Force between parallel currents Example calculations of B from the Biot- Savart field law Ampère s Law Example calculations

Transcription

1 Today in Physics 1: finding B Force between parallel currents Example calculations of B from the Biot- Savart field law Ampère s Law Example calculations of B from Ampère s law Uniform currents in conductors? André-Marie Ampère, for whom the law, and the amp, are named. 30 October 01 Physics 1, Fall 01 1

2 Brief reminders from the recent past Force laws: df d B dbsinnˆ F d B B sin nˆ d F QvB point charges Cross products: ab ba ab sinnˆ ab0 if a b ab abnˆ if ab Cross products of Cartesian unit vectors: x ˆy ˆ ˆ y ˆ ˆ x ˆ ˆx ˆ y ˆ currents B d Qvv or df or F Lay fingers of right hand along arc from d to B, pointing to B. Thumb points along df. 30 October 01 Physics 1, Fall 01

3 Brief reminders from the recent past (continued) r r Biot-Savart field law: d d rr r 0 db 4 r r r r db 7 1 where 4 10 T m A r 0 Note the ways the right-hand rule can help one work out the direction of B: r r d Fingers along db d arc from d to Thumb along d ; r r ; thumb fingers curl in points along db. direction of db. 30 October 01 Physics 1, Fall 01 3

4 Forces between currents Consider a section of a current with length L, parallel to a long wire carrying current 1 and lying a distance y away. What is the force on this wire? Let both currents flow in the direction, let the first one lie along the axis, and the second one lie a distance y away along the y axis. The field at the location of the second wire is uniform, as we found last time: 1 df dˆ B 01 d dˆ B xˆ y x y 30 October 01 Physics 1, Fall 01 4

5 Forces between currents (continued) So, from the Biot-Savart force law, F d B L 0 1 ˆxˆ d y L 0 1 ˆ y y That is, parallel currents attract each other with a magnetic force; antiparallel currents repel each other. 0 x 1 df dˆ B y 30 October 01 Physics 1, Fall 01 5

6 Biot-Savart vs. Coulomb The setup, and most of the execution, of B calculations from the Biot-Savart field law are the same as for E calculations l using Coulomb s law. That is, choose an appropriate coordinate system, dissect the source distribution into infinitesimal elements, use the symmetry of the source distribution to simplify the vector addition as much as possible, and then integrate the resulting expression. No new tricks are involved, and no new complications besides the intrusion i of cross products, and fields that t lie sideways with respect to the distance from the infinitesimal element. 30 October 01 Physics 1, Fall 01 6

7 Example calculations of B A current flows along two 90- degree arcs with radii a and b, and through the radii connecting them, as shown. Calculate l the magnetic field at point P, the center of the arcs. A Cartesian coordinate system with origin at P and axes along the straight segments seems appropriate; we can see a use for both x-y and polar coordinates. P y b a x 30 October 01 Physics 1, Fall 01 7

8 Example calculations of B (continued) The appropriate y infinitesimal current elements: ˆ 1 dyy 1 bd ˆ d dxxˆ 3 ˆ ad 4 b a 4 -r values (note r = 0): yyˆ 1 brˆ r arˆ 4 xxˆ 3 P 3 x 30 October 01 Physics 1, Fall 01 8

9 Example calculations of B (continued) For the straight segments 1 and 3, d r r dyˆ y yˆ 0, and dx ˆ ˆ 0. x x 0 B P d r r rr 4 0 ˆ rˆ 4 b 0 + ˆ rˆ 4 a 0 0 d d 30 October 01 Physics 1, Fall 01 9 P y 1 b a 4 3 x

10 Example calculations of B (continued) Note, for the polar unit vectors, rˆ ˆ ˆ ˆrˆ ˆ ˆ ˆ rˆ ˆ rˆ ˆ etc. So 0 ˆ 0 B P ˆ 4b 4a 0ˆ a b x ˆ ˆr ˆ y 30 October 01 Physics 1, Fall 01 10

11 Example calculations of B (continued) A current flows as shown in a circular loop of radius. Calculate the magnetic field a distance above the loop, along its axis. A coordinate system with origin at the center of the loop and axis perpendicular p to the loop seems good. nfinitesimal current element: d d ˆ ( 0 ) x d y 30 October 01 Physics 1, Fall 01 11

12 Example calculations of B (continued) db Distance from element: rr rˆcos ˆsin rˆ ˆ So the integral contains d rr d ˆ r ˆ ˆ d ˆ rˆ db db r db r r But ˆr points in opposite d directions for elements across the loop from one another, for which differs by. db r d 30 October 01 Physics 1, Fall 01 1

13 Example calculations of B (continued) db So the r components cancel out in pairs, and the components add in pairs. Keep only, multiply l by, and integrate from 0 to : 0 B d rr rr 4 0 ˆ 3 0 d 3 db d db r db r r r db 0 ˆ 3 d 30 October 01 Physics 1, Fall 01 13

