Lecture 27: MON 26 OCT Magnetic Fields Due to Currents II

Transcription

1 Physics 212 Jonathan Dowling Lecture 27: MON 26 OCT Magnetic Fields Due to Currents II Jean-Baptiste Biot ( ) Felix Savart ( )

2 Electric Current: A Source of Magnetic Field Observation: an electric current creates a magnetic field Simple experiment: hold a current-carrying wire near a compass needle! B B B I Wire with current INTO page

3 New Right Hand Rule! Point your thumb along the direction of the current in a straight wire The magnetic field created by the current consists of circular loops directed along your curled fingers. B i The magnetic field gets weaker with distance: For long wire it s a 1/R Law! You can apply this to ANY straight wire (even a small differential element!) What if you have a curved wire? Break into small elements. Direction of B! i

4 Field of a Straight Wire!!! ids r db = 3 4 π r db = 4π r ids( r sinθ ) 3 sin θ = R / r r = ( s R ) 1/ 2 B i ds( r sinθ ) = 3 4π r = i Rds 3/ 4π + R 2 ( 2 2 s ) = i Rds = 2π R ir s 2π 2 R R ( 2 2 s + ) 3/ 2 ( 2 2 s + ) 1/ 2 = i 2πR

5 Biot-Savart Law A circular arc of wire of radius R carries a current i. What is the magnetic field at the center of the loop?!! ids r db = 3 4 π r db B = idsr = irdφ 4π 3 R 4π 2 R idφ iφ = = 4π R 4π R Direction of B?? Not another right hand rule?! TWO right hand rules!: If your thumb points along the CURRENT, your fingers will point in the same direction as the FIELD. If you curl our fingers around direction of CURRENT, your thumb points along FIELD! i

6 Forces Between Wires Magnetic field due to wire 1 where the wire 2 is, L I 1 I 2 B 1 = I 1 2π d F Force on wire 2 due to this field, F = L I 21 2 B1 = LI 1 I 2 2π d d eharmony s Rule for Currents: Same Currents Attract! Opposite Currents Repel!

7 29.3: Force Between Two Parallel Wires:

8 F 21 = Li 1 i 2 2π d = Force on 2 due to 1. Like currents attract & Opposites Repel. 1/r double the distance halve the force. F!" ac a F!" net a F!" ab b c F!" bc a b F!" ba c F!" net b a b c F!" cb F!" ca F b net > F c net > F a net F!" net c

9 ICPP The drawing represents a device called Roget s Spiral. A coil of wire hangs vertically and its windings are parallel to one another. One end of the coil is connected by a wire to a terminal of a battery. The other end of the coil is slightly submerged below the surface of a cup of mercury. Mercury is a liquid metal at room temperature. The bottom of the cup is also metallic and connected by a wire to a switch. A wire from the switch to the battery completes the circuit. What is the behavior of this circuit after the switch is closed? a) When current flows in the circuit, the coils of the wire move apart and the wire is extended further into the mercury. b) Nothing happens to the coil because there will not be a current in this circuit. c) A current passes through the circuit until all of the mercury is boiled away. F 21 = Li 1 i 2 2π d = Force on 2 due to 1. Like currents attract & Opposites Repel. 1/r double the distance halve the force. d) When current flows in the circuit, the coils of the wire move together, causing the circuit to break at the surface of the mercury. The coil then extends and the process begins again when the circuit is once again complete.

10 The drawing represents a device called Roget s Spiral. A coil of wire hangs vertically and its windings are parallel to one another. One end of the coil is connected by a wire to a terminal of a battery. The other end of the coil is slightly submerged below the surface of a cup of mercury. Mercury is a liquid metal at room temperature. The bottom of the cup is also metallic and connected by a wire to a switch. A wire from the switch to the battery completes the circuit. What is the behavior of this circuit after the switch is closed? a) When current flows in the circuit, the coils of the wire move apart and the wire is extended further into the mercury. b) Nothing happens to the coil because there will not be a current in this circuit. c) A current passes through the circuit until all of the mercury is boiled away. d) When current flows in the circuit, the coils of the wire move together, causing the circuit to break at the surface of the mercury. The coil then extends and the process begins again when the circuit is once again complete.

