Lecture 35: MON 17 NOV��� CH32: Maxwell’s Equations I

James Clerk Maxwell (1831-1879)

Physics 2113 Jonathan Dowling

Maxwell I: Gauss’ Law for E-Fields: ���charges produce electric fields, ���

field lines start and end in charges

!E ⋅d!A = qins / ε0

S"∫

S

S S S

Maxwell II: Gauss’ law for B-Fields: ���

field lines are closed���or, there are no magnetic monopoles

!B id!A = 0

S"∫

S

S

S

S

S

S

Φ = BA∑ = 0

ΦaT&B = 2 × 6 − 4 × 3= 12 −12 = 0

ΦaSide = −Φa

T&B = 0

ΦaSide = 0

ΦbT&B = −2 ×1− 4 × 2 = −2 − 8 = −10

ΦbSide = −Φb

T&B = 10

ΦbSide = 10

ΦcT&B = −2 × 6 + 2 × 8 = −12 +16 = +4

ΦcSide = −Φc

T&B = −4

ΦcSide = 4

ΦdT&B = +2 × 3+ 3× 2 = +6 + 6 = +12

ΦdSide = −Φd

T&B = −12

ΦdSide = 12

d > b > c > a = 0

Maxwell III: Ampere’s law: ���electric currents produce magnetic fields

!B id!s = µ0iins

C"∫

C

Maxwell IV: Faraday’s law: ���changing magnetic fields produce (“induce”) ���

electric fields

!E id!s

C"∫ = − d

dt!B id!A

S∫ = − dΦB

dt

Maxwell Equations I – IV:

!E id!A = qins / ε0

S"∫

!B id!A = 0

S"∫

!B id!s = µ0iins

C"∫

!E id!s

C"∫ = − d

dt!B id!A

S∫ = − dΦB

dt

In Empty Space with No Charge or Current

…very suspicious… NO SYMMETRY!

? q=0��� i=0

!B id!A = 0

S"∫

!B id!s = 0

C"∫ !?

!E id!A = 0

S"∫

!E id!s

C"∫ = − d

dt!B id!A

S∫ = − dΦB

dt

Maxwell’s Displacement Current If we are charging a capacitor, there is a current left and right of the capacitor. Thus, there is the same magnetic field right and left of the capacitor, with circular lines around the wires. But no magnetic field inside the capacitor? With a compass, we can verify there is indeed a magnetic field, equal to the field elsewhere. But Maxwell reasoned this without any experiment! But there is no current producing it! ?

E B B

The missing Maxwell Equation!

Maxwell’s Fix

i = dqdt

= d(CV )dt

= C dVdt

= ε0Ad

d(Ed)dt

= ε0d(EA)dt

= ε0dΦE

dt

We can write the current as:

We calculate the magnetic field produced by the currents at left and at right using Ampere’s law :

!B id!s = µ0iins

C"∫

E

id=ε0dΦE/dt

q =VC C = ε0A / d V = Ed ΦE =

!E id!A

S"∫ = EA

B!"

ids"= µ0

C#∫ ε0

ddt"E id"A

S∫ = µ0ε0

dΦE

dtB !

E

i

B

i

B

Displacement “Current” Maxwell proposed it based on symmetry and math — no experiment!

!B id!s ≠ 0

C"∫

Changing E-field Gives Rise to B-Field!

Maxwell’s Equations I – V:

∫ =•S

qdAE 0/ε

∫ =•S

dAB 0

idAEdtddsB

SC000 µεµ +•=• ∫∫

∫∫ •−=•SC

dABdtddsE

I

II

III

IV

V

Maxwell Equations in Empty Space:

Fields without sources? Changing E gives B.

Changing B gives E.

!B id!A = 0

S"∫

!B id!s = + d

dt!E id!A

S∫

C"∫ = + dΦE

dt

!E id!A = 0

S"∫

!E id!s

C"∫ = − d

dt!B id!A

S∫ = − dΦB

dtPositive Feedback Loop!

32.3: Induced Magnetic Fields:

Here B is the magnetic field induced along a closed loop by the changing electric flux FE in the region encircled by that loop.

Fig. 32-5 (a) A circular parallel-plate capacitor, shown in side view, is being charged by a constant current i. (b) A view from within the capacitor, looking toward the plate at the right in (a).The electric field is uniform, is directed into the page (toward the plate), and grows in magnitude as the charge on the capacitor increases. The magnetic field induced by this changing electric field is shown at four points on a circle with a radius r less than the plate radius R.

dΦE

dt= ddtEA = A dE

dt∝ slope

Ba > Bc > Bb > Bd = 0

32.3: Induced Magnetic Fields: Ampere Maxwell Law:

Here ienc is the current encircled by the closed loop. In a more complete form, When there is a current but no change in electric flux (such as with a wire carrying a constant current), the first term on the right side of the second equation is zero, and so it reduces to the first equation, Ampere’s law.

32.3.2. When a parallel-plate capacitor is charging, there is both an electric field and an induced magnetic present between the plates. After some time, the charging stops, which of the following statements is true concerning the fields within the capacitor?

a) The magnetic field is zero, but the electric field is constant. b) The magnetic field is zero; and the electric field slowly decreases to

zero over time. c) Both the electric and magnetic fields are equal to zero. d) The electric field is zero; and the magnetic field slowly decreases to

zero over time. e) The electric field is zero, but the magnetic field is constant.

32.3.2. When a parallel-plate capacitor is charging, there is both an electric field and an induced magnetic present between the plates. After some time, the charging stops, which of the following statements is true concerning the fields within the capacitor?

a) The magnetic field is zero, but the electric field is constant. b) The magnetic field is zero; and the electric field slowly decreases to

zero over time. c) Both the electric and magnetic fields are equal to zero. d) The electric field is zero; and the magnetic field slowly decreases to

zero over time. e) The electric field is zero, but the magnetic field is constant.

Example, Magnetic Field Induced by Changing Electric Field:

Example, Magnetic Field Induced by Changing Electric Field, cont.:

32.4: Displacement Current:

Comparing the last two terms on the right side of the above equation shows that the term must have the dimension of a current. This product is usually treated as being a fictitious current called the displacement current id: in which id,enc is the displacement current that is encircled by the integration loop. The charge q on the plates of a parallel plate capacitor at any time is related to the magnitude E of the field between the plates at that time by in which A is the plate area. The associated magnetic field are:

AND

Example, Treating a Changing Electric Field as a Displacement Current:

32.5: Maxwell’s Equations: