This document discusses Maxwell's correction to Ampere's circuital law. It notes that Ampere's law was incomplete as it did not account for changing electric fields. Maxwell added a "displacement current" term to account for this. His full corrected law states that the curl of the magnetic field equals the permeability times the sum of the conduction current and the displacement current. This resolved inconsistencies in Ampere's law and completed the description of classical electromagnetism.

1. MAXWELL’S CORRECTION IN
AMPERE’S CIRCUITAL LAW
- Kamran Ansari
CBS 5th semester
(Physics Stream)

2. Contents
• Electrodynamics before Maxwell
• Stoke’s theorem
• Problem with Ampere’s circuital law
• How Maxwell fixed Ampere’s circuital law
• Electrodynamics after Maxwell

3. • Gauss’s law
𝑆
𝐸. 𝑑 𝑠 =
1
𝜀0 𝑉
𝜌 𝑑𝜏
• No monopole
𝑆
𝐵. 𝑑 𝑠 = 0
• Faraday’s law
𝐶
𝐸. 𝑑 𝑟 = 𝑆
−
𝜕𝐵
𝜕𝑡
. 𝑑 𝑠 = −
𝜕ϕ 𝐵
𝜕𝑡
• Ampere’s circuital law
𝐶
𝐵. 𝑑 𝑟 = 𝜇0 𝑆
𝐽. 𝑑 𝑠 = 𝜇0 𝐼𝑒𝑛𝑐
• Lorentz force
𝐹 = 𝑞𝐸 + 𝑞( 𝑣 x 𝐵)
Electrodynamics before Maxwell
In integral form

4. In differential form
• Gauss’s law
𝛻. 𝐸 =
𝜌
𝜀0
• No monopole
𝛻. 𝐵 = 0
• Faraday’s law
𝛻 x 𝐸 = −
𝜕𝐵
𝜕𝑡
• Ampere’s circuital law
𝛻 x 𝐵 = 𝜇0J
• Lorentz force
𝐹 = 𝑞𝐸 + q( 𝑣 x 𝐵)

5. Stokes’ theorem
• The circulation of 𝐹 = 𝑎 𝑖 + 𝑏 𝑗 + 𝑐 𝑘 around
the boundary C of an oriented surface S in
the direction counterclockwise with respect
to the surface’s unit normal vector 𝑛 equals
to the integral of 𝛻 x 𝐹 . 𝑛 over S.
𝐶
𝐹. 𝑑 𝑟 = 𝑆
( 𝛻 X 𝐹). 𝑛 𝑑𝑎
counterclock wise Curl integral
circulation
𝑛

6. Problem with Ampere’s circuital law
• Mathematically, divergence of curl of any vector is always zero. Applying this rule to electric field
𝛻. (𝛻 x 𝐸) = 𝛻. −
𝜕𝐵
𝜕𝑡
= −
𝜕
𝜕𝑡
( 𝛻. 𝐵 )
= 0
and if we apply thus rule to magnetic field than,
𝛻. 𝛻 x 𝐵 = 𝜇0( 𝛻. J )
but in general, the divergence of J is not zero. For steady current, the divergence of 𝐽 is zero but for non
steady current, 𝛻. J = −
𝜕𝜌
𝜕𝑡
𝛻. 𝛻 x 𝐵 = −𝜇0
𝜕𝜌
𝜕𝑡
≠ 0
so that when we go beyond magnetostatics Ampere’s circuital law cannot be right.

7. • Ampere’s law in integral form,
𝐶
𝐵. 𝑑 𝑟 = 𝜇0 𝑆
𝐽. 𝑑 𝑠 = 𝜇0 𝐼𝑒𝑛𝑐
In the process of charging up a capacitor,
Applying Ampere’s law in this amperian loop than in this case the simplest surface S in the plane of loop, 𝐼𝑒𝑛𝑐 = 𝐼
so 𝐶
𝐵. 𝑑 𝑟 = 𝜇0 𝐼 . Fine but if we draw instead the balloon shaped surface S’’ than value of 𝐶
𝐵. 𝑑 𝑟 must be
same by Stokes’ theorem but no current passes through this surface so 𝐼𝑒𝑛𝑐 = 0 and 𝐶
𝐵. 𝑑 𝑟 = 0. So that for
non steady current “the current enclosed by the loop” is an ill defined notion.
So there is need to correct Ampere’s law
which must be correct also for non steady current.
Amperian loop
Capacitor
I
I

