Transcript ch-29-magnetic fields due to currents

Chapter 29: Magnetic Fields due to Currents
Introduction
What are we going to talk about in chapter 30:
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H
ow do we calculate magnetic fields for any distribution of
currents?
•
What is the magnetic field due to a straight long wire carrying a
current?
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What is the magnetic field due to a loop carrying a current.
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H
ow much is the force between two parallel current carrying
wires?
•
What is Ampere’s law about?
•
What is the field inside a solenoid?
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What is a magnetic dipole?
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What is the torque on a current loop?
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29.2: Calculating the magnetic field due to a current:
Biot-Savart law:
dB = (mo/4p) i dsxř/r2
mo = 4p 10-7 T m/A
BSL is an inverse square law!!
If ds and ř are parallel, the contribution is zero!
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Special cases:
For a straight wire making angles q1 and q2:
B = (mo i/4pa)(cos q1- cos q2)
Prove it!
The direction is found through the right hand rule:
grasp the element in your right hand with your extended thumb pointing
in the direction of the current. Your fingers will then naturally curl
around in the direction of the magnetic field lines due to that element.
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For infinite straight wire:
B = (mo i/2pa)
At the center of a circular arc with angle f:
B = (mo i f/4pR)
The radial part extending from the arc does not contribute; why?
At the center of a circular loop:
B = (mo i/2R)
Checkpoint #1
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29.3: Force between two parallel currents:
For two long, straight parallel wires a distance [d] apart, the
magnetic field created by current ia at wire #b is:
Ba = moia/(2pd)
Therefore, the force felt at wire #b is:
Fba = Ib L x Ba
L and B are perpendicular, therefore Fba = ib L Ba
Fba/L = moiaib/(2pd)
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What about the force which wire #a feels?
The same!! Why?
Fab/L = moiaib/(2pd)
In which direction are the forces?
Towards one another if the currents are parallel and away from one
another if they are anti-parallel.
In this way we can define the ampere!!
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The rail gun:
Is a device in which a magnetic force can accelerate (~106 g) a
projectile to a high speed (~10 km/s) in a short time (~ 1 ms).
Checkpoint #2
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29-4: Ampere’s law:
As we learned in section 30.1, if a current carrying wire is
grasped with the right-hand with the thumb in the direction of
the current, the fingers curl in the direction of B.
Ampere’s law states that: the closed path line integral of B . ds
around a circle concentric with the current equals moienc.
 
 B  ds  moi
enc
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Special cases:
The magnetic field outside a long (thick/ circular) straight wire:
B = mo i/(2p r)
for r > R
The magnetic field inside a long (thick/ circular) straight wire with
uniform current density:
B = mo i r/(2p R2)
for r < R
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What is the magnetic field created by an infinite uniform current
sheet Js, with a current i over a perpendicular length of the sheet L
such: i = Js/L?
B = mo Js/2
Interaction:
Prove it!
Checkpoint #3
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29-5: Solenoids and Toroids:
The field inside the solenoid is ~ uniform. The field between the turns
tend to cancel. The field outside the solenoid is weak!
The field of a solenoid is similar to that of a bar magnet!
An ideal solenoid is one for which the turns are
closely spaced and the length is long compared
to the radius.
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Applying Ampere’s law to an ideal solenoid gives:
B = mo (N/L) I = mo n I
inside
B= 0
outside
So, now we know how to create strong uniform magnetic fields!!
Why do we use superconducting coils?
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What is the magnetic field created by a toroid?
B = mo N i/(2pr)
inside
B=0
outside
Note that B is not everywhere constant inside the toroid!!
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29-6: A current carrying coil as a magnetic dipole:
We have already seen that if a circuit of magnetic dipole m is
situated in a magnetic field B, the circuit experiences a torque t
produced such that:
t=mxB
Moreover, one can show that for points on the central axis (take it
to be the z-axis) of a single circular loop, a circulating current [i]
produces a magnetic field:
B(z) = (mo/2) i R2/(R2+z2)3/2
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For points far from the loop (still on the z-axiz), this can be
cast in the form:
B(z) = (mo/2p) m/z3
Notice the similarity with the electric field and the electric dipole!!
The circular loop current acts like a magnet:
Checkpoint #4
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