Lecture 16. Ampere`s Law
advertisement
Lecture 16. Ampere’s Law The second common hour in Physics 227 will be held Thursday, November 12 from 9:50 to 11:10 PM at night in 4 locations on two campuses, Livingston and Busch. You should go to the exam location assigned to the first letters of your last name. Note that the exam locations have changed since the first exam. Make sure you go to the correct exam location where you will find your exam. Aaa-Gzz Haa-Mzz Naa-Shz Sia-Zzz BE Aud LSH Aud PHY LH SEC 111 If you have a conflict for this exam you must send your request for a conflict and your entire schedule for the week of November 8 to Professor Cizewski Cizewski@physics.rutgers.edu no later than 5:00 PM on Wednesday, November 4. If you do not request a conflict exam before the deadline you will have to take the exam as scheduled on November 12. 1 Circulation of B The magnetic field at a distance r from a straight wire with current : 2 1 4 π΅ π = 3 π0 πΌ 2ππ π0 πΌ π΅ π = 2ππ οΏ½ π΅ πβ β ππβ = π0 πΌ ππππ The circulation of the field B ≠0 if the enclosedby-the-loop current ≠0. οΏ½ π΅ πβ β ππβ = οΏ½ π΅ πβ β ππβ + οΏ½ π΅ πβ β ππβ + οΏ½ π΅ πβ β ππβ + οΏ½ π΅ πβ β ππβ ππππ 1 2 π0 πΌ π0 πΌ =0+ π+0− π=0 2π 2π 3 4 For a loop that doesn’t enclose any current, the circulation is 0. Ampere’s Law Magnetostatics: both B and I are time-independent. Circulation of the magnetic field around a loop οΏ½ π΅ π β ππβ = π0 πΌππππ ππππ (“Discrete”) enclosed currents: πΌππππ = οΏ½ πΌπ π The line integral could go around the loop in either direction (clockwise or counterclockwise); the sign of currents is determined by the right-hand rule: the curled fingers are aligned along ππβ, the thumb points in the direction of “positive” currents . οΏ½ πΌπ = πΌ1 + πΌ3 − πΌ2 π 3 Continuous Current Distribution π₯β πβ - local current density πΌππππ = οΏ½ π π’π’π’π’π’π’ π₯β πβ β ππ΄β οΏ½ π΅ π β ππβ = π0 ππππ οΏ½ π π’π’π’π’π’π’ the flux of the current vector field π₯β πβ β ππ΄β There are infinitely many possible surfaces that have the loop as their border. Which of those surfaces is to be chosen? It does not matter; for all these π₯β πβ β ππ΄β would be the same (due to the continuity equation surfaces ∫ π π’π’π’π’π’π’ for charge). 6 Gauss’ Law vs. Ampere’s Law Similar to Gauss‘ Law, Ampere’s Law is very useful whenever it is possible to reduce a 3D (vector) problem to a 1D (scalar) problem. The key is the proper symmetry of a problem. οΏ½ π π π π π π π πΈ π β ππ΄β = πππππ π0 Symmetry: if a charge distribution is unchanged by rotations, translations, and reflections, then the E field is also unchanged by the same transformation. οΏ½ π΅ π β ππβ = π0 πΌππππ ππππ Symmetry: if a current distribution is unchanged by rotations and translations, then the B field is also unchanged by the same transformation. Exception: reflections. «Useful» symmetries: - Axial + translational (an infinite - Axial + translational (an infinite cylinder) cylinder) - Spherical - Infinite solenoid - Infinite slab (plane) - Infinite slab (plane) 7 Mirror Reflection Symmetry uniformly charged circle πππ πβ − ππ 1 πΈ π = οΏ½ 3 4ππ0 πβ − ππ πΈ is parallel to the mirror plane at any point on the plane. top view π0 πΌππβ × πβ − ππ π΅ π = οΏ½ 3 4π πβ − ππ π΅ is perpendicular to the mirror plane at any point on the plane. The Mirror Rule for Magnetic Fields: if we can slice a current distribution with a mirror in such a way that the distribution looks exactly the same after we insert the mirror as before, then π© at any point on the mirror’s surface will be perpendicular to that surface. 8 Axial + Translational Symmetry π΅ π × An infinite cylinder carrying a current whose density depends (at most) on the distance r from the axis. Symmetries of the current distribution: π - rotations around the axis; - translations along the axis; π π΅ π π0 πΌ 2ππ 0 π=π - reflections across any plane containing the axis. π0 πΌ 2ππ π΅ π is tangent to a circle centered at the axis and depends only on the distance from the axis. π Amperian loop: a circle centered at the axis οΏ½ π΅ π β ππβ = π0 πΌππππ ππππ Example: A circular wire of radius R with a uniform current density π = πΌ/ ππ 2 : π < π : π2 π΅ π 2ππ = π0 πΌ 2 π ππ π°π π© π = ππ πΉπ π > π : π΅ π 2ππ = π0 