8. How Maxwell fixed Ampere’s circuital law
Applying the continuity equation,
𝛻. J = −
𝜕𝜌
𝜕𝑡
=−
𝜕
𝜕𝑡
(𝜀0 𝛻. 𝐸 )
𝛻. 𝐽 = −𝛻 . (𝜀0
𝜕𝐸
𝜕𝑡
)
combine 𝜀0
𝜕𝐸
𝜕𝑡
with J in Ampere’s law than Ampere’s law with Maxwell’s correction
𝛻 x 𝐵 = 𝜇0J + 𝜇0 𝜀0
𝜕𝐸
𝜕𝑡
and in integral form,
𝐶
𝐵. 𝑑 𝑟 = 𝜇0 𝑆
𝐽. 𝑑 𝑠 + 𝜇0 𝜀0 𝑆′′
𝜕 𝐸
𝜕𝑡
. 𝑑 𝑠
for magnetostatics or when 𝐸 is constant, 𝛻 x 𝐵 = 𝜇0J (Ampere’s Law)
( By Gauss’s law )
𝐶
𝐵. 𝑑 𝑟 = 𝜇0
𝑆
𝐽. 𝑑 𝑠

9. The extra term in Maxwell’s correction in Ampere’s law
𝐼 𝑑 = 𝜀0 𝑆′′
𝜕 𝐸
𝜕𝑡
. 𝑑𝑎 = 𝜀0( 𝜕ϕ 𝐸
𝜕𝑡
)
is called displacement current and its density is
𝐽 𝑑 = 𝜀0
𝜕𝐸
𝜕𝑡
Now by Maxwell’s correction in Ampere’s law, calculating the value of divergence of curl of
magnetic field will be zero,
𝛻. 𝛻 x 𝐵 = 𝜇0( 𝛻. J ) + 𝜇0 𝜀0
𝜕
𝜕𝑡
(𝛻. E)
𝛻. 𝛻 x 𝐵 = 0

10. and in the process of charging up a capacitor
if we choose the flat surface S then E=0 and 𝐼𝑒𝑛𝑐=I,
and in the balloon-shaped surface S’’ then 𝐼𝑒𝑛𝑐= 0 and 𝑆′′
𝜕 𝐸
𝜕𝑡
. 𝑑 𝑎 =
𝐼
𝜀0
so we get same answer in both surfaces 𝐶
𝐵. 𝑑 𝑟 = 𝜇0 𝐼
Amperian loop
Capacitor

11. • Maxwell’s term has a certain aesthetic appeal: Just as a changing
magnetic field induces an electric field (Faraday’s law) So,
A changing electric field induces a magnetic field.

12. • Gauss’s law
𝑆
𝐸. 𝑑 𝑠 =
1
𝜀0 𝑉
𝜌 𝑑𝜏
• No monopole
𝑆
𝐵. 𝑑 𝑠 = 0
• Faraday’s law
𝐶
𝐸. 𝑑 𝑟 = 𝑆
−
𝜕𝐵
𝜕𝑡
. 𝑑 𝑠 = −
𝜕ϕ 𝐵
𝜕𝑡
• Ampere’s circuital law
𝐶
𝐵. 𝑑 𝑟 = 𝜇0 𝑆
𝐽. 𝑑 𝑠
• Lorentz force
𝐹 = 𝑞𝐸 + 𝑞( 𝑣 x 𝐵)
+ 𝜇0 𝜀0 𝑆
𝜕 𝐸
𝜕𝑡
. 𝑑 𝑎
Electrodynamics before MaxwellElectrodynamics after Maxwell
In integral form

13. In differential form
• Gauss’s law
𝛻. 𝐸 =
𝜌
𝜀0
• No monopole
𝛻. 𝐵 = 0
• Faraday’s law
𝛻 x 𝐸 = −
𝜕𝐵
𝜕𝑡
• Ampere’s circuital law
𝛻 x 𝐵 = 𝜇0J
• Lorentz force
𝐹 = 𝑞𝐸 + q( 𝑣 x 𝐵)
+ 𝜇0 𝜀0
𝜕𝐸
𝜕𝑡

14. THANK YOU