1. Chapter 28--Examples

2. Problem
Two long, parallel transmission lines, 40 cm apart, carry 25 A and 75 A currents. Find all locations where the net magnetic field of these wires is zero if these currents are a) in the same direction and b) in opposite directions

3. Parallel Directions
Based on the figure, the only place that the fields can be zero is between wires.
Assume that the zero point is a distance x from the 75 A wire. So now set B75=B25
O O O
X X X
75A x

4. Opposite Directions Directions
Based on the figure, there are two regions outside the wires.
Since one current is smaller than the other, we must choose the region closest to the smaller current.
Assume that the zero point is a distance x from the 25 A wire. So now set B75=B25 x
O O O
X X X
X X X
O O O
25A
75A

5. Problem
Consider the figure below. The curved segments are arcs of circles of radii a and b. The straight segments are along radii. Find the magnetic field at point P (the center of the curvature) assuming i as the current. q a b i

6. Biot-Savart
Based on Biot-Savart, the current segments along the radial vector do not contribute since they are parallel with r.
Now we must integrate across each arc. The integration variable is q.
ds=rd q q a b i

7. Direction
By the RH Rule, the current from b is out of the page at Point P.
The current from a is into the page.
We will choose out of the page as positive and the magnetic field at P is q a b i

8. Problem
Four long copper wires are parallel to each other, their cross-sections forming the corners of a square with sides a=20 cm. A 20 A current exists in each wire in the direction shown in the figure below. What are the magnitude and direction of B at the center of the square. X a

9. RH Rule X a
The lower right and upper left corners each point to the upper right.
The fields from the lower left and upper right point to the upper left.
Thus the total field point up the page

10. The Magnitude The distance to the center from any corner is a/sqrt(2).
All of the horizontal components have added out. Only the four vertical components sum to

11. Problem
A solid conductor with radius a is supported by insulating material on the axis of a conducting tube with inner radius b and outer radius c (see figure below). The central conductor and the outer conductor carry an equal current i in opposite directions. However, the magnitude of current density in the inner conductor is expressed by J=kr where k is an appropriate constant.
Derive an expression for the magnetic field at all points (r<a, a<r<b, b<r<c, r>c)
Derive an expression for current density in the outer conductor.
Show that each boundary is matched.

12. First, r<=a At r=a, total current is i so i=J*A
J=ka and A=pa2 so i=kpa3 or k=i/pa3
Using RH rule, with our thumb pointing in direction of current, our fingers are matching direction of integration so this is a positive direction

13. Next, a<=r<b

14. b<=r<=c
J’=-i/A
A=(pc2)-(pb2)
J’=-i/[(pc2)-(pb2)]

15. r>c
Recall for a coaxial cable with equal static charges, E=0 outside as well