Instructor's Solution Manual for Introduction to Electrodynamics 4th Ed by Griffiths

Instructor’s Solution Manual
Introduction to Electrodynamics
Fourth Edition
David J. Griffiths
2014

2
Contents
1 Vector Analysis 4
2 Electrostatics 26
3 Potential 53
4 Electric Fields in Matter 92
5 Magnetostatics 110
6 Magnetic Fields in Matter 133
7 Electrodynamics 145
8 Conservation Laws 168
9 Electromagnetic Waves 185
10 Potentials and Fields 210
11 Radiation 231
12 Electrodynamics and Relativity 262
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2012 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is
protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.

3
Preface
Although I wrote these solutions, much of the typesetting was done by Jonah Gollub, Christopher Lee, and
James Terwilliger (any mistakes are, of course, entirely their fault). Chris also did many of the figures, and I
would like to thank him particularly for all his help. If you find errors, please let me know (griffi[email protected]).
David Griffiths
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2012 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is
protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.

4CHAPTER 1. VECTOR ANALYSIS
Chapter 1
Vector Analysis
Problem 1.1
CHAPTER 1. VECTOR ANALYSIS 3
Chapter 1
Vector Analysis
Problem 1.1
✲
✒
✯
✣
A
B
C
B+C
! "# $
|B|cos θ1!"# $ |C|cos θ2
}
|B|sin θ1
}
|C|sin θ2
θ1
θ2
θ3
(a) From the diagram, |B+C|cos θ3=|B|cos θ1+|C|cos θ2.
|A||B+C|cos θ3=|A||B|cos θ1+|A||C|cos θ2.
So: A·(B+C) = A·B+A·C. (Dot product is distributive)
Similarly: |B+C|sin θ3=|B|sin θ1+|C|sin θ2. Mulitply by |A|ˆn.
|A||B+C|sin θ3ˆn =|A||B|sin θ1ˆn +|A||C|sin θ2ˆn.
If ˆn is the unit vector pointing out of the page, it follows that
A×(B+C) = (A×B) + (A×C). (Cross product is distributive)
(b) For the general case, see G. E. Hay’s Vector and Tensor Analysis, Chapter 1, Section 7 (dot product) and
Section 8 (cross product)
Problem 1.2
✲A=B
✻
C
❂
B×C❄A×(B×C)
The triple cross-product is not in general associative. For example,
suppose A=Band Cis perpendicular to A, as in the diagram.
Then (B×C) points out-of-the-page, and A×(B×C) points down,
and has magnitude ABC. But (A×B) = 0, so (A×B)×C= 0 ̸=
A×(B×C).
Problem 1.3
✲y
✻
z
✰
x
✣B
❲
A
θ
A= +1 ˆx + 1 ˆy −1ˆz;A=√3; B= 1 ˆx + 1 ˆy + 1 ˆz;
A·B= +1 + 1 −1 = 1 = AB cos θ=√3√3 cos θ⇒cos θ.
θ= cos−1%1
3&≈70.5288◦
Problem 1.4
The cross-product of any two vectors in the plane will give a vector perpendicular to the plane. For example,
we might pick the base (A) and the left side (B):
c
⃝2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is
protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.
(a) From the diagram, |B+C|cos ✓3=|B|cos ✓1+|C|cos ✓2. Multiply by |A|.
|A||B+C|cos ✓3=|A||B|cos ✓1+|A||C|cos ✓2.
So: A·(B+C)=A·B+A·C. (Dot product is distributive)
Similarly: |B+C|sin ✓3=|B|sin ✓1+|C|sin ✓2. Mulitply by |A|ˆn .
|A||B+C|sin ✓3ˆn =|A||B|sin ✓1ˆn +|A||C|sin ✓2ˆn .
If ˆn is the unit vector pointing out of the page, it follows that
A⇥(B+C)=(A⇥B)+(A⇥C). (Cross product is distributive)
(b) For the general case, see G. E. Hay’s Vector and Tensor Analysis, Chapter 1, Section 7 (dot product) and
Section 8 (cross product)
Problem 1.2
CHAPTER 1. VECTOR ANALYSIS 3
Chapter 1
Vector Analysis
Problem 1.1
✲
✒
✯
✣
A
B
C
B+C
! "# $
|B|cos θ1! "# $ |C|cos θ2
}
|B|sin θ1
}
|C|sin θ2
θ1
θ2
θ3
