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Maxwell’s Equations Integral Form | Maxwell’s Equations Class 12

Maxwell’s Equations Integral Form | Maxwell’s Equations Class 12 :- Maxwell gave 4 mathematical equations for electricity and magnetism which are called Maxwell’s equations. Electromagnetic phenomena can be explained on the basis of these equations. Maxwell’s equations are as follows :-

Maxwell’s Equations Integral Form

1. Gauss’s law for constant electric field

According to this rule, the total electric flux Φ_E emanating from a closed surface is equal to the ratio of the sum Σq of charges enclosed by that surface and the electric permittivity ε₀ of the vacuum. In other words

$\displaystyle {{\phi }_{E}}=\oint\limits_{S}{\overrightarrow{E}.\overrightarrow{dS}}=\frac{\Sigma q}{{{\varepsilon }_{0}}}\text{ }.....(1)$

2. Gauss’s law for constant magnetic field

Because the poles of a magnet cannot be separated, that is, there is no independent north pole or independent south pole, hence the value of total magnetic flux emanating from a closed surface is zero. Therefore Gauss’s law for magnetism will be as follows :-

$\displaystyle {{\phi }_{B}}=\oint\limits_{S}{\overrightarrow{B}.\overrightarrow{dS}}=0\text{ }.....(2)$

3. Faraday Henry’s law

Electromotive force induced by Faraday’s second law,

$\displaystyle e=-\frac{d{{\phi }_{B}}}{dt}$

The total magnetic flux emitted from a surface,

$\displaystyle {{\phi }_{B}}=\oint\limits_{S}{\overrightarrow{B}.\overrightarrow{dS}}$

Hence the induced electromotive force,

$\displaystyle e=-\frac{d}{dt}\left[ \oint\limits_{S}{\overrightarrow{B}.\overrightarrow{dS}} \right]$

Induced emf for a time varying magnetic field,

$\displaystyle e=-\oint\limits_{S}{\frac{d\overrightarrow{B}}{dt}.\overrightarrow{dS}}\text{ }.....(a)$

We know that

$\displaystyle E=-\frac{dV}{dl}$

Therefore, if there is an electric field E in the conductor, then the induced electromotive force,

$\displaystyle e=\oint\limits_{L}{\overrightarrow{E}.\overrightarrow{dl}}\text{ }.....(b)$

Now from equations (a) and (b)

$\displaystyle \oint\limits_{L}{\overrightarrow{E}.\overrightarrow{dl}}=-\oint\limits_{S}{\frac{d\overrightarrow{B}}{dt}.\overrightarrow{dS}}\text{ }.....(3)$

According to equation (3), the electric field generated due to the changing magnetic field is shown. This equation is called Faraday Henry’s law.

4. Ampere-Maxwell’s law

From Ampere’s circuital law :-

$\displaystyle \oint\limits_{L}{\overrightarrow{B}.\overrightarrow{dl}}={{\mu }_{0}}I$

The above equation is true for the conduction current I_c caused by the flow of free electrons in the conductor.

If such components are also present in the circuit in which electric current flows in the form of displacement current I_d due to the changing electric field, then,

$\displaystyle \oint\limits_{L}{\overrightarrow{B}.\overrightarrow{dl}}={{\mu }_{0}}({{I}_{c}}+{{I}_{d}})$

Since displacement current

$\displaystyle {{I}_{d}}={{\varepsilon }_{0}}\frac{d{{\phi }_{E}}}{dt}$

Therefore

$\displaystyle \oint\limits_{L}{\overrightarrow{B}.\overrightarrow{dl}}={{\mu }_{0}}({{I}_{c}}+{{\varepsilon }_{0}}\frac{d{{\phi }_{E}}}{dt})$

Now

$\displaystyle {{\phi }_{E}}=\oint\limits_{S}{\overrightarrow{E}.\overrightarrow{dS}}$

Therefore

$\displaystyle \oint\limits_{L}{\overrightarrow{B}.\overrightarrow{dl}}={{\mu }_{0}}({{I}_{c}}+{{\varepsilon }_{0}}\oint\limits_{S}{\frac{d\overrightarrow{E}}{dt}.\overrightarrow{dS}})\text{ }.....(4)$

According to equation (4), the magnetic field generated due to the changing electric field is shown. This equation is called Ampere-Maxwell’s circuital law.

Maxwell’s Equations Integral Form | Maxwell’s Equations Class 12

Maxwell’s Equations Integral Form

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