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Chapter 6: Q16P (page 335)

Question:What is wrong with the following 鈥減roof鈥� that there are no magnetic fields? By electromagnetic theory,鈭嚶� B = 0, and B =鈭嚸桝. (The error is not in these equations.) Using them, we find

鈭�鈭�鈭�鈭�炉路Bdt=0=鈭�鈭�B路苍诲蟽=鈭�鈭�(鈭�脳A)路苍诲蟽=鈭�A路dr

Since鈭�A路dr=0, A is conservative, or A =鈭囅�. Then B=鈭�脳A=鈭�脳鈭�蠄=0,so B = 0.

Short Answer

Expert verified

鈭�A路dr=0,the integral line (the boundary line of the surface S) is not closed.

Step by step solution

01

Given Information.

The given expression is鈭�鈭�鈭�鈭�炉路Bdt=0=鈭�鈭�B路苍诲蟽.

02

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines it magnitude.

03

Apply Stokes’s theorem.

The problem is with the conclusion after鈭�A路dr=0 as the true reason why the integral equals zero is the surface integral (resulted from the divergence theorem) is closed one with no boundaries. in other words it can be solved as shown below.

鈭�evB路苍诲蟽=鈭�螕A路dr=0as螕=0

Meaning that the last integral cannot surround a volume violating the requirements for Gauss' theorem

Hence,鈭�A路dr=0 the integral line (the boundary line of the surface S) is not closed.

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Most popular questions from this chapter

Evaluate each of the integrals in Problems 3to 8as either a volume integral or a surface integral, whichever is easier.

鈭�r脳苍诲蟽over the entire surface of the cone with base x2+y2鈮�16,z=0 and vertex at(0,0,3) where r=ix+jy+kz

(a) Suppose that a hill (as in Fig. 5.1) has the equation 32-x2-4y2, where z=heightmeasuredfromsomerefrencelevel(in hundreds of feet). Sketch acontour map (that is, draw on one graph a set of curvesz=const.); use the contours z=32,19,12,7,0(b) If you start at the point(3,2)and in the directioni+j, are you going up hillor downhill, and how fast?

Given 蠒=x2-yz and the point (3,4,1) find

(a) 鈭�蠁 at P ;

(b) a unit vector normal to the surface 蠁=5 at P ;

(c) a vector in the direction of most rapid increase of 蠁 at P;

(d) the magnitude of the vector in (c);

(e) the derivative of at in a direction parallel to the liner=i-j+2k+(6i-j-4k)t.

鈭�F.draround the circle over the curved part of the hemisphere in Problem 24, if x2+y2+2x=0, where F=yi-xj.

In the discussion of Figure 3.8, we found for the angular momentum, the formula L=mrx(蝇虫谤).Use (3.9) to expand this triple product. If ris perpendicular to 蝇, show that you obtain the elementary formula, angular momentum =mvr.
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