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 Hafsa Khattak 2 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, βπ ∈= βπ‘ Where, βπ = β π΅. π΄ ππ π΄ βπ΅ . So, the above equation becomes, π¨ βπ© ∈= ………………….. ( I) βπ 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. Hafsa Khattak Hafsa Khattak 4 d= 2ππ Then work done becomes, π = πΉ 2ππ π€βπππ, πΉ = ππΈ π π, π = ππΈ ( 2ππ) We also know that electric potential or EMF (∈) is equals to; π ∈= π Now putting W in above equation, we get; ∈= ππΈ ( 2ππ) π Or, ∈ = ππ ππ¬ …............ (II) Now compare eq. (I) and (II) we get, Hafsa Khattak 5 ∈= π΄ βπ΅ ………………….. ( I) βπ‘ ∈ = 2πππΈ …............ (II) 2πππΈ = π΄ βπ΅ βπ‘ or, π¨ βπ© π¬= ππ π βπ This equation shows that, change in magnetic field produces electric field in a coil. Hafsa Khattak 6 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. Hafsa Khattak Hafsa Khattak 7 As we know that for charging a capacitor the formula is π = πΆπ And πΆ= π΄ππ , π Then above eq. becomes, π΄ππ π= π π But we know that π½ = π¬π , so, Q becomes, π΄ππ π= πΈπ π Or, πΈ = π¬π¨πΊπ ………………… (III) Hafsa Khattak 8 (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. Hafsa Khattak 9 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, βπ βπΌ = βπ‘ But from eq (III) we can substitute value of Q in above eq, β(πΈπ΄ππ ) βπΌ = βπ‘ Hafsa Khattak 10 Or, πΌ= ππ β(π΄πΈ) βπ‘ But as we know that the electric flux is electric field lines passing through certain area. βπ = β πΈ. π΄ So the above equation can be written as, π°= πΊπ β(π) ………………. (IV) βπ 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, Hafsa Khattak 11 π΅ = ππ ππΌ Putting value of current from equation (IV) we get, πΊπ β(π) π© = ππ π βπ 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. Hafsa Khattak 12 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 Hafsa Khattak 13 Thank You Hafsa Email Address: hafsakhattak35@gmail.com Hafsa Khattak 14