Gauss's law of electrostatics states that the net electric flux passing through a closed surface is equal to the net charge enclosed in the surface. This law was derived in both integral and point form. Ampere's law relates the magnetic field around a closed loop to the current passing through the loop. It too was expressed in both integral and point forms. Finally, Faraday's law explains how a time-varying magnetic field produces an electromotive force. It links the induced electric field to the rate of change of the magnetic flux through a surface.

1. EAST AFRICA UNIVERSITY. SOMALIA
ELECTROMAGNETICS
Static and Time varying Maxwell equations
ABDIASIS ABDALLAH JAMA
10/4/2017
This notes are designed for electromagnetic students. It intuitively explains the underlying
principle of static and time varying Maxwell equations. It is written in simplified form and
hope students will benefit from it.

2. ©ABDIASIS ABDALLAH
In electromagnetics we are dealing with electric and magnetic fields. We can study them in the
following ways
 Static electric fields which does not vary with time. These fields are produced by point
charges and charge distributions
 Static magnetic field which does not vary with time. These fields are produced by steady
current.
 Time varying electromagnetic field that is produced by time varying currents. In this case
time varying electric field produces magnetic field (Ampere’s law) and time varying
magnetic fields produce electric fields (Faradays law)
For static electric fields we have
D is the electric flux density in Coulomb per meter square.
Let us take a uniform volume and place a lot of charges inside the volume V
Gauss law states that:
Electric flux passing through a closed surface is equal to total
charge enclosed in the surface.
∫ 𝑫. 𝒅𝒔 = 𝑸 (𝟏)
++++++++++
+++++++++++
++++++++++
Surface S
Volume V

3. ©ABDIASIS ABDALLAH
Let us define a new quantity called charge density inside the volume V
𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑐ℎ𝑎𝑟𝑔𝑒
𝑣𝑜𝑙𝑢𝑚𝑒
Since we have volume, the charge density becomes VOLUME CHARGE DENSITY
𝝆 𝑽 =
𝑸
𝑽
𝑸 = 𝝆 𝑽 𝑽
In the example above, we assumed uniform volume like sphere which contain charges. Other
uniform shapes include box and cylinder.
For non-uniform volumes, the volume cannot be obtained from a simple formula like sphere.
We need to use calculus.
We take small section of the volume and call it dv or differential volume. This differential
volume contains small subset of the charges dQ.
𝑑𝑄 = 𝜌 𝑣 𝑑𝑣
We can continue to divide the total volume into differential volumes and each will contain small
subset of the total charges.
Total charge is obtained from integrating the differential volume charges.
If you equate equation (1) and equation (2) we get
Equation (3) is GAUSS law in electrostatics.
In electromagnetics, the concepts of CURL and DIVERGENCE are very useful in calculations and
writing Maxwell’s equation in compact form.
If we have a vector field A such as electric field vector or magnetic field vector
𝑸 = ∫ 𝝆 𝒗 𝒅𝒗 (𝟐)
∫ 𝑫. 𝒅𝒔 = ∫ 𝝆 𝒗 𝒅𝒗 (𝟑)
Curl of A is written as 𝛁 𝒙 𝑨
Divergence of A is written as 𝛁 . 𝑨

4. ©ABDIASIS ABDALLAH
Physical meaning of Curl is circulation of the vector field around a point. Example is circulation
or rotation of magnetic field around a current carrying wire (Ampere’s law).
Physical meaning of divergence is a vector field emerging and spreading from a point. Example
is electric field lines diverging from point charges (Gauss law).
Two useful theorems from vector fields
If an object moves between two points along straight line of length l
Work done by the object = Force x distance
𝑊 = 𝐹 𝑥 𝑙
What if the distance moved is not straight line but a curve.
You could divide the curve into small straight line segments of length dl, multiple the
differential distance by the force to get differential work done. Again take another differential
distance dl multiply by force to get work and so on.
But you want total work done by the object right?
You can use calculus to find the total work along the total distance moved
𝑊 = ∫ 𝐹. 𝑑𝑙
This type of integration is called line integration.
You could similarly perform surface and volume integration.
Given a vector field A we can perform three
types of integrations.
∫ 𝑨. 𝒅𝒍 𝒍𝒊𝒏𝒆 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒐𝒏
∫ 𝑨. 𝒅𝑺 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒐𝒏
∫ 𝑨. 𝒅𝒗 𝒗𝒐𝒍𝒖𝒎𝒆 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒕𝒊𝒐𝒏

