Hints

Ch.30

 2 11 16 17 21 SP1
Answers

Related Lab: B-Plot
Chapter Summary:
(a)

 

 

Use RHR . Wrap r-hat into ds-vector with your right fingers. Thumb points in,  like  the dB-vector.

(b) RHR for long wires.

(c) Force between 2 wires.

(d)  Ampere's Law

Use RHR for long wires under (b) above !
2. Take out your bookmarks, and compare fig. 30.4 in Ex. 30.2 and fig. P30.2. Also use eqn. 30.3  as a reference. The magnetic field contributions due to the straight elements are zero because the ds-vector is parallel to the r-unit vector. That is,

On the other hand, each element along the  arc gives a dB-vector that points into the paper at P. Note that at every point on this arc, the ds-vector is perpendicular to the r-unit vector. Thus,
.
Thus,

Note that in the above expression, one power of r cancels. (See ex. 30.2)
Integrate this expression with respect to the angle over the appropriate range. (What is the range of the angle for this arc?) Also, state the direction of the net B-vector.
11. Study fig. P30.11 This problem is virtually identical to problem 2. That's why I assigned it ! So you can get practice ! So you won't be surprised should such a problem appear on the final exam !! Using the same arguments as in the previous problem, we get that:
.
Integrate this expression with respect to the angle over the appropriate range. (What is the range of the angle for a quarter-circle ?) Also, state the direction of the net B-vector.
16. Please do this problem assuming that the two currents are in the opposite direction (anti-parallel), in contrast to the example from sec. 30.2 in class where the currents were parallel.  In the case of anti-parallel currents, do the wires attract or repel? Review your notes.
17. Study fig. P30.17. Point your right thumb in the direction of the current in the long wire, which is upward in the figure. Note that your right fingers point into the page on the side of the long wire where the rectangular loop is positioned. Thus the B-field due to the long wire is in. Consider the top and bottom segments of the rectangle.
For the top, the L-vector points to the right. Take the cross product,
.
Here, the subscript t represents "top." The B-vector points in. The L-vector points right. Wrapping the L-vector into the B-vector, you see that your thumb points upward. Performing a similar procedure on the bottom segment, your will see that your thumb points down.
Thus, we see that the two forces cancel out.

Now consider the force on the side of the rectangle parallel with and closest to the long wire.
We know that along this side the magnetic field due to the long wire is in and has the magnitude given by,
quiz4b1.jpg (1383 bytes).
Now, evaluate:

Note that the L-vector points up in the same direction as the current. The B-vector points in.
Wrap your right fingers into the B-vector. Your thumb points left, in the direction of the force.
Note that the magnitude of this force is:

Now consider the force on the side of the rectangle parallel with and farthest from the long wire. Along this segment of the rectangle:
.
Now evaluate:

Again the B-vector points in. But now the L-vector points down in the direction of the current.
Wrap your right fingers into the B-vector and you will se that your thumb points right, in the direction of the force. Note again that the magnitude of this force is also given by:
.
But this time the force has a different value of magnitude (and points right.)

Calculate the difference in the magnitudes of the two forces to get the net force. What direction does the net force point ? Left or right ?

21. From Ampere's law, the value of the magnitude of the magnetic field at point A is given by:

Here the current is the net current flowing through the area of the circle of radius 1 mm. The current is 1 A, out of the page. This is the current in the inner conductor. To find the direction of the magnetic field, point your thumb out of the page, in the direction of the current.
Clearly, your right fingers will wrap in the counterclockwise direction. This means that at point A, the B-field points up.

At point B:

.
Here the net current flows through the area of the circle with radius 3 mm. Taking out as positive, clearly the net current points in the negative direction, inward. Find the value of the current by taking the difference of the two currents given in the problem. Now, find the direction of the magnetic field at point B. Your thumb points into the page. Your fingers wrap in the clockwise direction. What is the direction of the B-field at point B, up or down?
SP1. Redo example 30.4 . Assume that the wire radius R = 1.0 m and that the total current = 1.0 A. Assume that the current points out of the page. Calculate the magnitude and direction (CCW or CW) of the B-vector at:

(a) r = 0.5 m

(b) r = 2.0 m.