Ampere's Law: Superposition of Magnetic Fields

A conducting path (circuit) is created by carving out a portion of an otherwise circular wire (of radius R) and laying the pieces side-by-side to conduct a current I (into the page on the right and out of the page on the left). The centers of the wires (had they remained circular) are displaced by a vector distance d pointing from left to right. Find the magnetic field in the shaded, nonconducting region. Figure 1



QUESTIONS

(Score is number right minus number wrong.)

Stokes' Theorem (Equation (2)) is not true unless the line integral is done over a circular path and, hence, could not be applied to a path around the irregularly-shaped conductor.
True
False
Stokes' Theorem can easily be used to find the magnetic field inside a a circular conductor if the current in the wire is uniform.
True
False
For a single wire with circular cross-section carrying a uniform current, the magnitude of the magnetic field INSIDE the wire can be obtained from Stokes' Theorem in the form,
Equation (3a)
Equation (3b)
Equation (3c)
Equation (3d)
Equation (3e)
The magnetic field inside a single wire with circular cross-section carrying a uniform current I is,
Equation (4a)
Equation (4b)
Equation (4c)
Equation (4d)
Equation (4e)
The distances s and angles phi in Equation (4) must be measured in a cylindrical coordinate system centered on the circular wire.
True
False
The superposition of two fully circular wires carrying current I in opposite directions and imagined positioned as shown in the figure is equivalent to the conducting path created by the hollowed-out wires.
True
False
Because the angles and radial distances must be measured from a system centered on a circular wire, we define two cylindrical coordinate systems, a left (L) and a right (R) system for the two wires. See Equation (5).

If we express the fields of the two wires as vector expressions, we can simply add them together as follows,
Equation (6a)
Equation (6b)
Equation (6c)
Equation (6d)
Equation (6e)
It is true that,
Equation (7a)
Equation (7b)
Equation (7c)
Equation (7d)
In the shaded region, the magnetic field is constant and given correctly by Equation (8)
True
False
EQUATIONS

Equations


Equations









Username



Register BEFORE taking the first Quiz!
(Click here to return to the Table of Contents)