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\begin{document}
\newcommand{\bold}[1]{\mbox{\boldmath $#1$}}
\begin{bf}
\noindent
Institutt for fysikk, NTNU

\noindent
TFY4155/FY1303: Elektrisitet og magnetisme

\noindent
V{\aa}r 2004

\end{bf}

\begin{center}
\begin{large}
{\O}ving 8
\end{large}
\end{center}

\noindent
Guidance: 23.02, (24.02), 25.02, 26.02, 27.02, 01.03, 03.03, 04.03

\noindent
To be delivered by: Thursday March 4 kl. 1200 (Table for your answers on the last page.)

\vspace{0.3cm}
\noindent
Information:
\begin{itemize}
\item Unless otherwise stated, it is assumed that the system is in electrostatic equilibrium.
\item Unless otherwise stated, ``potential'' means ``electrostatic potential'',
and correspondingly for ``potential energy''.
\item Unless otherwise stated, zero (electrostatic) potential and potential energy is chosen
infinitely far away.
\item You may need some of these: $1/4\pi\varepsilon_0 = 9\cdot10^9$ Nm$^2$/C$^2$,
$e=1.6\cdot10^{-19}$ C, $m_e = 9.11\cdot10^{-31}$ kg, $m_p = 1.67\cdot10^{-27}$ kg, $g = 9.8$ m/s$^2$
\item Symbols are given in italics (e.g. $V$ for potential) while units are given without italics
(e.g. V for volt).
\end{itemize}

\vspace{0.5cm}
\noindent
$1)$ An arbitrarily shaped conductor has net charge $Q$. What happens in
the point $P$ if the charge on the conductor is increased to $2Q$?

\noindent
\parbox{10cm}{
\begin{tabular}{cl}
A & Only the potential is doubled. \\
B & Only the electric field strength is doubled. \\
C & Both the potential and the electric field  \\
  & strength are doubled. \\
D & Neither the potential nor the electric field \\
  & strength is doubled. \\
E & Both the potential and the electric field \\
  & strength are halved.
\end{tabular}
}
\parbox{5cm}{
%%FIGURE
\centerline{\epsfxsize=1.8in \epsfbox{ov8_fig1.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$2)$ A compact metal sphere has positive charge $Q$. The distance from the center of the
sphere to the point $A$ is half the distance to point $B$. Zero potential is chosen
at infinity. Then we have for the electric field strength $E$ and the potential $V$ in
the two points:

\noindent
\parbox{10cm}{
\begin{tabular}{cl}
A & $E_A = 4E_B$, $V_A = 4V_B$ \\
B & $E_A = 4E_B$, $V_A = 2V_B$ \\
C & $E_A = 4E_B$, $V_A = V_B$ \\
D & $E_A = 2E_B$, $V_A = 2V_B$ \\
E & $E_A = E_B$, $V_A = V_B$ 
\end{tabular}
}
\parbox{5cm}{
%%FIGURE
\centerline{\epsfxsize=1.8in \epsfbox{ov8_fig2.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$3)$ Three isolated metal spheres 1, 2, and 3 (i.e., they do not influence
{\em each other}) each have a positive charge $Q$. The diameter of the spheres
is $2a$, $3a$, and $5a$, respectively. How is the potential on the three spheres
related to each other? ($V(\infty)=0$)

\noindent
\parbox{10cm}{
\begin{tabular}{cl}
A & $V_1 = V_2 = V_3$ \\
B & $V_1 < V_2 < V_3$ \\
C & $V_1 > V_2 > V_3$ \\
D & $V_1 = V_2 < V_3$ \\
E & $V_1 = V_2 > V_3$
\end{tabular}
}
\parbox{5cm}{
%%FIGURE
\centerline{\epsfxsize=1.8in \epsfbox{ov8_fig3.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$4)$ Imagine two different experiments, both starting with to isolated, identical point
charges $Q$: (i) A (positive) point charge $q$ is brought from infinitely far away to
a distance $r$ from one of the $Q$s. (ii) A (positive) point charge $2q$ is brought from 
infinitely far away to a distance $2r$ from the other $Q$. When these two experiments have
been performed, what is then {\em not} true?