5. ©ABDIASIS ABDALLAH
These types of integration are related. This means line integration can be converted into
surface integration and surface integration can be converted into volume integration.
To do this, Curl and Divergence will play a great role.
∫ 𝐴. 𝑑𝑙 = ∫(∇ 𝑥 𝐴). 𝑑𝑆 (4) 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑆𝑡𝑜𝑘𝑒′
𝑠 𝑡ℎ𝑒𝑜𝑟𝑜𝑚
This means if I integrate a vector field A over a length (path) l it is the same as integrating the
curl of A over surface S formed by the closed length (path).
Remember a closed path or length forms an open surface.
Stokes’ theorem will be useful for writing Ampere and Faraday’s law in point form.
∫ 𝐴. 𝑑𝑆 = ∫(∇. 𝐴)𝑑𝑣 (5) 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑡ℎ𝑒𝑜𝑟𝑜𝑚
This means if I integrate a vector field A over a surface A it is the same as integrating the
divergence of A over the volume enclosed by the surface.
Remember a closed surface forms a volume inside.
Divergence theorem will be useful in writing Gauss’s law in electric and magnetic fields in point
form.
Using equations (4) and (5) we can now write Gauss’s law (3) in point form
Recall from (3) Gauss’s law for electrostatics states that
∫ 𝑫. 𝒅𝒔 = ∫ 𝝆 𝒗 𝒅𝒗
The LHS contains surface integral. Using equation (5) we can convert it into volume integral as
∫(𝛁. 𝑫)𝒅𝒗 = ∫ 𝝆 𝒗 𝒅𝒗
For these two volume integrals to be equal, their integrands must be equal.
Hence
Line integral of vector field A Surface integral of vector field A
Surface integral of vector field A Volume integral of vector field A
𝛁. 𝑫 = 𝝆 𝒗 (6)

6. ©ABDIASIS ABDALLAH
This is Gauss law in point form.
Equation (3) and equation (6) are the same. Both are Gauss’s law. The first one is in integral
form while the second equation is in point form
Static magnetic fields
Static magnetic fields are produced by currents. It was discovered that a wire carrying current
produces magnetic field that will encircle the wire.
Current  Magnetic field
I  H
As you can see the magnetic field follows along a closed path around the wire. The current
enters and exits the path of the magnetic field formed.
Since the magnet field H is measured in ampere per meter (A/m) from Biot-Savart law, you
could say that
𝐼 = 𝐻. 𝑙
Where l is the length of the loop and it equal the circumference of the circle.
Hence
𝐼 = 𝐻. 2𝜋𝑟 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒
The path above is perfect circle. However in reality we are interested in more general and
irregular path like the one below
Amperian
Surface

7. ©ABDIASIS ABDALLAH
Divide this path into differential lengths each of straight line approximation of length dl.
Multiple each differential length dl by H.
𝐼 = 𝐻. 𝑑𝑙
And add them together using integral Calculus
N.B H is vector quantity and is given right hand rule. Also closed path l is a vector, and its
direction at each point of the direction of the vector tangent at that point.
Current flows along wires and conductors. In theory a wire is drawn as a line of length l and
zero thickness, hence zero surface area.
But practically a wire has a thickness and it has a surface area S. Current I follows along the
conductor surface.
It is good to know how much current passes along small subset of the area, and the quantity
that will help us determine this is surface current density J
𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑐𝑢𝑟𝑟𝑒𝑛𝑡
𝑎𝑟𝑒𝑎
𝐽 =
𝐼
𝑆
𝑖𝑛
𝐴
𝑚2
𝐼 = 𝐽. 𝑆
This expression is enough for simple surfaces such as cylinder surface, sheets and so on.
But in space we normally have an arbitrary surface of any shape that is irregular.
So we divide the surface into differential surfaces dS.
Each differential surface dS carries small subset of the total current.
𝑑𝐼 = 𝐽. 𝑑𝑆
Total current is obtained from integrating this equation over the entire surface.
𝐼 = ∫ 𝐽. 𝑑𝑆
Substitute this into equation (1)
𝑰 = ∫ 𝑯. 𝒅𝒍 𝐴𝑚𝑝𝑒𝑟𝑒′
𝑠 𝑙𝑎𝑤 (7)

