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Introduction to Maxwell equation
- 1. Manju V V 1
- 2. Manju V V 2
- 3. M3 Maxwell Equations • “Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of each other. • One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Manju V V 3
- 4. Manju V V 4
- 5. Physical quantities (Quantity that can be measured, Ex: T, L, M) Scalars Vectors Manju V V 5
- 6. Scalars 5KM 5KM 5KM 5 KM Magnitude Unit The Physical quantities, which require only magnitude for their complete specification called scalars. Ex: Mass, Temperature, speed etc.. Manju V V 6
- 7. Vectors5 KM + East Magnitude Unit The Physical quantities, which has both magnitude and direction is called Vectors. Ex: Velocity, Acceleration, Force, weight etc.. 5KM + East Direction Manju V V 7
- 8. Unit Vector and Base Vector Unit Vector indicates just the direction, its magnitude is always unity Base Vector are unit vectors but oriented strictly along the coordinates. Manju V V 8
- 9. Dot product or Scalar product B A θ O 𝑎 𝑏 |𝒂| 𝐜𝐨𝐬 𝜽 𝒂. 𝒃 = 𝒂𝒃 𝒄𝒐𝒔 𝜽 𝒂 = 𝒂x 𝒊 + 𝒂 𝒚 𝒋 + 𝒂 𝒛 𝒋 𝒃 = 𝒃x 𝒊 + 𝒃 𝒚 𝒋 + 𝒃 𝒛 𝒋 𝒂. 𝒃 = 𝒂x 𝒃x + 𝒂y 𝒃y + 𝒂z 𝒃zManju V V 9
- 10. Cross product or Vector product B A θ O 𝑎 𝑏 |𝒂| 𝒔𝒊𝒏 𝜽• 𝑎 × 𝑏 = 𝑐 • Magnitude of 𝒂 × 𝒃 | 𝒂 × 𝒃| = ab sin 𝜃 • 𝐼𝑓 𝜃 is zero or parallel vectors then 𝒂 × 𝒃 = 0 • Direction of 𝑎 × 𝑏 • 𝒂 × 𝒃 = 𝒄 will be perpendicular to 𝒂 and 𝒃 𝒂 × 𝒃 𝒃 × 𝒂 • Direction of thumb is vertically downward direction • 𝒂 × 𝒃 = −𝒃 × 𝒂 • Magnitude same but direction is negative. Manju V V 10
- 11. | 𝒂 × 𝒃| = 𝐚𝐛 𝐬𝐢𝐧 𝜽 𝒂 = 𝒂x 𝒊 + 𝒂 𝒚 𝒋 + 𝒂 𝒛 𝒋 𝒃 = 𝒃x 𝒊 + 𝒃 𝒚 𝒋 + 𝒃 𝒛 𝒋 𝑎 × 𝑏 = 𝑖 𝑗 𝑘 𝑎 𝑥 𝑎 𝑦 𝑎 𝑧 𝑏 𝑥 𝑏 𝑦 𝑏 𝑧 Manju V V 11
- 12. Scalar and Vector field • Scalar field: Function of space whose value at each point is scalar quantity • Ex: Potential set up by charge in space • Vector field: Function of space whose value at each point is vector quantity • The electric field at each and every point surrounding the charge could be represented by vectors and hence vector field Manju V V 12
- 13. 𝛁 𝐃𝐞𝐥 𝐎𝐩𝐞𝐫𝐚𝐭𝐨𝐫 It is an operator used in mathematics, in particular in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇ 𝛁 𝐃𝐞𝐥 • If T be the scalar function • 𝛁 𝐚𝐜𝐭𝐬 𝐨𝐧 𝐓 • 𝛁T called Gradient • If 𝑬 be the vector function • 𝛁 𝐚𝐜𝐭𝐬 𝐨𝐧 𝑬 via dot product • 𝛁. 𝑬 called Divergence • +ve, -ve and zero divergence • If 𝑯 be the vector function • 𝛁 𝐚𝐜𝐭𝐬 𝐨𝐧 𝑯 via cross product • 𝛁 × 𝑯 called Curl Gradient Divergence Curl 𝛁 = 𝒊 𝝏 𝝏𝒙 + 𝒋 𝝏 𝝏𝒚 + 𝒌 𝝏 𝝏𝒛 Manju V V 13
- 14. Integrals Volume IntegralSurface IntegralLine Integral Manju V V 14
- 16. Line Integral over closed loop 𝑨 𝑨 𝑨 𝑨 𝑨 𝑳 𝑨. 𝒅𝒍 𝑨 Manju V V 16
- 17. Surface Integral 𝑨 𝑨 𝑨 𝑨 𝑨 ds M 𝒏 Area Vector=ds 𝒏 𝑺 𝑨. 𝒅𝒔 𝑺 𝑨. 𝒅𝑺 𝒂𝒏𝒅 Manju V V 17
- 18. Volume Integral + + + + + + + M 𝝆 𝒗 dv 𝒗 𝝆 𝒗 𝒅𝒗 Manju V V 18
