Question: Two long straight thin wires with current lie against an equally long plastic cylinder, at radius from the cylinder’s central axis.
Figure 29-58a shows, in cross section, the cylinder and wire 1 but not wire 2. With wire 2 fixed in place, wire 1 is moved around the cylinder, from angle localid="1663154367897" to angle localid="1663154390159" , through the first and second quadrants of the xy coordinate system. The net magnetic field at the center of the cylinder is measured as a function of . Figure 29-58b gives the x component of that field as a function of (the vertical scale is set by ), and Fig. 29-58c gives the y component (the vertical scale is set by ). (a) At what angle is wire 2 located? What are the (b) size and (c) direction (into or out of the page) of the current in wire 1 and the (d) size and (e) direction of the current in wire 2?
a) The angle at which wire 2 is located is either .
b) The size of the current in wire 1 = 4.0 A.
c) The direction of the current in wire 1= out of the page
d) The size of the current in wire 2 = 2.0 A
e) The direction of the current in wire 2 is into the page.
The distance between the center of the cylinder and the wires is .
The wire 2 is fixed in place.
The magnetic field at a point due to a current carrying wire is determined using the Biot-Savart law. The direction of the magnetic field is decided by the right-hand rule. The net magnetic field at the center of the cylinder is the vector sum of the magnetic fields due to wire 1 and wire 2. Using this, we can answer the above questions.
The information about the net magnetic field at the center of the cylinder is given in figures (b) and (c). Using this, we can write the equations as
In the component form,
Using the values form the graph, we write
From these equations, we can find
Hence the wire 2 must be along the y axis direction, i.e., its position must be at or (i.e., )
To avoid the collision of two wires, wire 2 must be at either .
Hence, the angle at which wire 2 is located is either .
Hence, the size of the current in wire 1 = 4.0 A.
The y component of the magnetic field is positive when wire 1 is at . This is possible only if the current in the wire 1 is out of the page. All the other equations are also valid for this direction of the current.
Hence, the direction of the current in wire 1= out of the page
Hence, the size of the current in wire 2 = 2.0 A.
The x component of the current in wire 2 is positive. Hence as seen in part (a), if the wire is at , the current must be going into the page
Hence, the direction of the current in wire 2 is into the page.
A long, hollow, cylindrical conductor (with inner radius 2.0mm and outer radius 4.0mm) carries a current of 24A distributed uniformly across its cross section. A long thin wire that is coaxial with the cylinder carries a current of 24A in the opposite direction. What is the magnitude of the magnetic field (a) 1.0mm,(b) 3.0mm, and (c) 5.0mm from the central axis of the wire and cylinder?
Figure 29-82 shows, in cross section, two long parallel wires spaced by distance d=10.0cm; each carries 100A, out of the page in wire 1. Point P is on a perpendicular bisector of the line connecting the wires. In unit-vector notation, what is the net magnetic field at P if the current in wire 2 is (a) out of the page and (b) into the page?
Figure 29-87 shows a cross section of a hollow cylindrical conductor of radii a and b, carrying a uniformly distributed current i. (a) Show that the magnetic field magnitude B(r) for the radial distance r in the range is given by
(b) Show that when r = a, this equation gives the magnetic field magnitude B at the surface of a long straight wire carrying current i; when r = b, it gives zero magnetic field; and when b = 0, it gives the magnetic field inside a solid conductor of radius a carrying current i. (c) Assume that a = 2.0 cm, b = 1,8 cm, and i = 100 A, and then plot B(r) for the range .
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