Physik M Formula Collection

Mechanics of point-like objects

Given the position $\vec{r}$ [m], the velocity $\vec{v}$ [m/s], the acceleration $\vec{a}$ [m/s²], or the force $\vec{F}$ [N] as a function of time for an object, any other of these four quantities can be calculated by either integrating or differentiating.

$\large \vec{r}(t)=\int\limits_{t_0}^{t} \vec{v}(t')dt' + \vec{r}(t_0)= \int\limits_{t_0}^{t} \left( \int\limits_{t_0}^{t'} \vec{a}(t'') dt'' + \vec{v}(t_0)\right) dt' + \vec{r}(t_0)= \int\limits_{t_0}^{t} \left(\int\limits_{t_0}^{t'} \frac{\vec{F}(t'')}{m} dt''+ \vec{v}(t_0)\right) dt' + \vec{r}(t_0),$

$\large \vec{v}(t)=\frac{ d\vec{r}}{dt} = \int\limits_{t_0}^{t} \vec{a}(t') dt' + \vec{v}(t_0)= \int\limits_{t_0}^{t} \frac{\vec{F}(t')}{m} dt'+ \vec{v}(t_0), $

$\large \vec{a}=\frac{ d^2\vec{r}}{dt^2} = \frac{ d\vec{v}}{dt}=\frac{\vec{F}}{m}, $

$\large \vec{F}=m\frac{ d^2\vec{r}}{dt^2} = m\frac{ d\vec{v}}{dt}=m\vec{a}. $

Here $m$ is the mass of the object in kg.

Check your calculation with the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/numerical_integration/numerical_integration.en.php">Numerical integration and differentiation app</a>.

Apps: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/zero_force.en.php">Zero Total Force = Straight Line Motion</a>,  * 
 <a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/constant_force.en.php">Constant Force = Parabolic Motion</a>,  * 
 <a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/terminal.en.php">Terminal Velocity - Falling with a Linear Drag Force</a>,  * 
 <a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/harmonic.en.php">Harmonic motion</a>,  * 
<a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/circular_motion.en.php">Circular motion</a>,  * 
<a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/constant_E.en.php">Motion of a charged particle in a constant electric field</a>,  * 
 <a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/constant_B.en.php">Motion of a charged particle in a constant magnetic field</a>,  * 
<a href="http://lampz.tugraz.at/~hadley/physikm/apps/rvaf/constant_EB.en.php">Motion of a charged particle in constant electric and magnetic fields</a>

Sometimes none of the quantities $\vec{r}$, $\vec{v}$, $\vec{a}$, or $\vec{F}$ are given as simply a function of time but the force is known as a function of the time, the position, and the velocity. For instance, if a spring force is involved, the force depends on the position or if a drag force is involved, the force depends on the velocity. In such a case, Newton's law can be written as a vector differential equation,

\[ \begin{equation}
\large m\frac{d^2\vec{r}}{dt^2}=\vec{F}(\vec{v},\vec{r},t).
\end{equation} \]

This vector differential equation can be written as six first order differential equations.

$\large \frac{dx}{dt}=v_x$   $\large \frac{dv_x}{dt}=F_x(x,y,z,v_x,v_y,v_z,t)/m$
$\large \frac{dy}{dt}=v_y$   $\large \frac{dv_y}{dt}=F_y(x,y,z,v_x,v_y,v_z,t)/m$
$\large \frac{dz}{dt}=v_z$   $\large \frac{dv_z}{dt}=F_z(x,y,z,v_x,v_y,v_z,t)/m$

If $\vec{F}(\vec{v},\vec{r},t)$ is known and the initial position and velocity are known, these equations can be solved using the app: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/ode6xyz.en.php">Numerical solutions of sixth order differential equations</a>.

Sometimes the motion is restricted to one dimension such as when a ball is thrown straight up or when a weight hangs from a spring and oscillates up and down. In these cases, a second order differential equation can be used to describe the motion. This second order differential equation,

\[ \begin{equation}
\large m\frac{d^2x}{dt^2}=F_x(v_x,x,t).
\end{equation} \]

Any differential equation of this form can be solved numerically by the app: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/ode2nx.en.php">Numerical solutions of second order differential equations</a>. In addition, if the differential equation is linear, analytic solutions can be found with the app: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/2nd_ode_an.en.php">Second order linear differential equations</a>.

<a name="forces">
Examples of forces that depend on position or velocity are the gravitational force, the Coulomb force, the spring force (Hooke's law), the Lorentz force and the drag force.

Gravitational force: 
The gravitational force acting on a body of mass $m_1$ [kg] at position $\vec{r}_1$ [m] due to a body of mass $m_2$ [kg] at position $\vec{r}_2$ [m] is,

\[ \begin{equation}
\large \vec{F} = -\frac{Gm_1m_2}{|\vec{r}_1-\vec{r}_2 |^2}\hat{r}_{2\rightarrow 1} \hspace{1cm}\text{[N]}.
\end{equation} \]

Here $G$ = 6.6726×10-11 N m²/kg² is the gravitational constant and $\hat{r}_{2\rightarrow 1}=\frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|}$ is the unit vector pointing from $\vec{r}_2$ towards $\vec{r}_1$.

Coulomb force: 
The Coulomb force acting on a particle of charge $q_1$ [C] at position $\vec{r}_1$ [m] due to a particle of charge $q_2$ [C] at position $\vec{r}_2$ [m] is,

\[ \begin{equation}
\large \vec{F} = \frac{q_1q_2}{4\pi\epsilon_0 |\vec{r}_1-\vec{r}_2 |^2}\hat{r}_{2\rightarrow 1} \hspace{1cm}\text{[N]}.
\end{equation} \]

Here $\epsilon_0$ = 8.854187817×10-12 F/m is the permittivity constant and $\hat{r}_{2\rightarrow 1}=\frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|}$ is the unit vector pointing from $\vec{r}_2$ towards $\vec{r}_1$.