- 19. Theorems of Electrostatics, Electricity, Magnetism and Electromagnetic Induction 1. Gauss Flux theorem: 𝑸 = 𝑺 𝑫. 𝒅𝒔 = 𝒒 1.1. Gauss Divergence theorem: 𝑺 𝑫. 𝒅𝒔 = 𝑽 𝛁. 𝑫 𝒅 𝒗 Manju V V 19
- 20. 2. Stokes theorem: Statement: The surface integral of Curl of 𝐹 throughout a chosen surface is equal to the circulation of the 𝐹 around the boundary of the chosen surface. Mathematically 𝑺 (𝜵 × 𝑭). 𝒅𝑺 = 𝑳 𝑭. 𝒅𝒍 Manju V V 20
- 21. 3. Gauss law of Magneto statistics Here 𝐵 magnetic flux density. Applying Gauss Divergence Theorem, we get 𝑆 𝐵. 𝑑𝑆 = 𝑣 𝛻. 𝐵 𝑑𝑣 = 0 −− −(2) 𝑺 𝑩. 𝒅𝑺 = 𝟎 Hence it could be written 𝜵. 𝑩 = 𝟎 Manju V V 21
- 22. By applying Stokes’ theorem, we get 𝑺 𝜵 × 𝑯 . 𝒅𝑺 = 𝑰𝒊𝒏𝒄 −− −(𝟐) The equation for 𝑰 𝒆𝒏𝒄 could be obtained as 𝑰 𝒆𝒏𝒄 = 𝑺 𝑱. 𝒅𝑺 −− −(𝟑) Equating equations (1) and (2) we get 𝑺 𝜵 × 𝑯 . 𝒅𝑺 = 𝑺 𝑱. 𝒅𝑺 Thus, we get Amperes Law as 𝜵 × 𝑯 = 𝑱 Thus, Amperes circuital law and another Maxwell’s equation. 4. Amperes Law Statement: The circulation of magnetic field strength 𝑯 along a closed path is equal to the net current enclosed (𝑰 𝒆𝒏𝒄 )by the loop. 𝑳 𝑯. 𝒅𝒍 = 𝑰 𝒆𝒏𝒄 −− −(𝟏) Manju V V 22
- 23. The magnitude of the magnetic field 𝑑𝐻 is 1. 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑑𝑙 2. 𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝐼. 3. 𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑖𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝜃, 𝑠𝑖𝑛 𝜃 4. 𝐼𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑟. 5. Biot- Savart Law 𝑑𝐻𝛼 𝐼 𝑑𝑙 sin 𝜃 𝑟2 𝑑𝐻 = 𝐼 𝑑𝑙 sin 𝜃 4𝜋𝑟2 𝒅𝑯 = 𝑰 𝒅𝒍 × 𝒓 𝟒𝝅𝒓 𝟐 Manju V V 23
- 24. Faraday’s Laws of electro-magnetic induction Statement 1. Whenever there is a change in magnetic flux linked with the circuit an emf is induced and is equal to rate of change of magnetic flux. 2. The emf induced is in such a direction that is opposes the cause. 𝑒 = − 𝑑𝜙 𝑑𝑡 𝑒 = −𝑁 𝑑𝜙 𝑑𝑡 Manju V V 24
- 25. Faraday’s law in integral and differential forms. For a conducting loop linked with change in magnetic flux the rate of change flux is 𝑑𝜙 𝑑𝑡 = 𝑆 𝜕𝐵 𝜕𝑡 . 𝑑𝑆 −− −(3) 𝑒 = 𝐿 𝐸. 𝑑𝐿 −− −(4) 𝐿 𝐸. 𝑑𝐿 = 𝑆 𝜕𝐵 𝜕𝑡 . 𝑑𝑆 𝐿 𝐸. 𝑑𝐿 = 𝑆 𝛻 × 𝐸 . 𝑑𝑆 𝑆 𝛻 × 𝐸 . 𝑑𝑆 = − 𝑆 𝜕𝐵 𝜕𝑡 . 𝑑𝑆 𝛻 × 𝐸 = − 𝜕𝐵 𝜕𝑡Manju V V 25
- 26. Manju V V 26 Plane electromagnetic Waves in Vacuum 𝐸 = 𝐴 cos 2𝜋 𝜆 𝑥 − 𝑐𝑡 𝑖 −− − 1 𝐵 = 1 𝑐 𝐴 cos 2𝜋 𝜆 𝑥 − 𝑐𝑡 𝑗 −− − 2 𝐸 𝐵 = 𝑐
- 27. Polarization of EM waves Manju V V • Linear polarization: Ex and Ey are inphase • Circular polarization: Amplitude of Ex and Ey are equal and Phase difference 90 degree • Elliptical polarization: Amplitude of Ex and Ey are unequal and Phase difference 90 degree 27
- 28. Manju V V 28 Linear polarization: Ex and Ey are inphase Circular polarization: Amplitude of Ex and Ey are equal and Phase difference 90 degree Elliptical polarization: Amplitude of Ex and Ey are unequal and Phase difference 90 degree