(a) From the diagram, |B+C|cos θ3=|B|cos θ1+|C|cos θ2. Multiply by |A|.
|A||B+C|cos θ3=|A||B|cos θ1+|A||C|cos θ2.
So: A·(B+C) = A·B+A·C. (Dot product is distributive)
Similarly: |B+C|sin θ3=|B|sin θ1+|C|sin θ2. Mulitply by |A|ˆn.
|A||B+C|sin θ3ˆn =|A||B|sin θ1ˆn +|A||C|sin θ2ˆn.
If ˆn is the unit vector pointing out of the page, it follows that
A×(B+C) = (A×B) + (A×C). (Cross product is distributive)
(b) For the general case, see G. E. Hay’s Vector and Tensor Analysis, Chapter 1, Section 7 (dot product) and
Section 8 (cross product)
Problem 1.2
✲A=B
✻
C
❂
B×C❄A×(B×C)
The triple cross-product is not in general associative. For example,
suppose A=Band Cis perpendicular to A, as in the diagram.
Then (B×C) points out-of-the-page, and A×(B×C) points down,
and has magnitude ABC. But (A×B) = 0, so (A×B)×C= 0 ̸=
A×(B×C).
Problem 1.3
✲y
✻
z
✰
x
✣B
❲
A
θ
A= +1 ˆx + 1 ˆy −1ˆz;A=√3; B= 1 ˆx + 1 ˆy + 1 ˆz;B=√3.
A·B= +1 + 1 −1 = 1 = AB cos θ=√3√3 cos θ⇒cos θ=1
3.
θ= cos−1%1
3&≈70.5288◦
Problem 1.4
The cross-product of any two vectors in the plane will give a vector perpendicular to the plane. For example,
we might pick the base (A) and the left side (B):
A=−1ˆx + 2 ˆy + 0 ˆz;B=−1ˆx + 0 ˆy + 3 ˆz.
c
⃝2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is
protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.
The triple cross-product is not in general associative. For example,
suppose A=Band Cis perpendicular to A, as in the diagram.
Then (B⇥C) points out-of-the-page, and A⇥(B⇥C) points down,
and has magnitude ABC. But (A⇥B)=0, so (A⇥B)⇥C=06=
A⇥(B⇥C).
Problem 1.3
CHAPTER 1. VECTOR ANALYSIS 3
Chapter 1
Vector Analysis
Problem 1.1
✲
✒
✯
✣
A
B
C
B+C
! "# $
|B|cos θ1!"# $ |C|cos θ2
}
|B|sin θ1
}
|C|sin θ2
θ1
θ2
θ3
(a) From the diagram, |B+C|cos θ3=|B|cos θ1+|C|cos θ2.
|A||B+C|cos θ3=|A||B|cos θ1+|A||C|cos θ2.
So: A·(B+C) = A·B+A·C. (Dot product is distributive)
Similarly: |B+C|sin θ3=|B|sin θ1+|C|sin θ2. Mulitply by |A|ˆn.
|A||B+C|sin θ3ˆn =|A||B|sin θ1ˆn +|A||C|sin θ2ˆn.
If ˆn is the unit vector pointing out of the page, it follows that
A×(B+C) = (A×B) + (A×C). (Cross product is distributive)
(b) For the general case, see G. E. Hay’s Vector and Tensor Analysis, Chapter 1, Section 7 (dot product) and
Section 8 (cross product)
Problem 1.2
✲A=B
✻
C
❂
B×C❄A×(B×C)
The triple cross-product is not in general associative. For example,
suppose A=Band Cis perpendicular to A, as in the diagram.
Then (B×C) points out-of-the-page, and A×(B×C) points down,
and has magnitude ABC. But (A×B) = 0, so (A×B)×C= 0 ̸=
A×(B×C).
Problem 1.3
✲y
✻
z
✰
x
✣B
❲
A
θ
A= +1 ˆx + 1 ˆy −1ˆz;A=√3; B= 1 ˆx + 1 ˆy + 1 ˆz;
A·B= +1 + 1 −1 = 1 = AB cos θ=√3√3 cos θ⇒cos θ.
θ= cos−1%1
3&≈70.5288◦
Problem 1.4
The cross-product of any two vectors in the plane will give a vector perpendicular to the plane. For example,
we might pick the base (A) and the left side (B):
c
⃝2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is
protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.
A=+1ˆx +1ˆy 1ˆz ;A=p3; B=1ˆx +1ˆy +1ˆz ;B=p3.
A·B= +1 + 1 1=1=AB cos ✓=p3p3 cos ✓)cos ✓=1
3.
✓= cos11
3⇡70.5288
Problem 1.4
The cross-product of any two vectors in the plane will give a vector perpendicular to the plane. For example,
we might pick the base (A) and the left side (B):
A=1ˆx +2ˆy +0ˆz ;B=1ˆx +0ˆy +3ˆz .
c
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protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.