πΌ ππ π° π© π = ππ π 9 More Complex Cases Any charge distribution that can be considered as superposition of symmetrical charge distributions can be treated on the basis of Ampere’s Law . π0 πΌ 2ππ π0 πΌ 2ππ π΅ π ∝ 1/π ∝π Coaxial cable π < π < π π΅ π 2ππ = π0 πΌ − πΌ π π 2 −π2 π π 2 −π2 π0 πΌ π 2 − π 2 π΅ π = 2ππ π 2 − π 2 π π = ∝ −π 2 π π π 2 −π 2 π0 πΌ 2 2 π −π 10 Field of an Infinite Solenoid Approximation: the solenoid’s radius is much smaller than its length, and we evaluate the field far from the solenoid’s ends. The solenoid carries current I, the number of turns per unit length is n. Symmetries of the current distribution: - rotations around the axis; - translations along the axis; - reflections across any plane perpendicular to the axis. π© at any point is directed along the axis; it can depend only on the distance from the axis. mirror Loop 1: Loop 2: οΏ½ π΅ π β ππβ = π΅π‘π‘π‘ πΏ − π΅πππ πΏ = 0 ππππ οΏ½ π΅ π β ππβ = π΅π‘π‘π‘ πΏ ± π΅πππ πΏ = π0 πππ ππππ Ampere’s Law allows us to calculate only the combintion π΅π‘π‘π‘ ± π΅πππ = π0 ππ. Experimental fact: Boutside = 0. - field inside is uniform - field outside is also uniform π©ππππππ = ππ ππ° π©πππππππ = π 11 Field of a Finite Solenoid π΅πππ 1 = π0 πΌπΌ 2 - only for a “semiinfinite” solenoid 12 Field of a Toroidal Solenoid Symmetries of the current distribution: - rotations around the center in the plane of the toroid; - reflections across any plane perpendicular to the plane of the toroid and going through its center. π – total # of turns π1 - inner radius π2 - outer radius Loop 1 Loop 2 Loop 3 mirror the radial component of π΅ is zero; π΅ can depend only on the distance from the axis. Amperian loops: circles centered at the toroid’s axis π΅ π 2ππ = 0 π΅ π = 0 π < π1 π πΌπΌ π΅ π 2ππ = π0 πΌπΌ π΅ π = 0 2ππ π΅ π 2ππ = 0 π΅ π = 0 π > π2 π1 < π < π2 π0 πΌπ 2ππ1 π0 πΌπ 2ππ2 π΅ π π1 π2 13 π π – the current density π΅ mirror π¦ π§ × × × × × × × π₯ × × × × × × × 2π Infinite Slab × × × × × × × π × × × × × × × mirror π Symmetries of the current distribution: - translations along the plane; - reflections across any π₯π§ plane - reflections across the π¦π§ plane centered at the slab. π΅ is directed along π¦ everywhere, it is zero along the π¦π¦ plane centered at the slab. Amperian loop: rectangle in the π₯π¦ plane of length π along y, one side is centered at the slab (B=0 at the center due to symmetry) π΅π = π0 πππ −π π΅ −π0 ππ π΅ = −π0 πππ₯Μ π΅ π0 ππ π π₯ −π0 πΎ/2 π0 πΎ/2 π₯ For an infinitely thin slab with the linear current density πΎ = 2ππ (current per unit length) 14 Pressure on Solenoid Walls π΅ππ × × × × × × × π΅=0 π΅ππ = π0 ππΌ = π0 πΎ πΎ = ππ – linear current density (per unit length) Force per one wire of length dl due to the external field Bext produced by all other elements of the solenoid: - Total force per unit area 1π2 (pressure) on a current-carrying sheet: For a 10T solenoid: π= 107 8π πΉβ = πΌπΌπβ × π΅πππ π = πππ΅πππ = πΎπ΅πππ = πΎ 102 ππ ≈ 4 β 107 ππ ≈ 400bar π0 πΎ 2 ! = 1 π΅ππ 2 2π0 Ultra-strong mag. fields: problem of mechanical stability of solenoids 15 Axial + Translational Symmetry ππ π°π π < πΉ: π© π = ππ πΉπ ππ π° π > πΉ: π© π = ππ π Infinite Solenoid Toroidal Solenoid π©ππππππ = ππ ππ° π©πππππππ = π ππ π°π° π© π = πππ π1 < π < π2 Infinite Slab Infinitely thin slab with linear current density K π© = ππ π² π 16 Next time: Lecture 17: Magnetic Materials. Electromagnetic Induction, Faraday’s Law. §§ 28.8, 29.1-3 17 Appendix I. Magnetic Field as a Pseudovector The magnetic field, being a cross product of two polar (or true) vectors, ∝ π × π°β, is a pseudovector (or an axial vector). A pseudovector transforms like a true vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection (including inversion). Geometrically the pseudovector is the opposite of its mirror image (in contrast to a polar vector, which on reflection matches its mirror image). Examples of pseudovectors: angular velocity, torque, and angular momentum. E field of a point charge vs. B field of a current-carrying wire mirror Another example: magnetic field of a current-carrying wire loop. If the position and current of the wire are reflected across the dashed line, the magnetic field it generates would be reflected and reversed. 18
