Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Introduction to Electrodynamics
Found in: Page 55
Introduction to Electrodynamics

Introduction to Electrodynamics

Book edition 4th edition
Author(s) David J. Griffiths
Pages 613 pages
ISBN 9780321856562

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Compute the line integral of

v=(r cos2θ)r^-(r cosθsinθ)θ^+3rϕ^

around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates).Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes' theorem. [Answer: 3rr /2]

The line integral is evaluated to be 3π2 . The left and right side of the Stokes theorem gives same result. Hence strokes theorem is verified.

See the step by step solution

Step by Step Solution

Step 1: Describe the given information

The given vector is, v=(r cos2θ)r^-(r cosθsinθ)θ^+3rϕ^ . The line integral of the given vector has to be evaluated over the path drawn as follows:

Step 2: Define the Stokes theorem

The integral of curl of a function f (x,y,z) over an open surface area is equal to the line integral of the function (×v).ds= b v.dl.

Step 3: Compute the left side of strokes theorem

The formula of curl of a vector in spherical coordinates is

The curl of the vector v=(r cos2θ)r^-(r cosθsinθ)θ^+3rϕ^ is obtained as

×v=1r sinθ(sinθ(3r))θ-rcosθsinθϕr^+1r1sinθr cos2θϕ-(r(3r))rθ^+1r(r(r cosθsinθr-(r cos2θ)θϕ^

=1rsinθ(3r)cosθ r^+1r(-6r)θ^+0=3 cotθ r^-6^θ

The differential elemental area is da=rdrdθ ϕ^ . Substitute role="math" localid="1654664744003" 3cot r^- 6^θ for ×v , into the stokes theorem (×v).dτ=-v.da

role="math" localid="1654665787372" (×v).dτ=0+01 6rdr 0π2 dϕ =6r2210ϕ 0π2 =(3)π2 =3π2...................(1)

Step 4: Compute the right side of strokes theorem 

The differential length vector is given bydl=dr r^+rdθ θ^+r sinθ ϕ^ . Along the path, localid="1654672580270" θ=π2,θ=0 0 and r varies from 0 to 1.Hence the line integral becomes,

localid="1654672141711" v.dl=(r cos2 θr^-(r cosθ sinθ)θ^+3r ϕ^)(dr r^+ θ^+r sinθdϕϕ^) =(r cos2θ)dr-(r2 cosθ sinθ)+3r2sinθdϕ =r cos2π2dr-r2 cosπ2sin 3r2 sinπ23r2sinπ2 =0 dr-0+3r2

Simplify further as

localid="1654672150719" v.dl=3r2 =3r2ϕ =3r20 =0

Along the path (ii), θ=π2 ,r=1, and ϕ varies from 0 tolocalid="1654667564073" π2 .Hence the line integral becomes,

localid="1654672163671" v.dl=((r cos2 θ)r^-(r cosθ sinθ)θ^+3r ϕ^)(dr r^+ θ^+r sinθdϕϕ^) =(r cos2θ)dr-(r2 cosθ sinθ)+3r2sinθdϕ =(1cos2π2)dr-((12) cosπ2sin π23r2 sinπ2)+3r2sin(π2) =3

Simplify further as,

v.dl=0π23 =3(ϕ)0π2 =3π2

Along the path (iii), θvaries from π2 to localid="1654668630697" tan112, localid="1654668661756" ϕ=π2 and localid="1654668647962" r=1sinθ, such that localid="1654668615795" dr=-11sin2θcosθdθ Hence the line integral becomes,

localid="1654672691611" v.dl=((r cos2 θ)r^-(r cosθ sinθ)θ^+3r ϕ^)(dr r^+dr r^+rdθ θ^ r sinθdϕϕ^) =(r cos2θ)dr-(r2 cosθ sinθ)+3r2sinθdϕ =(1sinθcos2θ)(-1sinθcosθdθ)-(1sinθ2cosθsinθ)dθ+0

Simplify further as,

localid="1654670347504" v.dl=-cos3θsin3θ+cosθsinθ =cosθsinθcos2θ+sin2θsin2θ =π2tan-1122cosθsin3θ ...........(2)

Let localid="1654669903166" x=sin θ, then dx=cos2θdθ.Substitute x for localid="1654672592461" sinθ and dx for cos θdθ into equation (2)

v.dl=-v.dl=12sin2θ =12x2

Substitute back for into above result as,

localid="1654672710295" v.dl=12sin2 θ

Evaluate the limit as,

localid="1654672130422" v.dl=12sin2θx2tan-112 =12sin2tan-112-12sin2π2 =12(0.2)-12 =2

Along the path (iv), θ=tan-112,ϕ=π2 and r varies from localid="1654671277385" 5 to 0, Hence the line integral becomes,

localid="1654672112340" v.dl=rcos2θr^-(r cosθsinθ)θ^+3rϕ^)(dr r^+rdθ θ^ +rsinθdϕϕ^ =r cos2tan-112dr =500.8rdr =0.8r2205

Simplify further as,

v.d=0.80-522 =-2

The integral of all the four parts are added to give:

v.dl=0+3π2+2+-2 =3π2 .........(3)

From equation (1) and (3), the left and right side gives same result. Hence strokes theorem is verified.

Most popular questions for Physics Textbooks


Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Physics Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.