Download ʃ B . ds

Branch: Elctronics and Communication
Subject: Electromagnetic
Prepared by: Vaibhavi H Jaradi
Enrollment No. 130150111034
Maxwell’s Equations in Integral and Point Form
(Maxwell’s Equations for Time Varying Fields)
Maxwell’s Equations using
1.
2.
3.
4.
Faraday’s Law
Ampere’s Law
Gauss’s Law( E-Field)
Gauss’s Law(H-Field)
1. Faraday’s Law
Statement of Faraday’s Law:
In a closed path(loop) the electric potential(emf) is developed due to
time-varying magnetic field in the vicinity of that closed path. Mathematically,
emf= ʃ E . dl = -dΨm (V)
………(i)
c
dt
 The negative sign in the expression indicates a Lenz’s law. The Ψm is the
magnetic flux and can be related with the flux density B as:
Ψm = ʃ B . ds
……….(ii)
Then form (i) and (ii),
ʃ E . dl = -d ʃ B . ds
c
dt
1. Faraday’s Law
 Actually emf is generated by a time changing magnetic flux through a fixed path or by a
time varying path in a steady magnetic field. If we consider a stationary path, the
magnetic flux is only time varying quantity on right hand side and a parial derivatives
may be taken under the integral sign.
ʃ E . dl = - d ʃ B . ds
c
dt
 This is called as integral form of Maxwell’s eqation derived from Faraday’s Law. The dot
over B expresses partial differentiation with respect to t.
 Applying Stokes theorem to convert line integral on left side of above equation to surface
integral:
ʃ E. dl = ʃ (∇×E ) . ds
 This is a point form or differential form of Maxwell’s equation derived from Faraday’s
Law.
2. Ampere’s Law
Statement of Ampere’s Law:
The line integral of H around a closed path is equal to the current enclosed
by the path.
Mathematically,
ʃ H . dl = I
………..(i)
Where, both conduction and displacement currents are present, the current enclosed
is the total current.
I = ʃ J . ds = ʃ ( Jcond + Jdisp ) . ds ………..(ii)
 The first term on RHS is the conduction current and the second term is the
displacement current. Thus, the conduction current is
ic = ʃ Jcond . ds
s
 The displacement current is the second term i.e.
id = ʃ Jdisp . ds
s
2. Ampere’s Law
 The displacement current density is D, therefore the displacement current is also written as,
id = ʃ D . ds
Therefore,
I = ic + id = ʃ (J + D ) . ds
s
Putting the value in Equation (i)
ʃ H . dl = ʃ ( J + D) . ds
 This is the integral form of Maxwell’s equation derived from Ampere’s law.
Differential Form:
Using Stoke’s theorem , we get
∇×H = J + ∂ D
∂t
This is the point form or differential form of Maxwell’s equation derived from Ampere’s law.
3. Gauss’s Law( E- Field)
Statement of Gauss’s Law(E- Field):
The total flux crossing the closed surface is equal to the total charge enclosed
by the closed path.
Mathematically,
D . ds = Q
.……..…..(i)
 The charge ‘Q’ can be written in terms of ρv as :
Q = ʃ ρv dv
…………….(ii)
V
From equation (i) and (ii) , we have ,
ʃ D . ds = ʃ ρv dv
v
 This is called as integral form of Maxwell’s equation derived from Gauss’s law.
3. Gauss’s Law( E- Field)
 Applying divergence theorem to convert surface integral on the left side to
volume integral
ʃ D . ds = ʃ (∇. D) dv ……….(1)
Then from Equation (1) , ʃ (∇. D ) dv = ʃ ρv dv
 Both sides of the above equation are volume integrals, comparing
∇ . D = ρv
 This is called as point form of Maxwell’s equation derived from Gauss’s Law.
4. Gauss’s Law( H- Field)
 Statement of Gauss’s Law(H- Field):
In case of magnetic field the total outgoing flux( magnetic ) is equal to zero.
Mathematically,
ʃ B . ds = 0
…………(i)
 This is called as integral form of Maxwell’s equation from Gauss’s law for magnetic
field.
 This equation results from the fact that the magnetic flux lines are continuous.
Using divergence theorem to convert surface integral to volume integral Equation
(i) becomes
ʃ B . ds = ʃ ( ∇ . B ) dv
s
v
Therefore,
ʃ (∇. B ) dv = 0
But dv can not be zero. Therefore,
∇.B=0
……….(ii)
 Equation (ii) is called as point form or differential form of Maxwell’s equation
derived from Gauss’s law for magnetic field.
General Set of Maxwell’s Equations
Law
Ampere’s Law
Faraday’s Law
Integral Form
ʃ H . dl = ʃ ( J + D) . ds
ʃ E . dl = - ʃ B . ds
Gauss’s Law(E- Field)
ʃ D . ds = ʃ ρv . dv
Gauss’s Law(H- Field)
ʃ B . ds = 0
Point Form
∇×H = J + ∂D
∂t
∇×E = −∂B
∂t
∇·D = ρv
∇·B = 0