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slides_lecture11_summary.tex
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\renewcommand{\summarizedlecture}{11 }
%
%
%
\begin{frame}{Lecture \summarizedlecture - \lecturesummarytitle}
\begin{columns}
\begin{column}{0.25\textwidth}
\begin{center}
\includegraphics[width=0.90\textwidth]{./images/schematics/mutual_inductance_1.png}\\
\end{center}
\end{column}
\begin{column}{0.75\textwidth}
{\scriptsize
The flux through the surface of loop 2, of the magnetic field $\vec{B}_1$
produced by the current $I_1$ in loop 1 is:
\begin{equation*}
\Phi_2 = M_{21} I_1
\end{equation*}
The constant of proportionality ($M_{21}$) is known as {\bf mutual inductance}.\\
It is:
\begin{itemize}
\item purely geometrical, and
\item unchanged if one switches the roles of loop 1 and 2.\\
\end{itemize}
}
\end{column}
\end{columns}
\vspace{0.4cm}
{\scriptsize
So \underline{whatever} the shapes and positions of the loops,
the flux through loop 2 when we run a current I around loop 1 is
identical to the flux through loop 1 when we run the same current around loop 2.
\begin{equation*}
M_{21} = M_{12} = M
\end{equation*}
The SI unit of the mutual inductance is the {Henry} (H)
\begin{itemize}
\item A derived unit.
\item 1 H = $\displaystyle \frac{Wb}{A}$ = $\displaystyle \frac{V \cdot s}{A}$
\end{itemize}
}
\end{frame}
%
%
%
\begin{frame}{Lecture \summarizedlecture - \lecturesummarytitle (cont'd)}
{ \scriptsize
We also considered what happens if the current varies with time.
\begin{itemize}
{ \scriptsize
\item time-varying current $\rightarrow$ time-varying magnetic field
\item time-varying magnetic field $\rightarrow$ time-varying flux.
\item time-varying flux $\rightarrow$ EMF (Faraday's law)
}
\end{itemize}
\vspace{0.3cm}
{\bf A change in current flow in a conductor induces a voltage (EMF)}
\vspace{0.2cm}
\begin{itemize}
{ \scriptsize
\item in the same conductor (self-inductance): \\
$\displaystyle \mathcal{E} = - L \frac{dI}{dt}$
\vspace{0.2cm}
\item and in neighbouring conductors (mutual inductance):\\
$\displaystyle \mathcal{E}_{neighbouring\;loop} = - M \frac{dI}{dt}$
}
\end{itemize}
\vspace{0.2cm}
In both cases the inductance (mutual or self) is the {\bf constant of proportionality}
between the EMF developed and the rate of current change.
\vspace{0.2cm}
We also studied the solenoid and its inductance per unit length (far from the ends of the solenoid) is:
\begin{equation*}
L = \mu_0 \cdot n^2 \cdot A
\end{equation*}
where $A$ is the area of each winding, and $n$ the number of turns per unit length.
}
\end{frame}
%
%
%
\begin{frame}{Lecture \summarizedlecture - \lecturesummarytitle (cont'd)}
Then we studied DC circuits with resistors, capacitors and inductors.
\begin{columns}
\begin{column}{0.40\textwidth}
\begin{center}
\begin{circuitikz} [scale=0.7]
\draw
(0,0) to[battery=$\varepsilon$] (0,2)
to[short, -o] (0.75, 2.0);
\draw[very thick]
(0.78,2.0)--(1.22,2.0);
\draw
(1.25, 2.0) to [short, o-] (2,2)
to[R=$R$, i=$I$] (2,0)
to[L=$L$] (0,0);
\end{circuitikz}
\end{center}
\end{column}
\begin{column}{0.60\textwidth}
{\scriptsize
We studied an RL circuit and we saw that its behaviour is determined
by the following differential equation:
\begin{equation*}
\mathcal{E} -L \cdot \frac{dI}{dt} = I \cdot R
\end{equation*}
}
\end{column}
\end{columns}
\begin{columns}
\begin{column}{0.40\textwidth}
\begin{center}
\includegraphics[width=0.95\textwidth]{./images/misc/ItRL_2.png}\\
\end{center}
\end{column}
\begin{column}{0.60\textwidth}
{\scriptsize
We solved that equation which gave as the following solutions for the
current after connecting or disconnecting the EMF:
\begin{equation*}
{\color{magenta}
I(t) = \frac{\mathcal{E}}{R} \Big(1 - exp^{-\frac{t}{\tau}} \Big)
}
\;\;\; and \;\;\;
{\color{magenta}
I(t) = \frac{\mathcal{E}}{R} \cdot exp^{-\frac{t}{\tau}}
}
\end{equation*}
Note that times are measured from the corresponding point of
connecting or disconnecting the EMF.\\
}
\end{column}
\end{columns}
{\scriptsize
{\bf Inductance is a kind of inertia in the circuit}. \\
\vspace{0.1cm}
So it is {\bf no longer possible to just change the current instantaneously} (as when L=0).
}
\end{frame}
%
%
%
\begin{frame}{Lecture \summarizedlecture - \lecturesummarytitle (cont'd)}
{ \scriptsize
We saw that the energy stored in the magnetic field of an inductor is: $\displaystyle U_B = \frac{1}{2} LI^2$.\\
\vspace{0.2cm}
Then we studied LC and RLC circuits both qualitatively and quantitatively:
\begin{columns}
\begin{column}{0.50\textwidth}
\begin{center}
\begin{circuitikz} [scale=0.8]
\draw
(0,0) to[battery=$\varepsilon$] (0,2) to[short, -o] (0.75, 2.0);
\draw[very thick]
(0.78,2.0)--(1.18,2.3);
\draw
(1.25, 2.0) to [short, o-] (2,2) to[C=$C$] (2,0)--(0,0);
\draw
(2,2)--(4,2) to[L=$L$,i=$I$] (4,0) -- (2,0);
\end{circuitikz}
\end{center}
\end{column}
\begin{column}{0.50\textwidth}
\begin{center}
\begin{circuitikz} [scale=0.8]
\draw
(0,0) to[battery=$\varepsilon$] (0,2) to[short, -o] (0.75, 2.0);
\draw[very thick]
(0.78,2.0)--(1.18,2.3);
\draw
(1.25, 2.0) to [short, o-] (2,2) to[C=$C$] (2,0)--(0,0);
\draw
(2,2) to[R=$R$] (4,2) to[L=$L$,i=$I$] (4,0) -- (2,0);
\end{circuitikz}
\end{center}
\end{column}
\end{columns}
\vspace{0.2cm}
RL and RLC are described by the following differential equation (with R=0 for LC):
\begin{equation*}
L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C}q = 0
\end{equation*}
\vspace{0.2cm}
\begin{itemize}
\item
Which saw that for R=0 we have undamped oscillations of charge, current and voltage and
that the stored energy is transferred fully between the capacitor (electric field) and the
inductor (magnetic field).
\item
For R$\ne$0 we have damped oscillations as, on every iteration, a fraction of the available energy is
converted to heat.
\end{itemize}
}
\end{frame}