Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 18 (Text Book: Intro to Electrodynamics by Griffiths, 3rd Ed.) Anand Kumar Jha 09-Feb-2018 Summary of Lecture # 17: • The magnetic field produced by a steady line current × r = 4 r 0 ̂ � 2 ′ • The magnetic field produced by a surface current ( ) × r = 4 r′ 0 ̂ � 2 ′ • The magnetic field produced by a volume current ( ) × r = 4 ′r 0 ̂ � 2 ′ • The magnetic field due to a finite-size wire with current = sin + sin 4 0 2 1 � • The magnetic field above a loop with current = 2 + 2 / 0 2 2 3 2 • The Ampere’s Law × = = 2 0 � ⋅ 0enc Question/Clarification • Current = = • Surface Current = = Density This is per unit transverse length (not area) • Volume Current = = Density This is per unit transverse area (not volume) 3 The Divergence of The magnetic field produced by a volume current ( ) × r = 4 ′r 0 ̂ � 2 ′ r = + + ′ ′ ′ Take the divergence of− the �above magnetic− � field − (with � respect to the unprimed coordinates × r = 4 ′r 0 ̂ ⋅ � ⋅ 2 ′ Using × = × ( × ) ⋅ r⋅ − ⋅ r = × × 4 r 4 r 0 ̂ ′ ′ 0 ′ ̂ ⋅ � 2 ⋅ − � ⋅ 2 ′ × = since does not depend on r ′ ′ × = . (Prob. 1.62 Griffiths, 3rd ed.) r ̂ 2 = 0 The divergence of a magnetic field is zero. 4 ⋅ The Curl of The magnetic field produced by a volume current ( ) × r = 4 ′r 0 ̂ � 2 ′ r = + + ′ ′ ′ Take the curl of the above− magnetic� − field� (with −respect � to the unprimed coordinates × r × = × 4 ′r 0 ̂ � 2 ′ × × = + ( ) ×r r ⋅ − ⋅ r ⋅ − r ⋅ r × = + ′r r r r r ̂ ̂ ′ ′ ̂ ′ ̂ ̂ ′ 2 2 ⋅ − ⋅ 2 ⋅ 2 − 2 ⋅ = since does not depend on ′ ′ r ⋅ = since does not depend on r ̂ ′ ′ 2 ⋅ r r × = 4 r 4 r 0 ′ ̂ 0 ′ ̂ 5 � ⋅ 2 ′ − � ⋅ 2 ′ The Curl of r r × = 4 r 4 r 0 ′ ̂ 0 ′ ̂ �r ⋅ 2 ′ − � ⋅ 2 ′ = 4 (r) r ̂ 3 r × = ⋅ 2 4 r 0 ′ ̂ 0 − � ⋅ 2 ′ r r = r r ′ ̂ ′ ̂ − ⋅ 2 ⋅ ′ 2 = x + y + z r ′ r ′ r ′ ′ − − − ⋅ ′ 3 � 3 � 3 � = = r ′ r ′ r ′ r ′ ′ − ′ − − ′ ′ − ⋅ ′ 3 ⋅ 3 − 3 =⋅ +⋅ ( 3 ) Magnetostatics implies steady current. So =⋅ = 0 ⋅ ⋅ ′ d ⋅ − For large enough integration = = 0 r ′ r ′ volume, all the currents are inside. ′ − ′ − So = 0 at the surface. � ⋅ 3 3 6 � ⋅ The Curl of r r × = 4 r 4 r 0 ′ ̂ 0 ′ ̂ �r ⋅ 2 ′ − � ⋅ 2 ′ = 4 (r) r ̂ 3 2 r × = ⋅ 4 r 0 ′ ̂ 0 − � ⋅ 2 ′ For large enough integration = = 0 r ′ r ′ volume, all the currents are inside. ′ − ′ − 3 3 So = 0 at the surface. � ⋅ � ⋅ So, r = 0 4 r 0 ′ ̂ ′ � ⋅ 2 × = Ampere’s Law 0 7 Ampere’s Law Ex. 5.9 (Griffiths, 3rd Ed. ): Calculate the magnetic field of a very long solenoid. ( closely wound turns per unit length, Radius , and a steady current . Can there be magnetic field in the direction? No. Can there be magnetic field in the direction? No. Can there be magnetic field in the direction? Let’s check For an Amperian loop outside = ⇒ = 0 ⇒ = B( ) But since = 0, = B = B = 0 � ⋅ 0enc − For an Amperian loop inside ∞ ∞ = ⇒ 0 = ⇒ = � ⋅ 0enc − 0 0 = 0 for ( > ) Magnetic field inside a solenoid is uniform---analogous to capacitor which = < ) for ( produces uniform electric fields. 8 0� Ampere’s Law Prob 5.15 (Griffiths, 3rd Ed. ): Two long coaxial solenoids. Current flows in opposite directions. Number of turns are (inner) and (outer). Find magnetic field (i) inside the inner solenoid, (ii) between the solenoids, and (iii) outside 1 2 both the solenoids . Field due to a solenoid = 0 for ( > ) = < ) for ( 0� Principle of linear superposition ⇒ (i) = + = ( ) 0 2� 0 1 −� 0 2−1 � (ii) = 0 2� (iii) = 0 9 Magnetostatics and Electrostatics 1 ( ) × r r = ( ) = 4 ′ 4 r 0 r ̂ ′ Biot̂ -Savart Law �Coulomb’s2 Law � 2 ′ 0 = Q Electric Force = Q( × ) Magnetic Force elec mag = Gauss’s Law = 0 No Name ⋅ 0 ⋅ Amperes’s Law × = 0 No Name × = 0 Maxwell’s equations (Electrostatics) Electric field Magnetic field curls diverges away from around a current. a positive charge = × × = 0 ⇒ = V = 0 ⇒ 10 Electric Potential (scalar) Magnetic Vector Potential − ⋅ Vector Potential (From Lecture # 4): If the divergence of a vector field is zero everywhere,( = 0), then: ⋅ (1) d is independent of surface. This is because of the divergence theorem = (2) ∫ ⋅ d = 0 for any closed surface. � ⋅ � ⋅ ∮ ⋅ ( 3) is the curl of a vector function: = × • This is because divergence of a curl is always zero × = 0 • The vector potential is not unique. A gradient ⋅ of a scalar function can be added to without affecting the curl, since the curl of a gradient is zero. 11 .