\vspace{0.3cm}
\noindent
\parbox{12cm}{
\begin{tabular}{cl}
A & The potential at distance $r$ (i.e., where the charge $q$ is) is the same \\
  & as the potential at distance $2r$ (i.e., where the charge $2q$ is). \\
B & In the two experiments, the work done is the same. \\
C & In the two experiments, we end up with \\
  & the same potential energy. \\
D & The charge $q$ at distance $r$ is influenced by a larger \\
  & repulsive force than the charge $2q$ at distance $2r$. \\
E & The ratio between the forces acting on $q$ and $2q$ is not \\
  & the same as the ratio between the electric field strengths \\
  & at distance $r$ and $2r$.
\end{tabular}
}
\parbox{5cm}{
%%FIGURE
\centerline{\epsfxsize=2.0in \epsfbox{ov8_fig4.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$5)$ The correct figure shows electric field lines in a plane passing
through the center of a metal sphere with net charge $Q > 0$.

\noindent
\parbox{6cm}{
\begin{tabular}{cl}
A & \\
B & \\
C & \\
D &
\end{tabular}
}
\parbox{10cm}{
%%FIGURE
\centerline{\epsfxsize=3.5in \epsfbox{ov8_fig5.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$6)$ The correct figure shows electric field lines in a plane passing
through the center of a charged metal sphere ($Q > 0$) uniformly covered
by a layer of (electrically neutral) plastic (with $\varepsilon > \varepsilon_0$).
The surrounding medium is air.

\noindent
\parbox{6cm}{
\begin{tabular}{cl}
A & \\
B & \\
C & \\
D & 
\end{tabular}
}
\parbox{10cm}{
%%FIGURE
\centerline{\epsfxsize=3.5in \epsfbox{ov8_fig6.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$7)$ The correct figure shows electric field lines for a parallel
plate capacitor that is half filled with a dielectric (i.e., the
hatched region has $\varepsilon > \varepsilon_0$). The linear dimension
of the plates is large compared to the distance between the plates.
The upper plate has negative charge $-Q$, the lower plate has positive
charge $Q$.

\noindent
\parbox{6cm}{
\begin{tabular}{cl}
A & \\
B & \\
C & \\
D &
\end{tabular}
}
\parbox{10cm}{
%%FIGURE
\centerline{\epsfxsize=3.5in \epsfbox{ov8_fig7.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$8)$ A compact metal sphere with radius $a$ has a net charge  $q>0$. It is
covered with a layer of (electrically neutral) plastic with thickness $a/2$. 
Outside follows an (electrically neutral) metallic spherical shell of
thickness $a/2$. Outside this, we have vacuum. The plastic is a dielectric with
permittivity $\varepsilon_1 = 10\varepsilon_0$. In which of the 4 labeled positions
{\bf A}, {\bf B}, {\bf C} or {\bf D} is the electric field strength largest?
The distance from the center of the sphere is in {\bf A}: $a/2$, {\bf B}: $5a/4$, 
{\bf C}: $7a/4$, {\bf D}: $5a/2$.

\vspace{0.5cm}
\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & \\
B & \\
C & \\
D &
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=1.7in \epsfbox{ov8_fig8.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$9)$ Four point charges are located in the $xy$ plane. Two of them have positive
charge $q$ and lie in $(x,y)=(a,-a)$ and $(-a,a)$. The two others have negative
charge $-q$ and lie in $(x,y)=(a,a)$ and $(-a,-a)$. What is the direction of the
electric field \bold{E} on the $x$ axis (assume $x>a$), i.e., in $(x,0)$?

