T09 - 1 TUTORIAL 9 1 B DUE TO A LONG STRAIGHT WIRE In...

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Unformatted text preview: 1 TUTORIAL 9 1. B DUE TO A LONG STRAIGHT WIRE In Tutorial 8, you showed (we hope) that the field due to a wire of length l directed along the z axis was given by ( ) B a = ! μ "# $ $ % 1 2 4 I cos cos where the angles were defined at the point of observation relative to a line parallel with the z axis. What is the magnetic flux density for an infinitely long wire? Use Ampere’s Law in integral form to deduce the 1/ ρ dependence of B . 2. INFINITELY LONG SOLENOID AGAIN In Lectures, this example was treated somewhat superficially. By more careful consideration of Ampere’s Law, show that: (a) the field inside the solenoid is uniform; (b) the field outside the solenoid is exactly zero. 3. PROOF OF MAXWELL’S THIRD EQUATION Use the Biot-Savart Law in the form ( ) B r J r a J r r r r r ( ) ( ) ( ) μ ! ! 2 3 4 4 = " # = " # $ " $ " % % R V V R dv dv , where it is understood that the integral is over the primed coordinates, to prove the assertion made in Lectures that ! " = B . 4. MAGNETISATION Prove that the magnetisation M and bound current density J b are related by ! " = M J b . You can find several derivations in different textbooks. 2 5. CONSERVATION OF FLUX In Question 3 above, you proved that ! " = B . In Lectures, we used this to show that the flux of B is conserved. What can you say about ! " H in general, and therefore, what can you say about the flux of H ? Can you think of any types of magnetic matter for which ! " = H ? 6. MAGNETOSTATIC BOUNDARY VALUE PROBLEMS Consider a type of magnetic matter for which ! " = H . In analogy with electrostatics, we saw that if ! " = = H J , then we can write H = !" V m . Derive Laplace’s equation for the magnetic scalar potential, i.e., ! = 2 V m . Hence, determine the magnetic flux density over all space when a sphere of linear magnetic material is placed in an initially uniform magnetic field with flux density B o . 7. MAGNETIC VECTOR POTENTIAL – MAGNETIC DIPOLE Recall that the vector magnetic potential is given by ! ! " = " " = " C V R d I v d R l J A # μ # μ 4 4 . We stated that the vector potential was useful in radiation problems (time-varying fields), and in deriving approximate quantities. Here, we put it to work to derive one such approximate quantity. Use the expression given here to derive the magnetic field intensity due to a magnetic dipole at distances great compared to the dimensions of the dipole. Again, this derivation can be found in many textbooks. 3 ELECTROMAGNETIC THEORY SOLUTIONS TO TUTORIAL 9 1. We set ! 1 = and ! " 2 = in B = μ I 4 "# cos $ 1 % cos $ 2 ( ) a & to obtain B a = μ !" # 2 I . Consider Ampere’s Law in integral form, B l J S ! = !...
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This homework help was uploaded on 04/17/2008 for the course ELEC 3305 taught by Professor - during the Spring '07 term at University of West Alabama-Livingston.

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T09 - 1 TUTORIAL 9 1 B DUE TO A LONG STRAIGHT WIRE In...

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