Ampere-Maxwell’s Law: Maxwell correction to Ampere’s Law Maxwell said that there has to be some current existing between plates. There are no conduction electrons between plates BUT there is time-varying E between plates, which generates a current. The E is directed from +ve plate to –ve plate, because the amount of charge on capacitor increases with time, AND The E between plates increases with time, as well. He called this new current as a displacement current. Here are links of other videos on Maxwell's equations: Electric Field due to a Solid Sphere of Charge (Gauss’s Law for Electrostatics: Applications) https://www.youtube.com/watch?v=sp5qV Electric Field due to Uniform Spherical Shell of Charge https://www.youtube.com/watch?v=AnPvK... Ampere's law: https://www.youtube.com/watch?v=y-DpDn6BI_Q Applications of Gauss's law: Electric field due to an infinitely large plane sheet of charge: https://www.youtube.com/watch?v=oJ7bA... Applications of Gauss’s law: Electric field of an infinitely long uniformly charged conductor https://www.youtube.com/watch?v=FwssH...

1. 4. Ampere-Maxwell’s Law
B dl I
   0
Ampere’s Law
‘The line integral of the magnetic field B around any closed
loop is 𝜇0 times the net steady current I enclosed by this
path”
Maxwell’s Equations
(1)
dl
I
p
dl
P
B
B
ds
B.dl = B dl cos 0= B dl
ds ds
As, B is uniform at each point on the field
line:
ds
B.dl = B (2π r) (2)
Equating (1) and (2):
B =
𝜇0 𝑰
2π r
i.e. the magnetic field a distance r from a long
straight wire carrying a steady current I.
https://www.youtube.com/channel/UC1hrAjhSKGAsWZ7LvR_rbRg
Here is a link of Youtube video. You can watch this lecture on Youtube, as
well.

2. 4. Ampere-Maxwell’s Law
B dl I
   0
Ampere’s Law
‘The line integral of the magnetic field B around any closed
loop is 𝜇0 times the net steady current I enclosed by this
path”
Maxwell’s Equations
(1)
dl
I
p
dl
P
B
B
ds
B.dl = B dl cos 0= B dl
ds ds
As, B is uniform at each point on the field
line:
ds
B.dl = B (2π r) (2)
Equating (1) and (2):
B =
𝜇0 𝑰
2π r
i.e. the magnetic field a distance r from a long
straight wire carrying a steady current I.
 Ampere’s law is true for steady currents
 The Magnetic field inside the loop remains constant
 I , arising due to flow of charges, is called the ’conduction current’ IC
B dl I
   0 c

3. 4. Ampere and Maxwell’s Law
B dl I
   0
Ampere’s Law
‘The line integral of the magnetic field B around any closed
loop is 𝜇0 times the net steady current I enclosed by this
path”
Maxwell’s Equations
(1)
dl
I
p
dl
P
B
B
ds
B.dl = B dl cos 0= B dl
ds ds
As, B is uniform at each point on the field
line:
ds
B.dl = B (2π r) (2)
Equating (1) and (2):
B =
𝜇0 𝑰
2π r
i.e. the magnetic field a distance r from a long
straight wire carrying a steady current I.
 Ampere’s law is true for steady currents
 The Magnetic field inside the loop remains constant
 I , arising due to flow of charges, is called the ’conduction current’ IC
B dl I
   0 c
dl
I
p
B
Let’s there is a capacitor and the Amperian surface is like a
bag and its top is open:
L A
Ampere’s Law

4. 4. Ampere and Maxwell’s Law
B dl I
   0
Ampere’s Law
‘The line integral of the magnetic field B around any closed
loop is 𝜇0 times the net steady current I enclosed by this
path”
Maxwell’s Equations
(1)
dl
I
p
dl
P
B
B
ds
B.dl = B dl cos 0= B dl
ds ds
As, B is uniform at each point on the field
line:
ds
B.dl = B (2π r) (2)
Equating (1) and (2):
B =
𝜇0 𝑰
2π r
i.e. the magnetic field a distance r from a long
straight wire carrying a steady current I.
 Ampere’s law is true for steady currents
 The Magnetic field inside the loop remains constant
 I , arising due to flow of charges, is called the ’conduction current’ IC
B dl I
   0 c
dl
I
p
B
Let’s there is a capacitor and the Amperian surface is like a
bag and its top is open:
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
L A
Ampere’s Law

5. 4. Ampere and Maxwell’s Law
B dl I
   0
Ampere’s Law
‘The line integral of the magnetic field B around any closed
loop is 𝜇0 times the net steady current I enclosed by this
path”
Maxwell’s Equations
(1)
dl
I
p
dl
P
B
B
ds
B.dl = B dl cos 0= B dl
ds ds
As, B is uniform at each point on the field
line:
ds
B.dl = B (2π r) (2)
Equating (1) and (2):
B =
𝜇0 𝑰
2π r
i.e. the magnetic field a distance r from a long
straight wire carrying a steady current I.
 Ampere’s law is true for steady currents
 The Magnetic field inside the loop remains constant
 I , arising due to flow of charges, is called the ’conduction current’ IC
B dl I
   0 c
dl
I
p
B
Let’s there is a capacitor and the Amperian surface is like a
bag and its top is open:
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
L A
Ampere’s Law

