10 Problem 5.13 A steady current I flows down a long cylindrical wire of radius a. Find the magnetic field, both inside and outside the wire, if (a) The current is uniformly distributed over the outside surface of the wire. (b) The current is distributed in such a way that J is proportional to s, the distance from the axis.Ia
11 Problem: 5.15 Two long coaxial solenoids each carry current I, but in opposite directions as shown. The inner solenoid (radius a ) has n1 turns per unit length, and the outer one(radius b) has n2 turns per unit length.abb
12 Problem: A very long straight conductor has a circular cross-section of radius R and carries a current Iin. Inside the conductor there is a cylindrical hole of radius a whose axis is parallel to the axis of the conductor and a distance b from it.aRbIinxyz
13 …find the magnetic field B at a point (a) on the x axis at x=2R and (b) on the y axis at y=2R. Iinxyz
15 Problem: 5.14 A thick slab extending from z=-a to z=+a carries a uniform volume current , Find the magnetic field, as a function of z,both inside and outside the slab.xzy+a-a
16 Problem:5.16 A large parallel plate capacitor with uniform surface charge density σ on the upper plate and –σ on the lower is moving with a constant speed v, as shown below.+σ-σv
17 Find, (a) The Magnetic field between the plates, (b) The Magnetic force per unit area on the upper plate & its direction, (c) …the speed v at which the magnetic force balances the electrical force.+σ-σv
18 Toriodal Coil which along wire is wrapped. …a circular ring, or “donut” aroundwhich along wire is wrapped...the winding is uniform and tight enoughso that each turn can be considered aclosed loop.
19 Problem: …the magnetic flux through the end face of a solenoid… K
20 Comparison of Magnetostatics and Electrostatics The divergence and curl of theelectrostatic field are:…together with the Boundary conditionsdetermine the field uniquely.
21 Comparison of Magnetostatics and Electrostatics The divergence and curl of the Magnetostatics field are:…together with the Boundary conditions determine the field uniquely.
22 Magnetic Vector Potential …permits us to introduce a Vector PotentialA in Magnetostatics:
23 Problem: A spherical shell, of radius R, carrying a uniform surface charge σ, is set spinning at angular velocity ω. Find the vector potential A it produces at point P.ωrzRxyΦ/Ө/Ψrsr/da/PωRrPz
25 …Expressions for the Magnetic Field Inside & Outside the Spherical Shell are: (Inside the Spherical Shell)
26 …Expressions for the Magnetic Field Inside & Outside the Spherical Shell are:
27 Problem: 5.42 Calculate the Magnetic Force of Attraction between the Northern and Southern Hemispheres of a Spinning Charged Spherical Shell(…of Radius R carrying a uniform charge density σ and spinning at an angular velocity ω).
35 Multipole Expansion of the Vector Potential rsOr/dr/=dl/I
36 Problem: 5.60 (a) Work out the Multipole expansion for the vector potential for a volume current J. (b) Write down the Monopole potential and prove that it vanishes. (c) Write the corresponding dipole moment m.
37 Problem: Find the magnetic dipole moment of the “bookend-shaped” loop as shown below. All sides have length w, and it carries a current I.zxyIw
39 Problem:5.34 Show that the magnetic field of a dipole can be written in coordinate free form as: ӨΦyzmrPx
40 Problem: 5.34 A circular loop of wire, with radius R, lies in the xy-plane, centered at the origin, and carries current I running counterclockwise as viewed from the positive z-axis.zR
41 (a) What is its magnetic dipole moment (a) What is its magnetic dipole moment? (b) What is the (approximate) magnetic field at points far from the origin? (c) Show that, for points on the z-axis, the answer is consistent with the exact field when z>>R.
42 Problem:5.35 A phonograph record of radius R, carrying a uniform surface charge σ, is rotating at constant angular velocity ω. Find its dipole moment.zωRyx
44 Problem: 5.55 A Magnetic dipole is situated at the origin, in an otherwiseuniform magnetic fieldShow that there exists a spherical surface,centered at the origin,through which no Magnetic field linepasses.
47 Thus: “The Perpendicular Component of the Magnetic Field is Continuous across a surface current.” Whereas “The Component of B that is parallel to the surface but perpendicular to the current is discontinuous by an amount μ0K.
48 These Results can be Summarized as: where, ‘n’ cap is theunit Vector perpendicularto theSurface pointing upwards.
49 Problem : Show that the Vector Potential A is continuous, but its derivative inherits a discontinuity across any boundary.
50 Problem: 5.24 (a)…find the vector potential a distance s from an infinite straight wire carrying a current I. (b) …find the vector potential inside the wire if it has a radius R and the current is uniformly distributed.