8. ©ABDIASIS ABDALLAH
N.B the surface S has an orientation is space. Hence it is vector quantity.
Do you remember Stokes’ theorem?
Well it converts line integral of a vector field into surface integral of the curl of the vector field
∫ 𝐻. 𝑑𝑙 = ∫(∇ 𝑥 𝐻). 𝑑𝑆
Using this, Ampere’s law in (2) now become
∫ 𝐽. 𝑑𝑆 = ∫(∇ 𝑥 𝐻). 𝑑𝑆
For these two integrals to be equal, the integrands must be equal
Equation (8) is point form of Ampere’s law and it is the third equation of Maxwell equations.
So far we have defined electric flux density D and used it to derive Gauss’s law. Now let us
defined magnetic flux density B.
An external bar magnet produces magnetic field which starts at North Pole and ends at South
Pole as shown below.
Let us place a close path (open surface) in front of the magnet and allow the fields to pass
through the surface S as shown below.
∫ 𝑱. 𝒅𝑺 = ∫ 𝑯. 𝒅𝒍 𝐴𝑚𝑝𝑒𝑟𝑒′
𝑠 𝑙𝑎𝑤
𝛁 𝒙 𝑯 = 𝑱 (8)

9. ©ABDIASIS ABDALLAH
As can be seen above, the magnetic flux enters and exits the surface. There is so source inside
that generates the flux. The flux comes from external magnet. This suggest that magnetic flux
through a surface is zero. You can drive the same conclusion if you replace the bar magnet with
current carrying wire.
This is Gauss’s law in static magnetic fields.
To write this into point form, we need to replace the surface integral with volume integral using
divergence theorem.
∫ 𝐵. 𝑑𝑆 = ∫(∇. 𝐵)𝑑𝑣 = 0
The only way the volume integral be zero is if the integrand (∇. 𝐵) is zero.
This is the second Maxwell equation.
The fourth Maxwell equation is based on Faraday’s law. Faradays law can be stated as
When a magnetic field changing with time passes through a coil of wire, an emf (voltage) will
be induced in the coil. For voltage to be induced, the magnetic field must be changing
Net magnetic flux passing through a surface is ZERO.
∫ 𝑩. 𝒅𝑺 = 𝟎 (𝟗)
𝛁. 𝑩 = 𝟎 (𝟏𝟎)

10. ©ABDIASIS ABDALLAH
(moving towards or away from the coil). The opposite will give same results if the magnet is
stationary and the coil is moved.
The magnetic flux linking the coil is a function of space and time. We can take partial derivative
of B with respect to time.
𝜕
𝜕𝑡
∫ 𝐵. 𝑑𝑆 = 𝑡𝑖𝑚𝑒 𝑣𝑎𝑟𝑦𝑖𝑛𝑔 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑓𝑙𝑢𝑥 𝑙𝑖𝑛𝑘𝑖𝑛𝑔 𝑡ℎ𝑒 𝑐𝑜𝑖𝑙
The voltage induced in the coil by the external magnetic field will produce and electric field.
Time varying magnetic field  voltage in coil  electric field in coil.
Electric field E and voltage V are related by the length of the coil loop in which the current
flows.
𝑉 = ∫ 𝐸. 𝑑𝑙
Since the changing magnetic field produced the voltage, the above two equations are equal
The minus sign is from Lenz’s law.
∫ 𝑬. 𝒅𝒍 = − ∫
𝝏𝑩
𝝏𝒕
. 𝒅𝑺 (𝟏𝟏)
Faraday’s law

