Magnetic Potentials
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This document contains lecture notes from a course on electromagnetic theory taught by Arpan Deyasi. It covers topics on magnetic scalar and vector potentials, including their definitions, properties, and applications to problems involving magnetic fields generated by currents. The notes provide the mathematical relationships between magnetic fields and potentials, and work through examples such as calculating the potentials for an infinite solenoid and current-carrying wire.
Magnetic Potentials
- 1. Course: Electromagnetic Theory paper code: EI 503 Course Coordinator: Arpan Deyasi Department of Electronics and Communication Engineering RCC Institute of Information Technology Kolkata, India Topic: Magnetostatics – Magnetic Potential 03-12-2021 Arpan Deyasi, EM Theory 1 Arpan Deyasi Electromagnetic Theory
- 2. 03-12-2021 Arpan Deyasi, EM Theory 2 Magnetic Scalar Potential Magnetic field is related with scalar potential by the relation m B = − where φm is the magnetic scalar potential Arpan Deyasi Electromagnetic Theory
- 3. 03-12-2021 Arpan Deyasi, EM Theory 3 Property of Magnetic Scalar Potential We know from solenoidal property of magnetic field that .B 0 = ( ) m . 0 − = 2 m 0 = Arpan Deyasi Electromagnetic Theory
- 4. 03-12-2021 Arpan Deyasi, EM Theory 4 Problem 1 Calculate magnetic scalar potential for infinite solenoid Soln ˆ B NIz = For infinite solenoid Where ‘N’ is the umber of turns, ‘I’ is current in the winding m B = − Also magnetic scalar potential is related with field as Arpan Deyasi Electromagnetic Theory
- 5. 03-12-2021 Arpan Deyasi, EM Theory 5 m m m 1 ˆ ˆ ˆ ˆ NIz z z = − + + m NI z = − m NIz = − Arpan Deyasi Electromagnetic Theory
- 6. Magnetic Vector Potential We know from solenoidal property of magnetic field that .B 0 = Let A be a differentiable vector ( ) . A 0 = Comparing B A = 03-12-2021 Arpan Deyasi, EM Theory 6 Arpan Deyasi Electromagnetic Theory
- 7. Expression of Magnetic Vector Potential According to Biot-Savart law 0 3 J r B dv 4 r = Now 3 1 r r r = − 03-12-2021 Arpan Deyasi, EM Theory 7 Arpan Deyasi Electromagnetic Theory
- 8. Expression of Magnetic Vector Potential 0 1 B J dv 4 r = Now ( ) J 1 1 J J r r r = + 03-12-2021 Arpan Deyasi, EM Theory 8 Arpan Deyasi Electromagnetic Theory
- 9. Expression of Magnetic Vector Potential Since J 0 = J 1 J r r = J 1 J r r = − 03-12-2021 Arpan Deyasi, EM Theory 9 Arpan Deyasi Electromagnetic Theory
- 10. Expression of Magnetic Vector Potential 0 J B dv 4 r = 0 J B dv 4 r = 03-12-2021 Arpan Deyasi, EM Theory 10 Arpan Deyasi Electromagnetic Theory
- 11. Comparing with B A = 0 J A dv 4 r = We get Expression of Magnetic Vector Potential 03-12-2021 Arpan Deyasi, EM Theory 11 Arpan Deyasi Electromagnetic Theory
- 12. Property of magnetic vector potential: Solenoidal 0 J A dv 4 r = 0 J .A . dv 4 r = 0 J .A . dv 4 r = 03-12-2021 Arpan Deyasi, EM Theory 12 Arpan Deyasi Electromagnetic Theory
- 13. Property of magnetic vector potential: Solenoidal Applying divergence theorem 0 J .A dS 4 r = By expanding the surface, we can make .A 0 = So, magnetic vector potential is solenoidal 03-12-2021 Arpan Deyasi, EM Theory 13 Arpan Deyasi Electromagnetic Theory
- 14. Problem 2 Calculate M.V.P in the region surrounding an infinitely long, straight, filamentary current I Soln For long straight wire carrying current I enc I ˆ B 2 = enc I ˆ A 2 = 03-12-2021 Arpan Deyasi, EM Theory 14 Arpan Deyasi Electromagnetic Theory
- 15. enc z A I A z 2 − = Since the filament is uniform along Z direction, so A should not be a function of Z enc z I dA d 2 − = Let at 0 z A 0 = = 03-12-2021 Arpan Deyasi, EM Theory 15 Arpan Deyasi Electromagnetic Theory
- 16. enc 0 z I A ln 2 = enc 0 z I ˆ A ln z 2 = 03-12-2021 Arpan Deyasi, EM Theory 16 Arpan Deyasi Electromagnetic Theory
- 17. B J = Magnetic vector potential is given by B A = ( ) B A = Ampere’s circuital law is given by Property of magnetic vector potential: Laplacian 03-12-2021 Arpan Deyasi, EM Theory 17 Arpan Deyasi Electromagnetic Theory
- 18. Property of magnetic vector potential: Laplacian Comparing ( ) A J = ( ) 2 .A A J − = Since .A 0 = 2 A J = − 03-12-2021 Arpan Deyasi, EM Theory 18 Arpan Deyasi Electromagnetic Theory
- 19. Problem 3 Magnetic vector potential is given by 2 ˆ A z Wb / m 4 = − Calculate total magnetic flux crossing the surface 1m<ρ<2m, φ=π/2, 0<z<5m Soln z A ˆ B A = = − 2 ˆ ˆ B 4 2 = − − = 03-12-2021 Arpan Deyasi, EM Theory 19 Arpan Deyasi Electromagnetic Theory
- 20. Magnetic flux m B.dS = 5 2 m z 0 1 ˆ ˆ .d dz 2 = = = 5 2 m z 0 1 dz d 2 = = = m 3.75 Wb = 03-12-2021 Arpan Deyasi, EM Theory 20 Arpan Deyasi Electromagnetic Theory