Lecture 8 Examples of Magnetic Fields

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Lecture 8Examples of Magnetic Fields
Chapter 19.7 ? 19.10
Outline

• Long Wire and Amperes Law
• Two Parallel Contours
• Solenoid

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Magnetic Field of a Wire
Charges flowing in a single wire produce magnetic
field, whose field lines circle the wire in the
direction according to the right hand rule.
The magnetic field strength (B) is the same at
all points of the circular path around the wire.
B at distance r from a wire carrying current I
is
?0 is the permeability of free space ?0 ? 4 ?
10?7 T m / A
?0 I B ?? 2 ? r
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Amperes Law
Andre-Marie Ampere proposed how to calculate the
magnetic field in an arbitrarily shaped wire.
Amperes circuital law states that over any
closed path around the wire ?(B??l) ?0 I
This law gives an identical result to that for a
long, straight current. Amperes circuital law is
only valid for the currents and fields constant
in time.
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Magnetic Field Between Two Parallel Conductors
Two current-carrying conductors exert magnetic
forces on each other.
Consider 2 long, straight wires of length l
carrying currents I1 and I2 and separated by a
distance d.
?0 I2 B2 ?? 2?d
?0 I2 ?0 I1
I2 F1B2 I1 l ?? I1 l ??? l
2?d 2?d
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Example
Two wires positioned one above another parallel
to the Earths surface have a weight per unit
length of 10?4 N/m each. If they are separated by
1 cm and not affected by Earths magnetic field,
what must be the current (same in both wires) to
keep the upper wire in the air?
mg ?0 I2 ? ?? l 2?d
?0 I1 I2 F2 ??? l 2?d
F1 mg
I ? (2?d mg/l?0) ? (107 dmg/2) 2.2 A
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Two Parallel Conductors
The magnetic force exerted by one wire onto the
other is directed toward the other wire.
F1 ?0 I1 I2 ? ??? l 2?d
Thus, the wires carrying currents in the same
direction attract each other. This result is
derived using the right-hand rule No 1 and in
compliance with Newtons third law.
Definition of the unit of current, ampere. 1 A is
the current in 2 wires 1 m apart attracting each
other with the force per unit length of 2 10?7
N/m.
All magnetic fields originate from moving
electric charges.
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Magnetic Field of a Solenoid
Solenoid is a coil of several closely spaced
loops.
Consider a solenoid of length l and total number
of turns N carrying a current I. The magnetic
field inside the solenoid will be
B ?0 n I
n N/l
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Current in a Solenoid

• Problem A superconducting solenoid is designed
  to generate a magnetic field of 5 T.
• If the solenoid winding has 1000 turns/m, what is
  the required current?

B I ?? ?0 n
n 1000
B ?0 n I
I 5/(4?10?7 103) 3978 A
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Amperes Law for a Solenoid
The magnetic field B inside a solenoid is uniform
and parallel to its axis. B outside a solenoid is
virtually zero.
? (B??l) BL ?0 N I
Picture of a solenoid field
N B ?0 ? I ?0 n I L
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Summary

• Amperes law is a relation between the current in
  an arbitrarily shaped wire and the magnetic field
  produced by the wire.

• Magnetic field of a closed loop increases the
  magnetic field set up by a piece of wire.