11 29.3: Force Between Two Parallel Wires, Rail Gun:

12 Remember Gauss Law for E-Fields? Given an arbitrary closed surface, the electric flux through it is proportional to the charge enclosed by the surface. q Flux =! q Φ E " Surface! E d! A = q ε

13 Gauss s Law for B-Fields! No isolated magnetic poles! The magnetic flux through any closed Gaussian surface will be ZERO. This is one of the four Maxwell s equations.!"!" Φ = B da = B # There are no SINKS or SOURCES of B-Fields! What Goes IN Must Come OUT! No isolated north or south monopoles.

14 ICPP Rank the magnitude of the magnetic flux Φ B = #!" B da!" for the three surfaces, greatest first. c Φ B = B!" da!" # Magnetic Flux is always what for a closed surface!? a b Φ a = Φ b = Φ c =

15 Ampere s law: Closed! B d s! " = i enclosed LOOP Loops i 4 The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current traversing the loop. loop i 2 i 3 i 1!! B ds = ds ( i + i i3) 1 2 Thumb rule for sign; ignore i 4 If you have a lot of symmetry, knowing the circulation of B allows you to know B.

16 Ampere s law: Closed! B d s! " = i enclosed LOOP Loops The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current piercing the loop.! B d! s = (i i ) 1 2 loop Thumb rule for sign; ignore i 3 If you have a lot of symmetry, knowing the circulation of B allows you to know B.

17 C a l c u l a t i o n o f i e n c. W e c u r l t h e f i n g e r s o f t h e r i g h t h a n d i n t h e d i r e c t i o n i n w h i c h t h e A m p e r i a n l o o p w a s t r a v e r s e d. W e n o t e t h e d i r e c t i o n o f t h e t h u m b. All currents inside the loop parallel to the thumb are counted as positive. All currents inside the loop antiparallel to the thumb are counted as negative. All currents outside the loop are not counted. In this example : i = i i. enc 1 2

18 ICPP: Two square conducting loops carry currents of 5. and 3. A as shown in Fig What s the value of B ds through each of the paths shown? Path 1: B ds = ( 5.A+3.A) Path 2: B ds = ( 5.A 5.A 3.A)

19 (a) i + i + i = i (b) i + + i = (c) i + + = i (d) + i + i = 2i d > a = c > b =

20 Ampere s Law: Example 1 Infinitely long straight wire with current i. Symmetry: magnetic field consists of circular loops centered around wire. So: choose a circular loop C so B is tangential to the loop everywhere! Angle between B and ds is. (Go around loop in same direction as B field lines!) Much Easier Way to Get B-Field Around A Wire: No Calculus!!! B ds = i C Bds = B( 2πR) = i C B = R i 2πR

21 29.4: Ampere s Law, Magnetic Field Outside a Long Straight Wire Carrying Current:

22 Ampere s Law: Example 2 Infinitely long cylindrical wire of finite radius R carries a total current i with uniform current density Compute the magnetic field at a distance r from cylinder axis for: r < a (inside the wire) r > a (outside the wire) Current out of page, circular field lines!! B ds = i C i

23 Ampere s Law: Example 2 (cont)!! B ds = i C B( 2π r) = B i enclosed i i = r 2 = enclosed B = i enclosed 2πr = J ( πr 2 ) π πr 2 ir 2πR 2 Need Current Density J! i r R = For r < R For r>r, i enc =i, so Current out of page, field tangent to the closed amperian loop B= i/2πr = LONG WIRE! 2 2

24 B-Field In/Out Wire: J is Constant i 2π R B O r < R r < R B r R B = ir 2π R 2 r > R r B 1 / r r > R B = i 2πr Outside Long Wire!

25 Example, Ampere s Law to find the magnetic field inside a long cylinder of current when J is NOT constant. Must integrate!