PHY 1361 Dr. Jie Zou 1

Chapter 30

Sources of the Magnetic Field (Cont.)

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Outline The magnetic force between two

parallel conductors (30.2) Ampère’s Law (30.3) The magnetic field of a solenoid

(30.4) Magnetic flux (30.5) Gauss’s law in magnetism (30.6) Magnetism in matter (30.8) (very

brief discussion)

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The magnetic force between two parallel conductors

Example (problem #17): In the figure shown, the current in the long,

straight wire is I1 = 5.00 A and the wire lies in the plane of the rectangular loop, which carries the current I2 = 10.0 A. The dimensions are c = 0.100 m, a = 0.150 m, and ℓ = 0.450 m. Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire.

What can we learn from this example: The force between parallel conductors is

attractive if the currents are parallel and repulsive if the currents are antiparallel.

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Ampère’s Law Ampère’s Law: The line integral of

Bds around any closed path equals 0I, where I is the total steady current passing through any surface bounded by the closed path.

Ampère’s Law is useful for calculating the magnetic field of current configurations having a high degree of symmetry.

Example (problem #21) The figure shown is a cross-sectional view of

a coaxial cable. In a particular application, the current in the inner conductor is 1.00 A out of the page and the current in the outer conductor is 3.00 A into the page. Determine the magnitude and direction of the magnetic field at points a and b.

Id 0sB

Orange: conductors

Green: rubber

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The magnetic field of a solenoid

A solenoid: a long wire wound in the form of a helix.

An ideal solenoid: when the turns are closely spaced and the length is much greater than the radius of the turns. Properties:

The external field is close to zero. The interior field is uniform and parallel to

the axis: B = 0(N/l)I = 0 nI. n = N/l: the number of turns per unit length.

Example (problem #29): What current is required in the windings of a

long solenoid that has 1000 turns uniformly distributed over a length of 0.400 m, to produce at the center of the solenoid a magnetic field of magnitude 1.00 × 10-4 T?

(a) A tightly wound solenoid of finite length; (b) Cross-sectional view of an ideal solenoid.

(b)(a)

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Magnetic flux General definition:

SI unit: weber (Wb); 1 Wb = 1 Tm2. Special case: a plane of area A

in a uniform field B. B = BA cos ; : the angle between

B and A. (a) If B // the plane, =90°, B=0;

(b) If B the plane, =0°, B=BA (the maximum).

Example (problem #33): A cube has edge length ℓ = 2.50

cm. A uniform magnetic field B = (5i + 4j + 3k) T exists throughout the region. (a) Calculate the flux through the shaded face. (b) What is the total flux through the six faces?

AB dB

(a) (b)

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Gauss’s law in magnetism Gauss’s law in magnetism: the

net magnetic flux through any closed surface is always zero, i.e.

Magnetic fields are continuous and form closed loops. Magnetic field lines do not begin or end at any point.

Electric field lines originate and terminate on electric charges.

Example (problem #35): The hemisphere is in a uniform

magnetic field that makes an angle θ with the vertical. Calculate the magnetic flux through (a) S1 and (b) S2.

0AB d