MAXWELL’S EQUATION (2ND year)
By Hafsa Khattak
Introduction
• We have studies Gauss’ Law, Faraday’s Law, Ampere’s Law.
• James Clerk Maxwell unifies these laws and introduce
electromagnetism in only four equations.
• First two equations represent that how magnetic field creates
electric field while later tow equations represents that how
electric field creates magnetic field
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Contributions
His major contributions were in field of electromagnetism. He enclosed the complex
mathematics into four set of equations. He shows that;
• Varying magnetic field produce electric field
• Varying electric field produce magnetic field
• Speed of light.
Varying magnetic field produce electric field
As we know from Faraday’s Law; that change in magnetic field
lines in a certain area induces EMF in that region. So,
∆𝜑
∈=
∆𝑡
Where, ∆𝜑 = ∆ 𝐵. 𝐴 𝑜𝑟 𝐴 ∆𝐵 . So, the above equation becomes,
𝑨 ∆𝑩
∈=
………………….. ( I)
∆𝒕
Now due to this EMF charges will start motion in circular loops of
solenoid and work will be done.
We know that;
W = F .d
But here the charges will move in a circle so the total distance
will be equal to circumference of the circle.
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d= 2𝜋𝑟
Then work done becomes,
𝑊 = 𝐹 2𝜋𝑟 𝑤ℎ𝑒𝑟𝑒, 𝐹 = 𝑞𝐸
𝑠𝑜, 𝑊 = 𝑞𝐸 ( 2𝜋𝑟)
We also know that electric potential or EMF (∈) is equals to;
𝑊
∈=
𝑞
Now putting W in above equation, we get;
∈=
𝑞𝐸 ( 2𝜋𝑟)
𝑞
Or,
∈ = 𝟐𝝅𝒓𝑬 …............ (II)
Now compare eq. (I) and (II) we get,
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∈=
𝐴 ∆𝐵
………………….. ( I)
∆𝑡
∈ = 2𝜋𝑟𝐸 …............ (II)
2𝜋𝑟𝐸 =
𝐴 ∆𝐵
∆𝑡
or,
𝑨 ∆𝑩
𝑬=
𝟐𝝅𝒓 ∆𝒕
This equation shows that, change in magnetic field produces electric field in a coil.
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Varying electric field produce magnetic field
• To explain this we will use Ampere’s Law.
• Consider the fig of charging a capacitor. An external battery is
connected to the parallel plates of capacitor. The source can be
of two types, (I) DC and (II) AC.
I.
Charging a capacitor using DC source.
As we know that in DC source current do not vary with time. So,
when the plates are connected to the DC source then charges will
start stored on its surface hence increasing the current. But when
the plates gets charged, then the motion of charges will stop
eventually by dropping the current across the plates.
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As we know that for charging a capacitor the formula is
𝑄 = 𝐶𝑉
And
𝐶=
𝐴𝜀𝑜
,
𝑑
Then above eq. becomes,
𝐴𝜀𝑜
𝑄=
𝑉
𝑑
But we know that 𝑽 = 𝑬𝒅, so, Q becomes,
𝐴𝜀𝑜
𝑄=
𝐸𝑑
𝑑
Or,
𝑸 = 𝑬𝑨𝜺𝒐 ………………… (III)
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(II) Charging a capacitor using AC source.
By using AC source a capacitor can not be charged because in AC source current varies with
time. Due the variable current charges stored on plates also varies. But here current flows
continuously without any decrease because at every instant charges get stored on the plates,
and similarly electric field also set up between the plates at every instant due to the potential
difference of plates as shown in fig.
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Origin of current between plates:
Maxwell’s for the first time introduced the origin of continuous current between the plates of
capacitor, when it is connected to the AC power supply. As we know that current produced
when ever the charge moves from one point to another but here between the plates of
capacitor there is an empty space which shows that charges can not move from one plate to
another. But AC source is time varying. So the electric field between the plates will vary with
time. Due to this variable electric field current is introduced inside the plates. Maxwell call this
current Displacement current. It is also time varying as the electric field between the plates is
time varying. So,
∆𝑄
∆𝐼 =
∆𝑡
But from eq (III) we can substitute value of Q in above eq,
∆(𝐸𝐴𝜀𝑜 )
∆𝐼 =
∆𝑡
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Or,
𝐼=
𝜀𝑜 ∆(𝐴𝐸)
∆𝑡
But as we know that the electric flux is electric field lines passing through certain area.
∆𝜑 = ∆ 𝐸. 𝐴
So the above equation can be written as,
𝑰=
𝜺𝒐 ∆(𝝋)
………………. (IV)
∆𝒕
Ampere’s Law and Displacement current:
As ampere’s law states that magnetic field is created around every type of current. So, in this
case magnetic field is set up due to the displacement current.
Mathematically it is stated that,
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𝐵 = 𝜇𝑜 𝑛𝐼
Putting value of current from equation (IV) we get,
𝜺𝒐 ∆(𝝋)
𝑩 = 𝝁𝒐 𝒏
∆𝒕
Above eq. shows that changing electric flux or varying electric field produces magnetic field,
similar to the above conclusion that varying magnetic field produces electric field.
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Summary
• We have learned about how changing magnetic
field produces electric field while varying electric
field produces magnetic field.
• Maxwell’s was the first scientist who encloses
electrodynamics into four set of equations.
• He also explained that light waves are
electromagnetic in nature meaning that time
varying electric and magnetic field produces light
waves
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Thank You
Hafsa
Email Address: hafsakhattak35@gmail.com
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