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & Along $\hat{x}$. \\
B & Along $-\hat{x}$. \\
C & Along $\hat{y}$. \\
D & Along $-\hat{y}$. \\
E & It depends upon $x$.
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.0in \epsfbox{ov8_fig9.eps}}
%%END FIGURE
}

\newpage
\noindent
$10)$ For the system with the four point charges in exercise 9, we choose
zero electric potential, $V=0$, at infinity. What is then $V(x,y=0)$,
i.e., on the $x$ axis?

\noindent
\begin{tabular}{cl}
A &  $V=0$ \\
B &  $V=q/4\pi\varepsilon_0 x$ \\
C &  $V=q/\pi\varepsilon_0 x$ \\
D &  $V=q/4\pi\varepsilon_0 \sqrt{(x-a)^2 + a^2}$ \\
E &  $V=-q/4\pi\varepsilon_0 \sqrt{(x-a)^2 + a^2}$
\end{tabular}

\vspace{1.1cm}
\noindent
$11)$ A metallic sphere with radius $R$ carries a net charge $Q$. What is
the potential energy $U$ of the sphere? (We choose $U=0$ when all infinitesimal
contributions to $Q$ are infinitely far away from each other.)

\vspace{0.5cm}
\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & $U=-Q^2/\pi\varepsilon_0 R$ \\
B & $U=-Q^2/8\pi\varepsilon_0 R$ \\
C & $U=Q^2/\pi\varepsilon_0 R$ \\
D & $U=Q^2/8\pi\varepsilon_0 R$ \\
E & $U=Q^2/4\pi\varepsilon_0 R^2$
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=1.1in \epsfbox{ov8_fig10.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$12)$ Two parallel plate capacitors are connected in {\em parallel}, as shown
in the figure. The upper metal plates are connected via an electric conductor
(e.g. a copper wire) so that these two plates "lie" on the same electric
potential. The same applies to the two lower metal plates. Therefore, the
potential difference (or {\em the voltage drop}) $\Delta V$ is the same
for the two capacitors. What is the total capacitance $C$ for two such
capacitors connected in parallel? (The two capacitors have capacitances
$C_1$ and $C_2$, respectively.)

\vspace{0.5cm}
\noindent
\parbox{7cm}{
\begin{tabular}{cl}
A & $C_1 - C_2$ \\
B & $C_1 + C_2$ \\
C & $(C_1 + C_2)/2$ \\
D & $(1/C_1 - 1/C_2)^{-1}$ \\
E & $(1/C_1 + 1/C_2)^{-1}$
\end{tabular}
}
\parbox{9cm}{
%%FIGURE
\centerline{\epsfxsize=3.0in \epsfbox{ov8_fig11.eps}}
%%END FIGURE
}

\newpage
\noindent
$13)$ Two parallel plate capacitors are connected in {\em series}, as shown
in the figure. The lower metal plate of capacitor 1 is connected to the
upper metal plate of capacitor 2 via a conductor (e.g. a copper wire) so that
these two plates "lie" on the same electric potential. The total potential
difference (or {\em voltage drop}) $\Delta V$ across the two capacitors equals
the sum of the voltage drops $\Delta V_1$ and $\Delta V_2$ across each of the two.
The net charge on the various metal plates is as shown in the figure. What is
the total capacitance $C$ for two such capacitors connected in series?
(The two capacitors have capacitances $C_1$ and $C_2$, respectively.)

\vspace{0.5cm}
\noindent
\parbox{7cm}{
\begin{tabular}{cl}
A & $C_1 - C_2$ \\
B & $C_1 + C_2$ \\
C & $(C_1 + C_2)/2$ \\
D & $(1/C_1 - 1/C_2)^{-1}$ \\
E & $(1/C_1 + 1/C_2)^{-1}$
\end{tabular}
}
\parbox{9cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig12.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$14)$ A parallel plate capacitor is made of two parallel metal
plates separated by a distance $d$. The two metal plates have charge
$Q$ and $-Q$, respectively. A metal sheet with thickness $h < d$ is
inserted in between the two plates without touching any of them. Then
the potential difference between the capacitor plates becomes

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & smaller. \\
B & bigger. \\
C & unchanged.
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig13.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$15)$ A parallel plate capacitor is made of two parallel metal
plates separated by a distance $d$. The two metal plates have charge
$Q$ and $-Q$, respectively. Which quantity remains unchanged if we
increase the distance between the plates?

