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

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IB

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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

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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:

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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

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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

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IB

IrBdlB

Rr

encl

2

2

0

0

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2

2

0

0

2

2

2

2

R

IrB

R

rIrB

IdlB

R

rII

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encl

encl

Magnetic Field Inside & Outside a Current Carrying Conductor

20

2 R

IrB

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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

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NIB

20