Short descriptions of Maxwell's equations

1. Integral form of Maxwell equations for time
varying field with interpretation.
BY
Sarole Yasir MohdWaseem
13ET41
Anjuman-I-Islam’s Kalsekar Technical
Campus, New Panvel.
13/02/2015

2. INDEX
•Introduction
•Equations for static fields
•Equations for time varying fields
•References

3. INTRODUCTION
• Vector equations which governs the electric and magnetic
fields.
• Generally developed for time varying fields.
• Can be reduced to static form.

4. Maxwell’s Equations for Static Fields
• Faraday’s Law:
Integral form of Maxwell’s equation.
Using Stoke’s Theorem,
∴
but, therefore
Point form of Maxwell’s equation.
𝑒𝑚𝑓 =
𝐸
.
𝑑𝑙
𝐸
.
𝑑𝑙
= 𝑠 𝛻 ×
𝐸
.
𝑑𝑠
𝑠 𝛻 ×
𝐸
.
𝑑𝑠
= 0
𝑑𝑠
⧧0 𝛻 ×
𝐸
= 0

5. • Ampere’s Law:
Integral form of Maxwell’s equation.
Using Stoke’s theorem,
∴
Or……
Point form of Maxwell’s equation.
𝐻
.
𝑑𝑙
= 𝐼 = 𝑠
𝐽
.
𝑑𝑠
𝐻
.
𝑑𝑙
= 𝑠 (𝛻 ×
𝐻
).
𝑑𝑠
𝑠 𝛻 ×
𝐻
.
𝑑𝑠
= 𝑠
𝐽
.
𝑑𝑠
𝛻 ×
𝐻
=
𝐽

6. • Gauss’s Law (For Electric):
𝐷
.
𝑑𝑠
= Q encl= ῥv dv
Integral form of the Maxwell’s equation.
Using Divergence theorem as, 𝑠
𝐷
.
𝑑𝑠
= 𝑣 (𝛻.
𝐷
)𝑑𝑣
∴ (𝛻.
𝐷
) 𝑑𝑣 = 𝑣 ῥv dv
or 𝛻.
𝐷
= ῥv
Point form of the Maxwell’s equation.

7. • Gauss’s Law (For Magnetic): 𝐵
.
𝑑𝑠
=0
Integral form of Maxwell’s equation.
Using Divergence theorem, 𝑠
𝐵
.
𝑑𝑠
= 𝑣 (𝛻.
𝐵
)𝑑𝑣
∴ 𝑣 𝛻.
𝐵
𝑑𝑣 = 0
or 𝛻.
𝐵
= 0
Point form of Maxwell’s equation.

8. • The Continuity Equation for Current: 𝐽
.
𝑑𝑠
= 0
Integral form of Maxwell’s equation.
Using Divergence theorem,
𝐽
.
𝑑𝑠
= 𝑣 (𝛻.
𝐽
)𝑑𝑣
∴ 𝑣 𝛻.
𝐽
𝑑𝑣 = 0
Or 𝛻.
𝐽
= 0
Point form of Maxwell’s equation.

9. Maxwell’s equations for static field are
summarized in below Table.

10. Maxwell’s Equations for Time Varying Fields
• Faraday’s Law: emf = 𝑐
𝐸
.
𝑑𝑙
= -
𝑑𝛹𝑚
𝑑𝑡
(V)
𝛻
×
𝐸
= -
𝐵
Point form or differential form of Maxwell’s eqn.

11. • Ampere’s Law: 𝐻 . 𝑑𝑙=I
I=
𝐽
𝑡𝑜𝑡𝑎𝑙.
𝑑𝑠
= (
𝐽
𝑐𝑜𝑛𝑑 =
𝐽
𝑑𝑖𝑠𝑝).
𝑑𝑠
.
𝐻
.
𝑑𝑙
= 𝑠 (
𝐽
+
𝐷
).
𝑑𝑠
Integral form of Maxwell’s equation.
Using Stoke’s Theorem, 𝐻
.
𝑑𝑙
= 𝑠(∇×
𝐻
).
𝑑𝑠
.
Or 𝑠(∇×
𝐻
).
𝑑𝑠
= 𝑠 (
𝐽
+
𝐷
).
𝑑𝑠

12. comparing we get,
.
∇×
𝐻
=
𝐽
+
𝐷
Point form or differential form of Maxwell’s equation.
.
𝐻
.
𝑑𝑙
= 𝑠 (
𝐽
+
𝐷
).
𝑑𝑠
.
∇×
𝐻
=
𝐽
+
𝐷

13. • Gauss’s Law (For Electric):
𝐷
.
𝑑𝑠
= Q encl= ῥv dv
Integral form of the Maxwell’s equation.
Using Divergence theorem as, 𝑠
𝐷
.
𝑑𝑠
= 𝑣 (𝛻.
𝐷
)𝑑𝑣
∴ (𝛻.
𝐷
) 𝑑𝑣 = 𝑣 ῥv dv
or 𝛻.
𝐷
= ῥv
Point form of the Maxwell’s equation.

14. • Gauss’s Law (For Magnetic): 𝐵
.
𝑑𝑠
=0
Integral form of Maxwell’s equation.
Using Divergence theorem, 𝑠
𝐵
.
𝑑𝑠
= 𝑣 (𝛻.
𝐵
)𝑑𝑣
∴ 𝑣 𝛻.
𝐵
𝑑𝑣 = 0
or 𝛻.
𝐵
= 0
Point form of Maxwell’s equation.

15. General Set of Maxwell’s Equations:

16. References:
Electromagnetic Waves – R K SHEVGAONKAR
Antenna & Wave Propagation – N.D. SADIKU