Chapter 1, Vector Operators Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

(1.1) $\#$ Show that the angle between $\boldsymbol{A}=2 \hat{\boldsymbol{x}}+3 \hat{\mathbf{y}}+\hat{z}$ and $\boldsymbol{B}=\hat{\boldsymbol{x}}-6 \hat{\mathbf{y}}+\hat{z}$ is $130.6^{\circ}$

Ashly Sunny

Numerade Educator

02:18

(1.1) (a) Show that $(\boldsymbol{A} \times \boldsymbol{B}) \cdot \boldsymbol{C}$ is the volume of the parallelepiped whose edges are $\boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C}$, when the vectors start from the same point.
(b) Show that $(\boldsymbol{A} \times \boldsymbol{C}) \cdot \boldsymbol{B}=-(\boldsymbol{A} \times \boldsymbol{B}) \cdot \boldsymbol{C}$. Observe how the sign changes when the cyclic order of the vectors changes.

Anthony Ramos

Numerade Educator

04:24

Problem 3

(1.1) Let $C$ be a plane closed curve. Prove that the area $\mathscr{A}$ enclosed by $C$ is given by
$$
\mathscr{A}=\frac{1}{2} \oint_{C} \boldsymbol{r} \times \boldsymbol{d} l
$$
where the vector $r$ goes from an arbitrary origin to the element $d l$ on the curve and where the positive directions for $\mathscr{A}$ and for $d l$ obey the right-hand screw rule. You can prove this as follows.
(a) The origin is at $O$, in the plane, and inside $C$. Show that the equation is valid.
(b) The origin is at $O^{\prime}$, again in the plane, but outside $C$. Show that the equation is still valid.
(c) The origin is at $O^{\prime \prime}$, at some point outside the plane. Show that the integral is again valid.

Carson Merrill

Numerade Educator

01:39

Problem 4

(1.2) The vector $r$ points from $P^{\prime}\left(x^{\prime}, y^{\prime}, z^{\prime}\right)$ to $P(x, y, z)$.
(a) Show that if $P$ is fixed and $P^{\prime}$ is allowed to move, then $\boldsymbol{V}^{\prime}(1 / r)=$ $\hat{\boldsymbol{r}} / r^{2}$, where $\hat{\boldsymbol{r}}$ is the unit vector along $\boldsymbol{r}$.
(b) Show that, similarly, if $P^{\prime}$ is fixed and $P$ is allowed to move, then $\boldsymbol{\nabla}(1 / r)=-\hat{\boldsymbol{r}} / r^{2}$

Manik Pulyani

Numerade Educator

01:34

Problem 5

(1.6) (a) Show that $\boldsymbol{\nabla} \cdot \boldsymbol{r}=3$.
(b) What is the flux of $r$ through a spherical surface of radius $a$ ?

Prabhu Ramji

Numerade Educator

09:57

Problem 6

(1.6) Show that
$$
\int_{v} \boldsymbol{\nabla} f d v=\int_{\omega} f \boldsymbol{d} \mathscr{A}
$$
where $\mathscr{A}$ is the area of the closed surface bounding the volume $v .$ You can prove this by multiplying both sides by $c$, where $c$ is any vector independent of the coordinates. Then use Identity 3 (from inside the front cover) and the divergence theorem.

Chris Trentman

Numerade Educator

02:13

Problem 7

$(1.8)$ Since $\boldsymbol{A} \times \boldsymbol{B}$ is normal to $\boldsymbol{B}$, it seems, offhand, that $\boldsymbol{\nabla} \times \boldsymbol{B}$ must be normal to $\boldsymbol{B}$. That is wrong.
As a counterexample, show that $(\boldsymbol{\nabla} \times \boldsymbol{B}) \cdot \boldsymbol{B}=-1$ if $\boldsymbol{B}=y \hat{\boldsymbol{x}}+\hat{z}$.

Zhumagali Shomanov

Numerade Educator

02:34

Problem 8

(1.11.1) (a) Check, by inspection of Fig. $1-10$, that the unit vectors in Cartesian and cylindrical coordinates are related as follows:
$$
\hat{\boldsymbol{\rho}}=\cos \phi \hat{\boldsymbol{x}}+\sin \phi \hat{\boldsymbol{y}}, \quad \hat{\boldsymbol{\phi}}=-\sin \phi \hat{\boldsymbol{x}}+\cos \phi \hat{\mathbf{y}}, \quad \hat{z}=\hat{\boldsymbol{z}}
$$
(b) Deduce from this set of equations that
$$
\hat{\boldsymbol{x}}=\cos \phi \hat{\boldsymbol{\rho}}-\sin \phi \hat{\boldsymbol{\phi}}, \quad \hat{\boldsymbol{y}}=\sin \phi \hat{\boldsymbol{\rho}}+\cos \phi \hat{\boldsymbol{\phi}}, \quad \hat{z}=\hat{\boldsymbol{z}}
$$
You can check this second set by inspection.