6. 4. Ampere and Maxwell’s Law
B dl I
   0
Ampere’s Law
‘The line integral of the magnetic field B around any closed
loop is 𝜇0 times the net steady current I enclosed by this
path”
Maxwell’s Equations
(1)
dl
I
p
dl
P
B
B
ds
B.dl = B dl cos 0= B dl
ds ds
As, B is uniform at each point on the field
line:
ds
B.dl = B (2π r) (2)
Equating (1) and (2):
B =
𝜇0 𝑰
2π r
i.e. the magnetic field a distance r from a long
straight wire carrying a steady current I.
 Ampere’s law is true for steady currents
 The Magnetic field inside the loop remains constant
 I , arising due to flow of charges, is called the ’conduction current’ IC
B dl I
   0 c
dl
I
p
B
Let’s there is a capacitor and the Amperian surface is like a
bag and its top is open:
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
As there is no current flowing inside the capacitor,
hence B.dl = 0. At the same point ‘P’ but with
different Amperian surfaces, the B is not same.
Conc: There are some gaps in the Ampere’s law.
Maxwell corrected and made Ampere’s law
consistent in all cases.
What happens between the plates while the current I is flowing?
L A
Ampere’s Law

7. Maxwell’s Equations
dl
I
p
B
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
E
Maxwell correction to Ampere’s Law
• There has to be some I existing between plates.
• There are no conduction electrons between plates BUT
• There is time-varying E between plates, which generates I.
• The E is directed from +ve plate to –ve plate, because
the amount of charge on capacitor increases with time, AND
• The E between plates increases with time, as well.
• He called this new current as a displacement current, Id .

8. Maxwell’s Equations
As the amount of charge on the capacitor increases with time,
hence the q charge, changes with time, produces a current:
Id =dq/dt
dl
I
p
B
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
E
q EA I
dq
dt
d EA
dt
  
 
0 0
( )
For a capacitor, , and
Maxwell correction to Ampere’s Law
• There has to be some I existing between plates.
• There are no conduction electrons between plates BUT
• There is time-varying E between plates, which generates I.
• The E is directed from +ve plate to –ve plate, because
the amount of charge on capacitor increases with time, AND
• The E between plates increases with time, as well.
• He called this new current as a displacement current, Id .
ΦE
d
i.e. Current is produced because of changing electric flux w.r.t
time OR Electric flux is generated due to the presence of E
between plates.

9. Maxwell’s Equations
As the amount of charge on the capacitor increases with time,
hence the q charge, changes with time, produces a current:
Id =dq/dt
dl
I
p
B
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
E
q EA I
dq
dt
d EA
dt
  
 
0 0
( )
For a capacitor, , and
Maxwell correction to Ampere’s Law
• There has to be some I existing between plates.
• There are no conduction electrons between plates BUT
• There is time-varying E between plates, which generates I.
• The E is directed from +ve plate to –ve plate, because
the amount of charge on capacitor increases with time, AND
• The E between plates increases with time, as well.
• He called this new current as a displacement current, Id .
ΦE
d
i.e. Current is produced because of changing electric flux w.r.t
time OR Electric flux is generated due to the presence of E
between plates.
CONC: Current is not ZERO between plates but it is Id AND
B is produced by the time varying E or Id

10. Maxwell’s Equations
As the amount of charge on the capacitor increases with time,
hence the q charge, changes with time, produces a current:
Id =dq/dt
dl
I
p
B
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
E
q EA I
dq
dt
d EA
dt
  
 
0 0
( )
For a capacitor, , and
Maxwell correction to Ampere’s Law
• There has to be some I existing between plates.
• There are no conduction electrons between plates BUT
• There is time-varying E between plates, which generates I.
• The E is directed from +ve plate to –ve plate, because
the amount of charge on capacitor increases with time, AND
• The E between plates increases with time, as well.
• He called this new current as a displacement current, Id .
ΦE
d
Conc: Bs are produced
both by a conduction
current and by a time-
varying field.
A changing electric field
induces a magnetic field
x
x
x
x
x
x
x
x
x
x
x
x
B
B dl
d
dt
I
E
d
   
  
0 0 0

B dl I I
B dl I d
dt
d
E


  
   

  
0
0 0 0
( )

Maxwell correction to Ampere’s Law is:
c
c
i.e. Current is produced because of changing electric flux w.r.t
time OR Electric flux is generated due to the presence of E
between plates
CONC: Current is not ZERO between plates but it is Id AND
B is produced by the time varying E or Id

11. Maxwell’s Equations
As the amount of charge on the capacitor increases with time,
hence the q charge, changes with time, produces a current:
Id =dq/dt
dl
I
p
B
+
+
+
+
+
-
-
-
-
-
-
-
-
-
-
+
+
+
+
+
E
q EA I
dq
dt
d EA
dt
  
 
0 0
( )
For a capacitor, , and
Maxwell correction to Ampere’s Law
• There has to be some I existing between plates.
• There are no conduction electrons between plates BUT
• There is time-varying E between plates, which generates I.
• The E is directed from +ve plate to –ve plate, because
the amount of charge on capacitor increases with time, AND
• The E between plates increases with time, as well.
• He called this new current as a displacement current, Id .
ΦE
d
Conc: Bs are produced
both by a conduction
current and by a time-
varying field.
A changing electric field
induces a magnetic field
x
x
x
x
x
x
x
x
x
x
x
x
B
B dl
d
dt
I
E
d
   
  
0 0 0

B dl I I
B dl I d
dt
d
E


  
   

  
0
0 0 0
( )

Maxwell correction to Ampere’s Law is:
c
c
i.e. Current is produced because of changing electric flux w.r.t
time OR Electric flux is generated due to the presence of E
between plates
CONC: Current is not ZERO between plates but it is Id AND
B is produced by the time varying E or Id