chapter 29: magnetic fields due to currents chapter 29 3 creating magnetic fields Îsources of...
TRANSCRIPT
PHY2049: Chapter 29 1
Chapter 29: Magnetic Fields Due to Currents
PHY2049: Chapter 29 2
Unlike the law of static electricity, comes in two pieces
Piece 1: Effect of B field on moving charge
Piece 2: B field produced by current
Biot-Savart Law Ampere’s LawProof of equivalence not in the book (Requires vector calculus and relies on the absence of magnetic monopoles)
Reminds you of similar equivalence between Coulomb’s law and Gauss’ law
Equivalent!
Law of Magnetism
BvqFrr
×= (Chapt. 28)
PHY2049: Chapter 29 3
Creating Magnetic FieldsSources of magnetic fields
Electric current (moving charges)Atomic orbits of electrons (angular momentum L > 0 only)Internal “spin” of elementary particles (mostly electrons)
Magnetic field produced by current is fundamentalHow about field produced by a bar magnet? Bar magnet ← magnetic ions ← orbital motion and spin of electrons in them ← they are microscopic currents
Three examples studied hereLong wireWire loopSolenoid
PHY2049: Chapter 29 4
B Field Around Very Long WireField around wire is circular, intensity falls with
distanceDirection given by RHR #2 (compass follows field lines)
02iBr
µπ
=
70 4 10µ π −= ×
Right Hand Rule #2
Derived from Ampere’s law
PHY2049: Chapter 29 5
(continued)Why does µ0 have such a simple value?
Magnetism is inseparable from electricity. This allows the units in electricity and magnetism (in particular coulomb and tesla) to be chosen so that only one constant, ε0, has a non-trivial value.
This example illustrates important general property of magnetic fields:
Magnetic field lines have no beginning/end, unlike electricfield lines.
PHY2049: Chapter 29 6
Long Wire B Field Example
I = 500 A toward observer. Find B vs rRHR #2 ⇒ field is counterclockwise
r = 1 mm B = 0.10 T = 1000 gaussr = 1 cm (~0.4”) B = 0.010 T = 100 gaussr = 10 cm (~4”) B = 0.001 T = 10 gauss
( )70
4 10 500 0.00012 2iBr r r
πµπ π
−×= = =
PHY2049: Chapter 29 7
Charged Particle Moving Near Wire
Wire carries current of 400 A upwardsProton moving at v = 5 × 106 m/s downwards, 4 mm from wireFind magnitude and direction of force on proton
SolutionDirection of force is to left, away from wireMagnitude of force at r = 4 mm
Iv
02IF evB evr
µπ
= =
( )( )7
19 6 2 10 4001.6 10 5 100.004
F−
− × ×= × ×
141.6 10 NF −= ×
PHY2049: Chapter 29 8
Ampere’s Law First (Biot-Savart law later)
Take arbitrary closed path around set of currentsLet ienc be total enclosed current (signs +/– according to RHR #2)Let B be magnetic field, and ds be differential length along path
Direction of field due to each current element obeys RHR #2
Only currents inside path count!5 currents inside path (included)1 outside path (not included)This does not mean that current outside path does not contribute to B (note similarity to Gauss’ law)
Not includedin iencenc0iµd =⋅∫ sB
PHY2049: Chapter 29 9
Let’s try this for long wire. Find B at distance at point P
According to RHR #2, B field has only azimuthalcomponent, no radial componentDraw circular path passing through P (radius r)From symmetry, strength of B must be constant along path
An easy derivation
Ampere’s Law For Straight Wire
r
P( ) 0
0
2
2
d B r i
iBr
π µ
µπ
⋅ = =
=
∫ B s∫ ⋅ sB d
PHY2049: Chapter 29 10
20
2 RπirµB =
Ampere’s Law: More Application
Find B vs r inside long wire, assuming uniform current
Wire radius R, total current iDraw circular path of radius r
Key fact: enclosed current ∝ areaInside
Outside: (derived in previous slide)
2
2
enc Rrii = r
R
02iBR
µπ
= On surface
( ) enc02 iµrπB =
rπiµB
20=
PHY2049: Chapter 29 11
Question 10Figure shows the magnitude of B field inside and
outside four long wires. Current is uniformly distributed in each wire. Which wire carries the largest current?
(a) 1(b) 1 and 2(c) 1 and 3(d) Insufficient info
In which wire is the current density the highest?
(a) 1(b) 1 and 2(c) 1 and 3(d) Insufficient info
r
B1
3
4
2
PHY2049: Chapter 29 12
Force Between Two Parallel CurrentsForce on I2 from I1
RHR ⇒ Force towards I1
Force on I1 from I2
Must be the same and towards I2
Why? Newton’s third lawOr view from behind the screen.(I1 is now on left, and I2 now on right.)
Magnetic forces attract two parallel currents
I1I2
0 1 0 1 22 2 1 2 2 2
I I IF I B L I L Lr r
µ µπ π
= = =
I1I2
PHY2049: Chapter 29 13
Force Between Two Anti-Parallel CurrentsForce on I2 from I1
RHR ⇒ Force away from I1
Force on I1 from I2
Must be the same and away from I2
Magnetic forces repel two antiparallel currentsI1I2
I1I2
0 1 0 1 22 2 1 2 2 2
I I IF I B L I L Lr r
µ µπ π
= = =
PHY2049: Chapter 29 14
Parallel Currents (cont.)Look at them edge on to see B fields more clearly
Antiparallel: repel
FF
Parallel: attract
F F
B
BB
B
2 1
2
2
2
1
11
PHY2049: Chapter 29 15
Question 6
OUT IN OUT IN OUT
OUT OUT IN IN IN
OUT IN OUT OUT IN
IN OUT IN IN IN
Long wires, carrying equal currents, are parallel to each other and equally spaced. In which arrangement is the net force on the central wire the largest?
(a) a(b) b(c) c(d) d (e) Insufficient info
Wires are of equal lengths.(a) a(b) b(c) c(d) d