\noindent
\begin{tabular}{cl}
A & The capacitance of the capacitor. \\
B & The electric field between the plates. \\
C & The potential energy stored in the capacitor. \\
D & The potential difference between the plates.
\end{tabular}

\newpage
\noindent
$16)$ A parallel plate capacitor is made of two parallel metal
plates separated by a distance $d$. The two metal plates have charge
$Q$ and $-Q$, respectively. A dielectric with permittivity 
$\varepsilon > \varepsilon_0$ fills the left half of the volume between
the capacitor plates, as shown in the figure. In the right half, we have
vacuuum. On the left half of the metal plates, we have charge
$Q_1$ and $-Q_1$, respectively, on the right half, we have charge
$Q_2$ and $-Q_2$, respectively. Then,

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & $Q_1 = Q_2$ \\
B & $Q_1 > Q_2$ \\
C & $Q_1 < Q_2$
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig17.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$17)$ A parallel plate capacitor is made of two parallel metal
plates separated by a distance $d$. The two metal plates have charge
$Q$ and $-Q$, respectively. A dielectric with permittivity
$\varepsilon > \varepsilon_0$ fills the left half of the volume between
the capacitor plates, as shown in the figure. In the right half, we have
vacuuum. Then the arrows in the figure denote field lines for

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & electric displacement $\bold{D}$ \\
B & electric field $\bold{E}$ \\
C & polarization $\bold{P}$
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig14.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$18)$ A parallel plate capacitor is made of two parallel metal
plates separated by a distance $d$. The two metal plates have an area $A$ and
charge $Q$ and $-Q$, respectively. A dielectric with permittivity
$\varepsilon = \varepsilon_r \varepsilon_0 > \varepsilon_0$ fills the left
half of the volume between the capacitor plates, as shown in the figure.
In the right half, we have vacuum. What is the capacitance $C$, expressed
in terms of $C_0 = \varepsilon_0 A/d$, which would have been the
capacitance without the dielectric?

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & $C = \left[2\varepsilon_r/(\varepsilon_r + 1)\right] C_0$ \\ 
B & $C = \left[\varepsilon_r/(\varepsilon_r + 1)\right] C_0$ \\ 
C & $C = \varepsilon_r C_0$ \\ 
D & $C = (\varepsilon_r + 1) C_0$ \\ 
E & $C = \left[(\varepsilon_r + 1)/2\right] C_0$
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig15.eps}}
%%END FIGURE
}

\newpage
\noindent
$19)$ A parallel plate capacitor is made of two parallel metal
plates separated by a distance $d$. The two metal plates have an area $A$ and
charge $Q$ and $-Q$, respectively. A dielectric with permittivity
$\varepsilon = \varepsilon_r \varepsilon_0 > \varepsilon_0$ fills the lower
half of the volume between the capacitor plates, as shown in the figure.
In the upper half, we have vacuum. What is the capacitance $C$, expressed
in terms of $C_0 = \varepsilon_0 A/d$, which would have been the
capacitance without the dielectric?

