Today in Physics 217: Ampère s Law

Transcription

1 Today in Physis 217: Ampère s Law Magneti field in a solenoid, alulated with the Biot-Savart law The divergene and url of the magneti field Ampère s law Magneti field in a solenoid, alulated with Ampère s law Summary of eletrostatis and magnetostatis so far C A d B d = J da= A C J enl 6 November 2002 Physis 217, Fall

2 Another Biot-Savart law example: the solenoid Griffiths problem 5.11: find the magneti field at point P on the axis of a tightly-wound solenoid (helial oil) onsisting of n turns per unit length wrapped around a ylindrial tube of radius a and arrying urrent. Express your answer in terms of θ1 and θ2(it s easiest that way). Consider the turns to be essentially irular, and use the result of example 5.6. What is the magneti field on the axis of an infinite solenoid? a θ 1 θ 2 P 6 November 2002 Physis 217, Fall

3 Reminder of the result of Example 5.6 Magneti field a distane z along the axis of a irular loop with radius R and urrent : 2π B = zˆ R 2 ( 2 2 z + R ) 32 db z r db db s z db 2 z db db s r db z d R d 6 November 2002 Physis 217, Fall

4 The solenoid (ontinued) a θ 1 θ 2 P z Suppose that n is so large that we an onsider the loops in the oil to be displaed infinitesimally; then the number of loops in a length dz is ndz, and db 2π ndz = zˆ a 2 ( 2 2 z + a ) 32 6 November 2002 Physis 217, Fall

5 The solenoid (ontinued) a θ 1 θ 2 P z Take so tanθ = a z dθ a tan θ + dθ = = dz= dz 2 2 os θ z a a dz = 2 sin θ ( 2 1 tan θ) 2 6 November 2002 Physis 217, Fall

6 The solenoid (ontinued) a θ 1 θ 2 P z 2πndz a 2πn adθ sin θ db = zˆ = zˆ 32 2 sin θ a ( 2 2 z + a ) 2 3 2π n = zˆ sin θdθ ; θ 2πn 2πn B = zˆ sinθdθ = zˆ ( osθ2 os θ1). θ November 2002 Physis 217, Fall

7 The solenoid (ontinued) a θ 1 θ 2 P z For an infinite solenoid, θ = 0 and θ = π, so 2 1 B π π = z 2 n 4 n ˆ ( os 0 os π ) = ˆ = µ 0n ˆ in MKS. z z 6 November 2002 Physis 217, Fall

8 The divergene and url of B Any vetor field is uniquely speified by its divergene and url. What are the divergene and url of B? Consider a volume V to ontain urrent, urrent density ( ) B r ( ) 1 J r rˆ = 2 r dτ Denote gradient with respet to the omponents of r and r by and. Now note that 1 1 = (beause r = r r ), r r 1 rˆ and =. 2 r r J( r ) 6 November 2002 Physis 217, Fall S r V r dτ r J( r ): P

9 The divergene and url of B (ontinued) With these, 1 rˆ 1 1 B( r) = J( r ) dτ = J( r ) dτ 2 r r V ( ) 1 J r = dτ ( remember, J f ( r) ). r V This is a useful form for B, whih we will use a lot next leture too (the integral turns out to be the magneti vetor potential, A). Take its divergene: V 1 J( r ) B( r) = d τ = r V 0. The divergene of any url is zero, remember. 6 November 2002 Physis 217, Fall

10 The divergene and url of B (ontinued) ntegrate this last expression over any volume: ( ) dτ B r = B da= 0. Compare these to the expressions for E in eletrostatis, and we see that magnetostatis involves no ounterpart of harge: there s no magneti harge. Now for the url: 1 J( r ) B( r) = dτ. r V Use Produt Rule #10: ( ) A= A 2 A : 6 November 2002 Physis 217, Fall

11 to write The divergene and url of B (ontinued) 1 J( r ) 1 2 J( r ) B( r) = dτ dτ r r V V 1 J( r ) = dτ J( r ) dτ. r r V V Now use your old friend Produt Rule #5, ( ) ( fa) f ( A) A ( f ) = +, =0 (J independent of r) J r dτ = J( r ) + J( r ) = J( r ) r r r r 6 November 2002 Physis 217, Fall

12 Also, The divergene and url of B (ontinued) ˆ = = r = 4 πδ 3 ( r), 2 r r r so 1 1 ( ) = ( ) τ + ( 3 B r J r d J r ) δ ( r r ) dτ r V 1 1 = J r + r V ( ) dτ J( r) Use Produt Rule #5 again, on the first term: ( ) 6 November 2002 Physis 217, Fall V. ( ) 1 J r 1 J r J( r ) = J( r ) = r r r r =0 in magnetostatis

13 So, The divergene and url of B (ontinued) 1 J( r ) π ( ) = τ 4 B r d + r V 1 J( r ) = da + J( r). r S But by definition J = 0 on the surfae, so the integral vanishes: This an be put into integral form by hoosing an area that some urrent flows through: ( ) J r π B( r) = 4 J( r). Ampère s Law 6 November 2002 Physis 217, Fall

14 A ( ) The divergene and url of B (ontinued) B da= J da C B d = A enlosed. Ampère s law is to magnetostatis what Gauss law is to J eletrostatis, exept that one uses an Ampèrean loop to enlose urrent, instead of a Gaussian surfae to enlose harge. The same triks we learned with Gauss law and superposition have analogues in magnetostatis. A d C 6 November 2002 Physis 217, Fall

15 Example: field in an infinite solenoid The symmetry of the oil ditates that the field must be along z, and must be a lot stronger inside than out, so if the number of turns per unit length is n, and the urrent is, C B d = J da= A enlosed n B z= n z B = zˆ [ = µ 0n in MKS ]. Same as before! 6 November 2002 Physis 217, Fall C z A

16 Maxwell s equations for eletrostatis and magnetostatis Note the similarities and differenes: Ε = ρ B = 0 E = 0 B = J E da= Q B da= 0 enl E d = 0 B d = enl 6 November 2002 Physis 217, Fall

17 Maxwell s equations for eletrostatis and magnetostatis Note the similarities and differenes (MKS): ρ Ε = ε0 B = 0 E = 0 B = µ J 0 1 E da= Qenl B da= 0 ε 0 E d = 0 B d = µ 0 enl 6 November 2002 Physis 217, Fall