
  tries to cancel the change in the flux.
• Here flux through loop due to bar magnet is
  increasing. Induced current produces flux to
  left.
• Force on bar magnet is to left.

17
Quick quiz

• What direction force do I feel due to Lenz law
  when I push the magnet down?

1. Up
2. Down
3. Left
4. Right

Strong magnet
Copper
18
Quick Quiz

• A conducting rectangular loop moves with constant
  velocity v in the x direction through a region
  of constant magnetic field B in the -z direction
  as shown.
• What is the direction of the induced loop current?

1. CCW
2. CW
3. No induced current

y
x
19
Quick Quiz

• Conducting rectangular loop moves with constant
  velocity v in the -y direction away from a wire
  with a constant current I as shown. What is the
  direction of the induced loop current?

I

1. CCW
2. CW
3. No induced current

v
B-field from wire into page at loopLoop moves to
region of smaller B, so flux decreasesInduced
loop current opposes this change, so must create
a field in same direction as field from wire -gt
CW current.
20
Motional EMF

• Conductor moving in uniform magnetic field
• / - charges in conductor are moving.
• Magnetic field exerts force.

Charges pile up at ends Static equilibrium
E-field generated canceling magnetic force
Solid conductor