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & $C = \left[2\varepsilon_r/(\varepsilon_r + 1)\right] C_0$ \\ 
B & $C = \left[\varepsilon_r/(\varepsilon_r + 1)\right] C_0$ \\ 
C & $C = \varepsilon_r C_0$ \\ 
D & $C = (\varepsilon_r + 1) C_0$ \\ 
E & $C = \left[(\varepsilon_r + 1)/2\right] C_0$
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig16.eps}}
%%END FIGURE
}

\vspace{1.1cm}
\noindent
$20)$ A cylindrical capacitor consists of two (thin) parallel concentric
metal cylinders, the inner one with radius $a$ and the outer one with
radius $b$. The two cylinders have length $L$ and charge pr unit length
equal to $\lambda$ (the inner one) and  $-\lambda$ (the outer one). A
dielectric with permittivity $\varepsilon$ fills the volume between
the inner and outer metal cylinder. The electric field in the region
$a < r < b$ is $\bold{E}(r) = (\lambda/2\pi\varepsilon r) \hat{r}$,
where $r$ denotes the distance from the axis of the cylinders, and
$\hat{r}$ is a unit vector directed perpendicularly away from the
cylinder axis. What is the capacitance $C$ of this cylindrical capacitor?
[Hint: The potential difference between the inner and the outer
cylinder is $\Delta V = V_a - V_b = -\int_b^a E(r)\;dr$.]

\noindent
\parbox{8cm}{
\begin{tabular}{cl}
A & $C = \pi\varepsilon L^2/b$ \\ 
B & $C = \pi\varepsilon L^2/a$ \\ 
C & $C = \pi\varepsilon L^3/ab$ \\ 
D & $C = 2\pi\varepsilon L/\ln (a/b)$ \\
E & $C = 2\pi\varepsilon L/\ln (b/a)$
\end{tabular}
}
\parbox{8cm}{
%%FIGURE
\centerline{\epsfxsize=2.5in \epsfbox{ov8_fig18.eps}}
%%END FIGURE
}


\newpage

\noindent
{\O}ving 8 i Elektromagnetisme / Elektrisitet og magnetisme v{\aa}ren 2004

\vspace{0.3cm}
\noindent
To be delivered by: Thursday March 4 kl. 1200.

\vspace{1cm}
\noindent
Navn:

\vspace{0.5cm}
\noindent
{\O}vingsgruppe:

\vspace{1cm}

\begin{tabular}{|r|c|c|c|c|c|} \hline
Oppgave &\hspace{0.5cm} A \hspace{0.5cm} &\hspace{0.5cm} B \hspace{0.5cm} &\hspace{0.5cm} C \hspace{0.5cm} &
\hspace{0.5cm} D \hspace{0.5cm} &\hspace{0.5cm} E \hspace{0.5cm} \\ \hline
1 \hspace{0.5cm} & & & & & \\ \hline
2 \hspace{0.5cm} & & & & & \\ \hline
3 \hspace{0.5cm} & & & & & \\ \hline
4 \hspace{0.5cm} & & & & & \\ \hline
5 \hspace{0.5cm} & & & & &\#\#\#\#\# \\ \hline
6 \hspace{0.5cm} & & & & &\#\#\#\#\# \\ \hline
7 \hspace{0.5cm} & & & & &\#\#\#\#\# \\ \hline
8 \hspace{0.5cm} & & & & &\#\#\#\#\# \\ \hline
9 \hspace{0.5cm} & & & & & \\ \hline
10 \hspace{0.5cm} & & & & & \\ \hline
11 \hspace{0.5cm} & & & & & \\ \hline
12 \hspace{0.5cm} & & & & & \\ \hline
13 \hspace{0.5cm} & & & & & \\ \hline
14 \hspace{0.5cm} & & & &\#\#\#\#\# &\#\#\#\#\# \\ \hline
15 \hspace{0.5cm} & & & & &\#\#\#\#\# \\ \hline
16 \hspace{0.5cm} & & & &\#\#\#\#\# &\#\#\#\#\# \\ \hline
17 \hspace{0.5cm} & & & &\#\#\#\#\# &\#\#\#\#\# \\ \hline
18 \hspace{0.5cm} & & & & & \\ \hline
19 \hspace{0.5cm} & & & & & \\ \hline
20 \hspace{0.5cm} & & & & & \\ \hline
\end{tabular}

\vspace{1.2cm}
\noindent
\#\#\#\#\#: not a relevant alternative
\end{document}