Kratika Bhadauria

Numerade Educator

04:12

Problem 9

(1.11.2) (a) Check, by inspection of Fig. 1-12, that the unit vectors in Cartesian and spherical coordinates are related as follows:
$$
\begin{gathered}
\hat{\boldsymbol{r}}=\sin \theta \cos \phi \hat{\boldsymbol{x}}+\sin \theta \sin \phi \hat{\boldsymbol{y}}+\cos \theta \tilde{\boldsymbol{z}} \\
\hat{\boldsymbol{\theta}}=\cos \theta \cos \phi \hat{\boldsymbol{x}}+\cos \theta \sin \phi \hat{\mathbf{y}}-\sin \theta \hat{z}, \quad \hat{\boldsymbol{\phi}}=-\sin \phi \hat{\boldsymbol{x}}+\cos \phi \hat{\boldsymbol{y}}
\end{gathered}
$$
(b) Show that
$$
\begin{gathered}
\hat{\boldsymbol{x}}=\sin \theta \cos \phi \hat{\boldsymbol{r}}+\cos \theta \cos \phi \hat{\boldsymbol{\theta}}-\sin \phi \hat{\boldsymbol{\phi}} \\
\hat{\boldsymbol{y}}=\sin \theta \sin \phi \hat{\boldsymbol{r}}+\cos \theta \sin \phi \hat{\boldsymbol{\theta}}+\cos \phi \hat{\boldsymbol{\phi}}, \quad \hat{\boldsymbol{z}}=\cos \theta \hat{\boldsymbol{r}}-\sin \theta \hat{\boldsymbol{\theta}} .
\end{gathered}
$$

Jose Hannan

Numerade Educator

02:22

Problem 10

(1.11.2) A vector $\boldsymbol{F}$ has the same magnitude and direction at all points in space. Choose the $z$-axis parallel to $\boldsymbol{F}$. Then, in Cartesian and in cylindrical coordinates, $\boldsymbol{F}=F \hat{z}$.
Express $\boldsymbol{F}$ in spherical coordinates.

Ernest Castorena

Numerade Educator

00:59

Problem 11

(1.11.2) Show, by differentiating the appropriate expressions for $\boldsymbol{r}$, that the velocity $\dot{r}$ in cylindrical coordinates is $\dot{\rho} \hat{\boldsymbol{\rho}}+\rho \dot{\phi} \hat{\boldsymbol{\phi}}+\dot{z} \hat{z}$, while in spherical coordinates it is $\dot{r} \hat{\boldsymbol{r}}+r \dot{\theta} \hat{\boldsymbol{\theta}}+r \sin \theta \dot{\phi} \hat{\boldsymbol{\phi}}$.

Raj Bala

Numerade Educator

07:00

Problem 12

(1.11.5) A force $\boldsymbol{F}$ is of the form $\left(K / r^{3}\right) \hat{\boldsymbol{r}}$ in spherical coordinates, where $K$ is a constant. Is the field conservative?

John Gehad

Numerade Educator

01:09

Problem 13

(1.11.6) Show that, in cylindrical coordinates,
(a) $\boldsymbol{\nabla} \rho=\hat{\boldsymbol{\rho}}$,
(b) $\boldsymbol{\nabla} \cdot(\rho \hat{\boldsymbol{\rho}})=2$
(c) $\boldsymbol{\nabla} \times(\rho \hat{\boldsymbol{\rho}})=0$,
(d) $\boldsymbol{\nabla} \times(z \hat{\boldsymbol{\rho}})=\rho \hat{\boldsymbol{\phi}}$,
(e) $\boldsymbol{\nabla}^{2} \rho=\frac{1}{\rho}$.

Raj Bala

Numerade Educator

03:29

Problem 14

In the coordinate systems that we have used until now, vectors and the operator $\boldsymbol{\nabla}$ all have three components. However, in relativity theory (Chaps. 13 to 17 ), it is often more convenient to consider only two components, one that is parallel to a given direction and one that is perpendicular. For example, one writes that $\boldsymbol{r}=\boldsymbol{r}_{\mathrm{H}}+\boldsymbol{r}_{\perp}$.
If the chosen direction is the $x$-axis, then
$$
\boldsymbol{r}_{\|}=x \hat{\boldsymbol{x}} \quad \text { and } \quad \boldsymbol{r}_{i}=y \hat{\boldsymbol{y}}+z \hat{\boldsymbol{z}}
$$
Also, $\boldsymbol{\nabla}=\boldsymbol{\nabla}_{\|}+\boldsymbol{\nabla}_{\perp}$, with
$$
\boldsymbol{\nabla}_{\|}=\hat{\boldsymbol{x}} \frac{\partial}{\partial x}, \quad \boldsymbol{\nabla}_{\perp}=\hat{\boldsymbol{y}} \frac{\partial}{\partial y}+\hat{z} \frac{\partial}{\partial z}
$$
Then
$$
\boldsymbol{\nabla} V=\boldsymbol{\nabla}_{\|} V+\boldsymbol{\nabla}_{\perp} V
$$
Show that
$$
\boldsymbol{\nabla} \cdot \boldsymbol{A}=\boldsymbol{\nabla}_{\|} \cdot \boldsymbol{A}_{\|}+\boldsymbol{V}_{1} \cdot \boldsymbol{A}_{\perp}, \quad \boldsymbol{\nabla} \times \boldsymbol{A}=\boldsymbol{\nabla}_{\|} \times \boldsymbol{A}_{\perp}+\boldsymbol{\nabla}_{\perp} \times \boldsymbol{A}_{\|}+\boldsymbol{\nabla}_{\perp} \times \boldsymbol{A}_{\perp}
$$

Ahmad Reda

Numerade Educator

Problem 1

Problem 2

Problem 3

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Problem 10

Problem 11

Problem 12

Problem 13

Problem 14