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Chapter 10: Q12 P (page 517)

F=q(E+v×B)

Short Answer

Expert verified

B is an axial vector.

Step by step solution

01

Given information.

Physics definitions are given.

02

Definition of Biot-Savart law.

Define the Biot Savart law.

B(r)=μ04π∫ΩJ(r')×(r-r')|r-r'|3d3r'

03

Write about the nature of various vectors.

Write about the nature of various vectors.

F, E are polar vectors. Therefore,V×Bis also a polar vector.

It is known that the vector product of a polar vector and an axial vector results in a polar vector. It is known that velocity is a polar vector. This implies that B is an axial vector.

04

Another method. Begin with the definition of the Biot Savart Law.

Define the Biot Savart Law.

B(r)=μ04π∫ΩJ(r')×(r-r')|r-r'|3d3r'

Since J is a polar vector in this equation, write the conclusion.

J(r')×(r-r')

Due to the reasoning explained above, the equation above is an axial vector.

05

Write the final conclusion.

Since, the above vector is an axial vector, B is an axial vector.

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Most popular questions from this chapter

Write and prove in tensor notation:

(a) Chapter 6, Problem 3.13.

(b) Chapter 6, Problem 3.14.

(c) Lagrange’s identity:(A×B)·(C×D)=(A·C)(B·D)-(A·D)(B·C).

(d), role="math" localid="1659335462905" (A×B)(C×D)=(ABD)C-(ABC)Dwhere the symbol(xyz) means the triple scalar product of the three vectors.

Write the equations in (2.16) and so in (2.17) solved for the unprimed components in terms of the primed components.

In equation(10.13),let the variables be rectangular coordinates x, y, z, and let x1,x2,x3 , be general curvilinear coordinates, orthogonal or not (see end of Section 8 ). Show that JTJ is the gijmatrix in(8.13) [or in (8.16) for an orthogonal system]. Thus show that the volume element in a general coordinate system is dV=gdx1dx2dx3 whereg=detgij , and that for an orthogonal system, this becomes [by (8.16) or (10.19) ], dV=h1h2h3dx1dx2dx3. Hint: To evaluate the products of partial derivatives in JTJ, observe that the same expressions arise as in finding ds2 . In fact, from (8.11) and (8.12) , you can show that row i times column j in JTJ is just ai.aj=gij in equations (8.11) to (8.14) .

Show that the transformation equation for a 2nd-rank Cartesian tensor is equivalent to a similarity transformation. Warning hint: Note that the matrix C in Chapter 3 , Section 11 , is the inverse of the matrix A we are using in Chapter 10 (comparer'=Arandr=Cr'). Thus a similarity transformation of the matrix T with tensor componentsTij isT'=ATA-1. Also, see “Tensors and Matrices” in Section 3 and remember that A is orthogonal.

Show that, in polar coordinates, theθcontravariant component of dsis which is unitless, the dθphysical component of ds is rdθwhich has units of length, and theθcovariant component of ds isr2dθwhich has units role="math" localid="1659265070715" (length)2 .

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