List of Questions and link to solutions

Chapter 1: Vector Analysis

Problem 1.1: Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams,
show that the dot product and cross product are distributive,
a) when the three vectors are co-planar.

b) in the general case. 

Solution

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Problem 1.2 Is the cross product associative?

$$(\overrightarrow{A}\times \overrightarrow{B})\times \overrightarrow{C}\stackrel{?}{=}\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})$$

If so, prove it; if not, provide a counterexample (the simpler the better). Solution

 

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Problem 1.3 Find the angle between the body diagonals of a cube. Solution

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Problem 1.4 Use the cross product to find the components of the unit vector $\hat{n}$
perpendicular to the shaded plane in Fig. 1.11. Solution 

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Problem 1.5 Prove the BAC-CAB rule by writing out both sides in component form.

Solution

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Problem 1.6 Prove that

$$[\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})]+[\overrightarrow{B}\times (\overrightarrow{C}\times \overrightarrow{A})]+[\overrightarrow{C}\times (\overrightarrow{A}\times \overrightarrow{B})]=0$$
Under what conditions does $\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})=(\overrightarrow{A}\times \overrightarrow{B})\times \overrightarrow{C}$? Solution

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Solution

 

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 Problem 1.8

(a) Prove that the two-dimensional rotation matrix ($\mathbf{E}\mathbf{q}\mathbf{.}\mathbf{1.29}$) preserves dot products.
(That is, show that $\overline{A_y{B}_y}+\overline{A_z{B}_z}=A_y{B}_y+A_z{B}_z$ .)
(b) What constraints must the elements ($R_{ij}$ ) of the three-dimensional rotation matrix
($\mathbf{E}\mathbf{q}\mathbf{.}\mathbf{1.30}$) satisfy, in order to preserve the length of $\mathbf{A}$ (for all vectors $\overrightarrow{A}$)?

 Solution

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Problem 1.11 Find the gradients of the following functions:

(a) $f(x,y,z)=x^2+y^3+z^4$

(a) $f(x,y,z)=x^2{y}^3{z}^4$

(a) $f(x,y,z)=e^x\sin (y)\ln (z)$ Solution

 

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Problem 1.12 The height of a certain hill (in feet) is given by
$$h(x,y)=10(2xy-3x^2-4y^2-18x+28y+12),$$
where $y$ is the distance (in miles) north, $x$ the distance east of South Hadley.
(a) Where is the top of the hill located?
(b) How high is the hill?
(c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile
east of South Hadley? In what direction is the slope steepest, at that point?

Solution 

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Solution

 

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Problem 1.15 Calculate the divergence of the following vector functions:

(a) $v_a=x^2\hat{i}+3x{z}^2\hat{j}-2xz\hat{k}$

(b) $v_b=xy\hat{i}+2yz\hat{j}+3zx\hat{k}$

(c) $v_c=y^2\hat{i}+(2xy+z^2)\hat{j}+2yz\hat{k}$

 Solution

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Problem 1.16 Sketch the vector function

$$\overrightarrow{v}=\frac{\hat{r}}{r^2}$$

and compute its divergence. The answer may surprise you.. can you explain it?

 Solution

 

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Calculate CURL of vector in different coordinate system? Article

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Problem 1.18 Calculate the CURL of the following vector functions:

(a) $v_a=x^2\hat{i}+3x{z}^2\hat{j}-2xz\hat{k}$

(b) $v_b=xy\hat{i}+2yz\hat{j}+3zx\hat{k}$

(c) $v_c=y^2\hat{i}+(2xy+z^2)\hat{j}+2yz\hat{k}$

 Solution

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Problem 1.27: Prove that the divergence of a curl is always zero. Check it for function
$v_a$ in Prob. 1.15. ($v_a=x^2\hat{x}+3x{z}^2\hat{y}-2xz\hat{z}$). Solution 

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 Problem 1.54: Check the divergence theorem for the function
$$v=r^2\cos \theta \hat{r}+r^2\cos \varphi \hat{\theta }-r^2\cos \theta \sin \varphi \hat{\varphi }$$ ,
using as your volume one octant of the sphere of radius $R$ (Fig. 1.48). Make sure
you include the entire surface. Solution 

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 Problem 1.55: Check Stokes’ theorem using the function $v=ay\overrightarrow{x}+bx\overrightarrow{y}$ ($a$ and $b$ are constants) and the circular path of radius $R$, centered at the origin in the $xy$ plane. Solution

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 Problem 1.56: Compute the line integral of
$$v=6\hat{x}+y{z}^2\hat{y}+(3y+z)\hat{z}$$
along the triangular path shown in Fig. 1.49. Check your answer using Stokes’
theorem.Solution

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 Chapter 2: Electrostatics 

Problem 2.10: A charge $q$ sits at the back corner of a cube, as shown in Fig. 2.17.
What is the flux of E through the shaded side? Solution

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 Problem 2.11: Use Gauss’s law to find the electric field inside and outside a spherical
shell of radius R that carries a uniform surface charge density σ. Compare your
answer to Prob. 2.7. Solution

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 Problem 2.12 Use Gauss’s law to find the electric field inside a uniformly charged
solid sphere (charge density $\rho$). Compare your answer to Prob. 2.8. Solution

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 Problem 2.13: Find the electric field a distance s from an infinitely long straight
wire that carries a uniform line charge $\lambda$. Compare Eq. 2.9. Solution

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 Problem 2.14 Find the electric field inside a sphere that carries a charge density proportional to the distance from the origin, $\rho =kr$, for some constant $k$. [Hint: This charge density is not uniform, and you must integrate to get the enclosed charge.] Solution

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 Problem 2.38: A metal sphere of radius $R$, carrying charge $q$, is surrounded by a
thick concentric metal shell (inner radius $a$, outer radius $b$, as in Fig. 2.48). The
shell carries no net charge.
(a) Find the surface charge density $\sigma$ at $R$, at $a$, and at $b$.
(b) Find the potential at the center, using infinity as the reference point.
(c) Now the outer surface is touched to a grounding wire, which drains off charge
and lowers its potential to zero (same as at infinity). How do your answers to
(a) and (b) change? Solution

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