# Derivation of maxwell equations notes in easiest way

Introduction

• In today’s article, we are going to study about ‘ Maxwell equations ‘ How beneficial it is and what is the basics of concept of these equations you should know before getting a detailed information on it. And hence We will also read about some more topics like what is Maxwell equations, Maxwell’s first equation, Maxwell’s second relation, Maxwell’s third relation, Maxwell’s fourth relation etc given in Detail.  So, take your notebooks in you hand and get ready to study physics in an easy and sorted way.

# Maxwell equations/ relation

The state of homogeneous system is completely determine if we know its mass and any two of the thermodynamic variables (p,v,t,u,s) maxwell’s relation relate the (p,v,t,u,s).

## Maxwell’s first relation – from first thermodynamic law

We have,

dU = dQ – dW

dU = dQ – PdV

And dU = TdS- PdV     …(1)

Internal energy is function of s and dV then by partial differentiation,

( ∂U /∂S)at constant V = T  …(2)

Similarly, ( ∂U /∂V)at constant S = -P  …(3)

By the property of exact Differentiation,

[ ∂/∂V (∂U/ ∂V) at constant V] at constant S = [ ∂/∂S (∂U/∂V)at constant S] at constant V

( ∂T /∂V)at constant S =- ( ∂P /∂S)at constant V  …(3)

## Maxwell’s second equation

By the definition of helmholtz free energy

F= U- TS   …(1)

For small changes equation first becomes,

dF= dU – SdT – TdS

We know dU – TdS = -PdV

dF=  – SdT – PdV

Thus f is a function of V and T, so that

( dF/dT)at constant V = -S     …(2)

And ( dF/dV)at constant T = -P  …(3)

Definition of exact differentiation

( [ ∂/∂V (∂F/ ∂T) at constant V] at constant T = [ ∂/∂T (∂F/∂V)at constant T] at constant V

-( ∂S /∂V)at constant T = -( ∂P /∂T)at constant V

( ∂S /∂V)at constant T = ( ∂P /∂T)at constant V

## Maxwell’s third relation

By the definition of enthalpy

H= U+ PV

For small changes,

dH= dU+ PdV+ VdP      …(2)

Now we know that TdS= dU+ PdV

dH= TdS+ VdP

H is a function of S And P

( ∂H /∂S)at constant P = T    …(3)

And ( ∂H /∂P)at constant S = V     …(4)

By the definition of exact differentiation equation

( [ ∂/∂P (∂H/ ∂S) at constant P] at constant S = [ ∂/∂S (∂H/∂P)at constant S] at constant P

( ∂T /∂P)at constant S = ( ∂V /∂S)at constant P

## Maxwell’s fourth relation

By the definition of Gibbs free energy,

G= H- TS     …(1)

G= U+ PV – TS

(∴ H= U+ PV)

dG= dU + PdV + VdP -TdS -SdT

dG= TdS – TdS + VdP -SdT

And hence we get,

dG= VdP- SdT

G is a function of T and P,

( ∂G /∂P)at constant T = V     …(2)

(∂G /∂T)at constant V = -S     …(3)

By the definition of exact differential equation

( [ ∂/∂P (∂G/ ∂T) at constant P] at constant T = [ ∂/∂T (∂G/∂P)at constant T] at constant P

(∂S /∂P)at constant T = – (∂V /∂T)at constant P

### Conclusion

So our article is finished and after completely reading this article, one can easily tell what is Maxwell equations. And we have also discussed some more topics like Maxwell equations, Maxwell’s first equation, Maxwell’s second relation, Maxwell’s third relation, Maxwell’s fourth relation etc.

‌So one can say that they got a detailed information about Maxwell equations and basic information about other topics.

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