CHAPTER 1. VECTOR ANALYSIS 5
A⇥B=
ˆx ˆy ˆz
120
103
=6ˆx +3ˆy +2ˆz .
This has the right direction, but the wrong magnitude. To make a unit vector out of it, simply divide by its
length:
|A⇥B|=p36 + 9 + 4 = 7. ˆn =A⇥B
|A⇥B|=6
7ˆx +3 7ˆy +2 7ˆz .
Problem 1.5
A⇥(B⇥C)=
ˆx ˆy ˆz
AxAyAz
(ByCzBzCy)(BzCxBxCz)(BxCyByCx)
=ˆx [Ay(BxCyByCx)Az(BzCxBxCz)] + ˆy () + ˆz ()
(I’ll just check the x-component; the others go the same way)
=ˆx (AyBxCyAyByCxAzBzCx+AzBxCz)+ˆy () + ˆz ().
B(A·C)C(A·B)=[Bx(AxCx+AyCy+AzCz)Cx(AxBx+AyBy+AzBz)] ˆx + () ˆy + () ˆz
=ˆx (AyBxCy+AzBxCzAyByCxAzBzCx)+ˆy () + ˆz (). They agree.
Problem 1.6
A⇥(B⇥C)+B⇥(C⇥A)+C⇥(A⇥B)=B(A·C)C(A·B)+C(A·B)A(C·B)+A(B·C)B(C·A)=0.
So: A⇥(B⇥C)(A⇥B)⇥C=B⇥(C⇥A)=A(B·C)C(A·B).
If this is zero, then either Ais parallel to C(including the case in which they point in opposite directions, or
one is zero), or else B·C=B·A= 0, in which case Bis perpendicular to Aand C(including the case B=0.)
Conclusion: A⇥(B⇥C)=(A⇥B)⇥C() either Ais parallel to C, or Bis perpendicular to Aand C.
Problem 1.7
r
= (4 ˆx +6ˆy +8ˆz )(2 ˆx +8ˆy +7ˆz )= 2ˆx 2ˆy +ˆz
r
=p4+4+1= 3
ˆ
r
=
r
r
=2
3ˆx 2 3ˆy +1 3ˆz
Problem 1.8
(a) ¯ Ay¯ By+¯ Az¯ Bz= (cos Ay+ sin Az)(cos By+ sin Bz)+(sin Ay+ cos Az)(sin By+ cos Bz)
= cos2AyBy+sincos (AyBz+AzBy) + sin2AzBz+sin
2AyBysin cos (AyBz+AzBy)+
cos2AzBz
= (cos2+ sin2)AyBy+ (sin2+ cos2)AzBz=AyBy+AzBz.X
(b) (Ax)2+(Ay)2+(Az)2=⌃
3
i=1AiAi=⌃
3
i=1 ⌃3 j=1Rij Aj⌃3
k=1RikAk=⌃
j,k (⌃iRij Rik)AjAk.
This equals A2
x+A2 y+A2 zprovided ⌃3
i=1Rij Rik =1if j =k
0if j 6=k
Moreover, if Ris to preserve lengths for all vectors A, then this condition is not only sufficient but also
necessary. For suppose A= (1,0,0). Then ⌃j,k (⌃iRij Rik )AjAk=⌃
iRi1Ri1, and this must equal 1 (since we
want A2
x+A2 y+A2 z= 1). Likewise, ⌃3
i=1Ri2Ri2=⌃
3
i=1Ri3Ri3= 1. To check the case j6=k, choose A=(1,1,0).
Then we want 2 = ⌃j,k (⌃iRij Rik)AjAk=⌃
iRi1Ri1+⌃ iRi2Ri2+⌃ iRi1Ri2+⌃ iRi2Ri1. But we already
know that the first two sums are both 1; the third and fourth are equal,so⌃iRi1Ri2=⌃
iRi2Ri1= 0, and so
on for other unequal combinations of j,k.XIn matrix notation: ˜ RR = 1, where ˜ Ris the transpose of R.
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2012 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is
protected under all copyright laws as they currently exist. No portion of this material may be
reproduced, in any form or by any means, without permission in writing from the publisher.