Ampère’s Law

Copyright © 2009 Pearson Education, Inc.

Ampère’s Law


The integral is taken around the outside edge of the closed loop.

Ampère’s Law relates the magnetic field B around a closed loop to the total current Iencl flowing through the loop:


so B = (μ0I)/(2πr), as before.

Example: Use Ampère’s Law to find the field around a long straight wire. Use a circular path with the wire at the center; then B is tangent to dl at every point. The integral then gives


A long, straight cylindrical wire conductor of radius R carries a current I of uniform current density in the conductor. Calculate the magnetic field due to this current at (a) points outside the conductor (r > R) (b) points inside the conductor (r < R).

Assume that r, the radial distance from the axis, is much less than the length of the wire. (c) If R = 2.0 mm & I = 60 A, Calculate B at r = 1.0 mm, r = 2.0 mm, & r = 3.0 mm.

Example: Field Inside & Outside a Current Carrying Wire


Field Due to a Long Straight WireFrom Ampere’s Law

Outside of the wire, r > R

so B = (μ0I)/(2πr), as before.


Inside the wire, r < R

Here, we need Iencl, the current inside the Amperian loop. The current is uniform, so

Iencl = (r2/R2)I

B(2r) = μ0(r2/R2)I or

B = (μ0Ir)/(2R2)


Field Due to a Long Straight Wire: Summary

The field is proportional to r inside the wire.

B = (μ0Ir)/(2R2)

The field varies as 1/r outside the wire.

B = (μ0I)/(2πr)


A Coaxial Cable is a single wire surrounded by a cylindrical metallic braid. The 2 conductors are separated by an insulator. The central wire carries current to the other end of the cable, & the outer braid carries the return current & is usually considered ground. Calculate the magnetic field (a) in the space between the conductors, (b) outside the cable.

Conceptual Example: Coaxial Cable.


Solving Problems Using Ampère’s Law• Ampère’s Law is most useful for solving

problems when there is considerable Symmetry.

Identify the Symmetry.• Choose an Integration Path that reflectsthe Symmetry (typically, the best path is along lineswhere the field is constant & perpendicular to the fieldwhere it is changing).

• Use the Symmetry to determine the direction ofthe field.

Calculate the Enclosed Current.


Magnetic Field of a Solenoid & a ToroidSolenoid A coil of wire with many loops. To find the field inside, use Ampère’s Law along the closed path in the figure.

B = 0 outside the solenoid, & the path integral is zero along the vertical lines, so the field is (n = number of loops per unit length):


Example: Field Inside a SolenoidA thin ℓ = 10 cm long solenoid has a total of N = 400 turns (n = N/ℓ) of wire & carries a current I = 2.0 A.

Calculate the magnetic field inside near the center.

A Toroid is similar to a solenoid, but it is bent into the shape of a circle as shown.

Example: ToroidUse Ampère’s Law to calculate the magnetic field

(a) inside &(b) outside a toroid.

Ampère’s Law

Documents

Transcript of Ampère’s Law