unit 4 day 8 – ampere’s law & magnetic fields thru solenoids & toroids
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Unit 4 Day 8 – Ampere’s Law & Magnetic Fields thru Solenoids & Toroids
• Definition of Current
• Ampere’s Law
• Magnetic Field Inside & Outside a Current Carrying Conductor
• Magnetic Field of a Solenoid
• Magnetic Field of a Toroid
Definition of Current• The unit of current, Ampere, is defined in terms of the
magnetic field it produces
μ0 was originally measured experimentally
• To define μ0, a standard was created using two parallel wires, each with a current of I = 1.0 A, separated by a distance d = 1.0 m
AmTwhere
r
IB
70
0
104
2
Definition of Current
• The force between the wires per unit length is:
using μ0 = 4π x 10-7 T·m/A exactly
• Therefore, 1A, by definition, is the current flowing in each of 2 long parallel wires, resulting in a magnetic force of 2.0 x 10-7N/m
• Then, 1C = 1A·s, and the values of k & ε0 were then obtained experimentally
mN
d
II
l
F 7210 100.22
Ampere’s Law• Remembering that the magnetic field in a long, straight
current carrying conductor is:
• This equation is only valid for long straight wires. In general the relationship between current in a wire of any shape, and its magnetic field around it was derived by Andre Marie Ampere.
• For any arbitrary closed path around a current enclosed by the area of the closed path:
r
IB
20
enclIldB 0
Ampere’s Law
• Where the integrand is taken around any closed loop, and Iencl is the current passing through the area enclosed by the closed path
• For a straight conductor:
enclIldB 0
r
IB
rBdlBdlBI
2
2
0
0
Magnetic Field Inside & Outside a Current Carrying Conductor
• Outside the conductor, the magnetic field is an inverse law:
• Inside the conductor, the magnetic field is linear because the current is uniformly distributed
r
IB
IrBdlB
Rr
encl
2
2
0
0
20
2
2
0
0
2
2
2
2
R
IrB
R
rIrB
IdlB
R
rII
Rr
encl
encl
Magnetic Field Inside & Outside a Current Carrying Conductor
20
2 R
IrB
r
IB
20
Magnetic Field of a Solenoid• A solenoid is a long coil of wire made
of many (N) loops, each producing a magnetic field
• Inside the solenoid, the magnetic field is parallel to the long axis
• Outside the solenoid, the magnetic field is zero
• The magnetic field on-axis is:
l
NIB 0
Magnetic Field of a Toroid• The magnetic field is confined to
being inside the ring only
• The magnetic field is not uniformly distributed inside the ring; it is largest along the inner edge of the ring, and smallest at the outer edge of the ring
• Outside the ring the magnetic field is zero
• The magnetic field is all inside the coil, made of N loops of wire
r
NIB
20
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