14 Example calculations of B (continued) The solenoid. Find the magnetic field at point P on the axis of a tightly-wound solenoid (helical coil) consisting of n circular turns per unit length wrapped around a cylindrical tube of radius and carrying current. Express your answer in terms of 1 and. What is the magnetic field on the axis of an infinite solenoid? 1 P 30 October 01 Physics 1, Fall 01 14

15 Example calculations of B (continued) 1 P Suppose that n is so large that we can consider the loops in the coil to be displaced d infinitesimally; it i then the number of loops in a length d is nd, and db 0 nd ˆ 3 30 October 01 Physics 1, Fall 01 15

16 Example calculations of B (continued) Substitute tan so 1 tan 1 d tan d d d cos d d sin 30 October 01 Physics 1, Fall P

17 Example calculations of B (continued) d 3 0nd 0n d sin ˆ ˆ 3 sin B ˆ 0n sin d ; 0n ˆ ˆ 0n B sind cos cos October 01 Physics 1, Fall P

18 Example calculations of B (continued) 1 P For an infinite solenoid, 0 and, so 1 0n B ˆ cos 0 cos 0nˆ 30 October 01 Physics 1, Fall 01 18

19 Ampère s law n 186, André-Marie Ampère (France) saw that Ørsted s result for the magnetic field of a long wire could be cast in a different, useful way. Handwaving derivation follows. (For a real one, click here, and start at page 8.) We saw last time that the long wire produced a magnetic field 0 ˆ r 0 ˆ B r x r ˆr ˆ B y 30 October 01 Physics 1, Fall 01 19

20 Ampère s law (continued) This can be written as B r ˆ ˆ ˆ 0 0 The lines of B are circles centered on the wire, in a plane perpendicular to the wire. The term r ˆ looks like an integral of displacement vectors around B the circle: ˆ r d rd y B rd ˆ B d 0 x ˆr ˆ 0 30 October 01 Physics 1, Fall 01 0

21 Ampère s law (continued) But B is constant in magnitude along the circles, so it can be taken inside the integral. Furthermore, is the current enclosed db by the circular loop. Thus Bd 0 encl Ampère s law B This relation, which h turns out to be quite general, can be used to find B for cases in which is symmetrically distributed, in much the same way that Gauss s law can be used to find E. x r ˆr ˆ y 30 October 01 Physics 1, Fall 01 1

22 Uniform cylindrical current Example. A long straight wire with radius carries a current which is uniformly distributed over its cross- sectional area. Calculate l B inside id the wire. We already know that B is uniform in magnitude on circles outside the wire: 0 B r ˆ r nside the current is still cylindrically ll symmetric, so we suspect B will be too. A circle is a good Ampèrean path here. r 30 October 01 Physics 1, Fall 01

23 Uniform cylindrical current (continued) Define the current density J (current per unit cross-sectional area of the wire): J A Uniform current means that J is uniform across the wire s cross section: the total current is J, and the current enclosed by a circle of radius r coaxial with the wire our Ampèrean path is encl J r r r 30 October 01 Physics 1, Fall 01 3

24 Uniform cylindrical current (continued) Now we apply Ampère s law, noting again that B has uniform magnitude along the circle: B d 0encl r B r 0 0r ˆ 0Jr B ˆ B increases linearly with radius, starting from ero, and matches up with the outside value at the wire s surface. B r 30 October 01 Physics 1, Fall 01 4

25 Aside: current distribution in conductors Note that didn t say that this uniform current is carried in a conductor! Parallel currents attract each other, as we showed earlier today. So any effort to make a uniform current in a conducting wire will result in the current tending to pile up at the center: current threads attract one another, and the charges are free within the conductor to move in the direction they re being pulled. The calculation of a self-consistent current distribution in a conducting wire is beyond the scope of PHY 1; you will learn how to do this in a course like PHY October 01 Physics 1, Fall 01 5

26 Field in an infinite solenoid, Ampère s version ectangular Ampèrean loop, as shown. The symmetry of the coil dictates that the field must be along, and must be a lot stronger inside than out, so if the number of turns per unit length is n,, and the current is, B d 0 enclosed B n n B B 0 0 ˆ Same as before. 30 October 01 Physics 1, Fall 01 6

27 Ampère vs. Gauss Here are the conditions under which it is profitable to use Ampère s law to find B, compared to Gauss s law to find E. Ampère Gauss nfinite linear current nfinite planar current nfinite cylindrical current, any radial dependence nfinite solenoid Toroid nfinite linear charge nfinite planar charge nfinite cylindrical charge, any radial dependence Spherically symmetric charge, any radial dependence 30 October 01 Physics 1, Fall 01 7