Slide 1 of 10 Electrodynamics:Lecture-4

Maxwell’s Equations

Ambar JainIISER Bhopal

Outline1. In this lecture we will review Maxwell’s equations in differential and integral form2. We will understand the apparent symmetry between the Electric and Magnetic fields

10-Aug-2018

References1. Introduction to Electrodynamics, DJ Griffiths

Slide 2 of 10 Maxwell’s equations: Differential Form

! Let’s review the famous Maxwell’s equations that summarize Gauss’s law, Ampere’s law, Faraday’s law etc. in the differential form.

∇ ·E =ρϵ0

(Gauss' law) (1)

∇ ·B = 0 (Gauss' law for Magnetism) (2)

∇ ×E =-∂B∂t

(Faraday' s law) (3)

∇ ×B = μ0 J +μ0 ϵ0∂E∂t

(Maxwell–Ampere law) (4)

! The above set of equations are established by careful experimentation, therefore we have to accept them to be correct unless shown otherwise. As a matter of fact, no experiment has shown Maxwell’s equations to be incorrect.

! The term marked in red is known as Maxwell’s correction to the Ampere’s law.

! Maxwell’s equations describe how electric and magnetic fields are produced from sources, that is electric charge density (ρ) and current density (J).

! Maxwell’s equations also say that magnetic charges (or magnetic monopoles) do not exist.

! Along with Lorentz force equation, they constitute a complete set of equations to understand dynamics of fields and charges.

Fq = q E + q v ×B (5)

! Lorentz force tells us how fields effect the charges while Maxwell’s equations tell us how charges effects fields.

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Slide 3 of 10 Group Problem - 1

(a) Maxwell’s equations constitute a total of how many scalar equations?

(b) Using Maxwell’s equations show that the charges and current’s obey the following continuity equation:

-∂ρ∂t

= ∇ · J (6)

Illustrate the equation above with a picture on the board near you.(c) Show that in the absence of Maxwell’s correction to the Maxwell-Ampere law, we obtain

∇ · J = 0 (7)which is correct only for steady currents.

(d) Show that the magnetic fields do no work on charges.

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Slide 4 of 10 Divergence and Curl: Demo

! If a vector field has no sources, sinks or singularities then divergence vanishes.

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choose field x2, y2

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

flow across the circle: 13.823flow around the circle: 0.

Demonstration taken from Wolfram Demonstrations site: https://demonstrations.wolfram.com/VectorFieldFlowThroughAndAroundACircle/

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Slide 5 of 10 Group Problem - 2

(a) Show that the electric and magnetic fields, given by the following equation, satisfy all the Maxwell’s equations:

E(r, t) =1

4 π ϵ0

q

r2θ(r - v t) r1

B(r, t) = 0(8)

Identify the physical situation that gives rise to these fields.

(b) Verify that your solution for ρ and J satisfy the continuity equation.

You may find the following formulas useful

∇ ·V =1

r2∂∂r

r2 Vr +1

r sin θ∂∂θ

(sin θ Vθ) +1

r sin θ∂Vϕ∂ϕ

(9)

∇ ×V =1

r sin θ∂∂θ

sin θ Vϕ -∂Vθ∂ϕ

r1 +1r

1sin θ

∂Vr∂ϕ

-∂∂r

r Vϕ θ1+

1r

∂∂r

(r Vθ) -∂Ar∂θ

ϕ1

(10)

Derivative of Heaviside θ function is ∂ θ(x - a)

∂x= δ(x - a) (11)

Solution

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J =q v

4 π r2δ(r - v t) r1 (12)

ρ =q

4 π r2δ(r - v t) (13)

Physical situation is expanding spherical shell of charge q. Its expanding at the rate ⅆrⅆt = v.

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Slide 6 of 10 Maxwell’s Equation in free space (without sources)

! In free space, set ρ and J to 0. We get:

∇ ·E = 0 (14)

∇ ×E =-∂B∂t

(15)

∇ ·B = 0 (16)

∇ ×B = μ0 ϵ0∂E∂t

(17)

! There is symmetry to these set of equation. Consider the transformation

E → B and B → -μ0 ϵ0 E (18)! First two Maxwell’s equation in free space transform as

∇ ·E = 0 E→B ∇ ·B = 0 (19)

∇ ×E =-∂B∂t

B→-μ0 ϵ0 E ∇ ×B = μ0 ϵ0∂E∂t

(20)

! Similarly the second pair of equations become the first pair. Homework: Verify this statement.

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Slide 7 of 10 Magnetic Charge

! The apparent symmetry of free space Maxwell' s equations are suggestive of symmetry in existence of sources.

! We are tempted to claim that magnetic charges and magnetic current might exist and act also as sources for electro-magnetic fields.

! We add missing source terms to equations without source terms to get:

∇ ·E =ρeϵ0

(21)

∇ ×B = μ0 Je + μ0 ϵ0∂E∂t

(22)

∇ ·B = μ0 ρm (23)

∇ ×E = -μ0 Jm +-∂B∂t

(24)

! Notation:ρe = electric charge density (25)

Je ⩵ electric current density (26)

ρm = magnetic charge density (27)

Jm = magnetic current density (28)

! We add missing source terms to equations without source terms to get:

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-∂ρe∂t

= ∇ · Je (29)

-∂ρm∂t

= ∇ · Jm (30)

! A single magnetic charge is found would be called a magnetic monopole.

! Even after a rigorous search, no one has ever found a magnetic monopole, magnetic current or any other magnetic charge distribution.

! Thus, we conclude that the electric field and magnetic field are not really at an equal footing, although they appear to be so in the free space.

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Slide 8 of 10 Gauss Divergence Theorem

! According to Gauss divergence theorem, over a closed surface S enclosing volume V and for a vector field F:

∇ ·F ⅆV = S

F · ⅆS = S

F · n1 ⅆS (31)

Image Courtesy: Wikipedia

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Slide 9 of 10 Stokes Theorem

! According to Stokes theorem, over a closed surface S enclosing volume V and for a vector field F:

Σ

∇ ×F · ⅆS = Σ

∇ ×F · n1 ⅆS = ∂Σ

F · ⅆ ℓ (32)

Image Courtesy: Wikipedia

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Slide 10 of 10 Maxwell’s Equation in the Integral Form

! Homework: Using Gauss Divergence Theorem and Stoke’s Theorem. Obtain Maxwell’s equations in the integral form. Provide an interpretation to each equation. The results are given below for verification.

! The Maxwell’s equations in the integral form are given as:

! Gauss’ law can be written as

S

E · ⅆS =1ϵ0

ρ(r) ⅆV =

Qencϵ0

(33)

! Gauss’ law for magnetic field says

S

B · ⅆS = 0 (34)

! Faraday’s law can be written as

∂Σ

E · ⅆ ℓ =-∂ΦB∂t (35)

! Ampere-Maxwell law can be written as

∂Σ

B · ⅆ ℓ = μ0 Ienc + μ0 ϵ0∂ΦE∂t (36)

! Here ΦE and ΦB are electric and magnetic flux passing through the closed loop represented by ∂Σ bounding the surface Σ through which flux is passing.

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