11. ©ABDIASIS ABDALLAH
We can write this equation in point form using Stokes’ theorem.
∫ 𝐸. 𝑑𝑙 = ∫(∇ 𝑥 𝐸). 𝑑𝑆
∫(∇ 𝑥 𝐸). 𝑑𝑆 = − ∫
𝜕𝐵
𝜕𝑡
. 𝑑𝑆
Hence
The four electromagnetic quantity E, D, B, H are related by permittivity and permeability
𝑫 = 𝜺𝑬
𝑩 = 𝝁𝑯
𝛁 𝒙 𝑬 = −
𝝏𝑩
𝝏𝒕
(𝟏𝟐)
Faraday’s law in point form

12. ©ABDIASIS ABDALLAH
Section Summary
Maxwell equation in
point form
Maxwell equation in
integral form
Law it based on
𝛁. 𝑫 = 𝝆 𝒗 ∫ 𝐷. 𝑑𝑆 = ∫ 𝜌 𝑣 𝑑𝑣
Gauss’s law of electrostatics:
Net electric flux passing through as closed
surface is equal to net charge enclosed in
the surface.
D = electric flux density in
𝐶
𝑚2
𝜌 𝑣 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑖𝑛
𝐶
𝑚3
𝛁. 𝑩 = 𝟎
∫ 𝐵. 𝑑𝑆 = 0
Gauss’s law in magnetostatics:
Net magnetic flux crossing a given surface
is zero. This mean there are no magnetic
monopoles (point charges)
B = magnetic flux density in Tesla
𝛁 𝒙 𝑯 = 𝑱 +
𝝏𝑫
𝝏𝒕
∫ 𝐻. 𝑑𝑙
= ∫(𝐽 +
𝜕𝐷
𝜕𝑡
). 𝑑𝑆
Ampere’s law:
Current carrying wire produces magnetic
field around the wire.
Magnetic field around the path enclosing
the surface is the same current density
distributed over the surface
H = magnetic field intensity in
𝐴
𝑚
J = surface current density in
𝐴
𝑚2
𝛁 𝒙 𝑬 = −
𝝏𝑩
𝝏𝒕
∫ 𝐸. 𝑑𝑙 = −
𝜕
𝜕𝑡
∫ 𝐵. 𝑑𝑆
Faraday’s law:
A time varying magnetic field linking a coil
of wire induces and electric field in the
wire.
E = electric field intensity in
𝑉
𝑚

13. ©ABDIASIS ABDALLAH
There is a useful identity that can be stated as follows
∇. (∇ 𝑥 𝐴) = 0
This says that the divergence of the curl of any vector field is zero.
Let us apply it to Ampere’s law
∇ 𝑥 𝐻 = 𝐽
∇. (∇ 𝑥 𝐻) = ∇. 𝐽
The RHS is zero. This gives us
∇. 𝐽 = 0
This last equation says, the divergence of the current density is zero.
This equation is inconsistent with the continuity equation
∇. 𝐽 = −
𝜕𝜌 𝑣
𝜕𝑡
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑧𝑒𝑟𝑜
To eliminate this inconsistency, we add another term to Ampere’s law
∇ 𝑥 𝐻 = 𝐽 + 𝐽 𝑑
And again take divergence on both sides
∇. (∇ 𝑥 𝐻) = ∇. (𝐽 + 𝐽 𝑑) = ∇. 𝐽 + ∇. 𝐽 𝑑 = 0
∇. 𝐽 𝑑 = −∇. 𝐽 = −(−
𝜕𝜌 𝑣
𝜕𝑡
)
∇. 𝐽 𝑑 =
𝜕𝜌 𝑣
𝜕𝑡
Now Gauss law say that
∇. 𝐷 = 𝜌 𝑣
Hence
∇. 𝐽 𝑑 =
𝜕
𝜕𝑡
(∇. 𝐷) = ∇.
𝜕𝐷
𝜕𝑡
From this expression we finally have displacement current
𝑱 𝒅 =
𝝏𝑫
𝝏𝒕
𝑡ℎ𝑖𝑠 𝑡𝑒𝑟𝑚 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑎𝑑𝑑𝑒𝑑 𝑡𝑜 𝐴𝑚𝑝𝑒𝑟𝑒′
𝑠 𝑙𝑎𝑤