Linear spring force: 
A linear spring exerts a force that is proportional to the displacement $x$ [m] of the spring from its equilibrium position $x_0$.

\[ \begin{equation}
\large F = -k(x - x_0) \hspace{1cm}\text{[N]}.
\end{equation} \]

Here $k$ is the spring constant measured in units of N/m.

Lorentz force: 
The force on a particle of charge $q$ [C] moving at velocity $\vec{v}$ [m/s] in an electric field $\vec{E}$ [V/m] and a magnetic field $\vec{B}$ [T] is,

\[ \begin{equation}
\large \vec{F} = q(\vec{E}+\vec{v}\times\vec{B}) \hspace{1cm}\text{[N]}.
\end{equation} \]

Drag force: 
The drag force is a frictional force that points in the opposite direction as the velocity $\vec{v}$ of an object,

\[ \begin{equation}
\large \vec{F}_{drag} = -b_1\vec{v} - b_2\vec{v}|\vec{v}|.
\end{equation} \]

Here $b_1$ [N s/m] and $b_2$ [N s²/m²] are constants. For a low <a href="http://en.wikipedia.org/wiki/Reynolds_number">Reynolds number</a>, the linear term $-b_1\vec{v}$ usually dominates whereas for a high Reynolds number, the quadratic term $-b_2\vec{v}|\vec{v}|$ dominates.

Work and Energy

The work performed by pushing a object from position $\vec{r}_1$ to position $\vec{r}_2$ is,

\[ \begin{equation}
\large W = \int\limits_{\vec{r}_1}^{\vec{r}_2}\vec{F}\cdot d\vec{r}\hspace{1cm}\text{[J]}.
\end{equation} \]

This is a line integral. The expression for work can be written out in terms of its the three components.

\[ \begin{equation}
\large W = \int\limits_{x_1}^{x_2}F_x dx + \int\limits_{y_1}^{y_2}F_y dy+\int\limits_{z_1}^{z_2}F_z dz\hspace{1cm}\text{[J]}.
\end{equation} \]

If the force is known in terms of the position (for instance when the only force is a spring force or a gravitational force) then it is possible to perform the integrals. Often however, the position is known in terms of the time and the force depends also on the velocity (such as a drag force).Then it is possible to integrate over time where $dx=\frac{dx}{dt}dt = v_xdt$, $dy=\frac{dy}{dt}dt = v_xdt$, and $dz=\frac{dz}{dt}dt = v_zdt$.

\[ \begin{equation}
\large W=\int\limits_{t_1}^{t_2} F_xv_xdt + \int\limits_{t_1}^{t_2} F_yv_ydt + \int\limits_{t_1}^{t_2} F_zv_zdt.
\end{equation} \]

Power $P$ is the work that is exerted per second,

\begin{equation}
\large P = \frac{dW}{dt}=  F_xv_x +  F_yv_y +  F_zv_z\hspace{1cm}\text{[W]}.
\end{equation}

In general, the work can depend on the path that is taken to get from $\vec{r}_1$ to $\vec{r}_2$. If the path does not matter for a certain force, then that force is a conservative force and a potential energy $U$ for that force can be defined as, $\Delta U=-W$.

The kinetic energy of a particle is, $E_{kin} = \frac{mv^2}{2}$ [J] where $m$ [kg] is the mass of the particle and $v$ [m/s] is its velocity.

There is conservation of energy. The work performed equals the change in kinetic energy plus the change in potential energy plus the work performed against any nonconservative (frictional) forces.

Gravitational force: 
The gravitational force is a conservative force. The potential energy is

\[ \begin{equation}
\large U = -\frac{Gm_1m_2}{|\vec{r}_1-\vec{r}_2 |} +U_0\hspace{1cm}\text{[J]}.
\end{equation} \]

Here $U_0$ is an arbitrary constant. Near the surface of the earth, the force on a weight mass $m$ [kg] is $-\frac{Gm_1m_{\text{earth}}}{r_{\text{earth}}}\approx -9.81 m$. The potential energy of the weight when it is a distance $h$ [m] above the surface of the earth is $mgh$ [J], where $g=$ 9.81 [m/s²] is the acceleration of gravity at the earth's surface.

Coulomb force: 
The Coulomb force is a conservative force. The potential energy is,

\[ \begin{equation}
\large U = \frac{q_1q_2}{4\pi\epsilon_0 |\vec{r}_1-\vec{r}_2 |}+U_0 \hspace{1cm}\text{[J]}.
\end{equation} \]

Here $U_0$ is an arbitrary constant.

Linear spring force: 
A linear spring exerts a conservative force that is proportional to the displacement of the spring from its equilibrium position $x_0$. The potential energy is,

\[ \begin{equation}
\large U = \frac{k(x - x_0)^2}{2} \hspace{1cm}\text{[J]}.
\end{equation} \]

Lorentz force: 
The magnetic field exerts a force that is perpendicular to the motion of the charged particle so $\vec{F}_{\text{magnetic}}\cdot d\vec{r}=0$ and the magnetic force does not change the potential energy of the particle. The electric force is a conservative force and the potential energy associated with it is,

\[ \begin{equation}
\large U = -q\int\vec{E}\cdot d\vec{r}+U_0 \hspace{1cm}\text{[J]}.
\end{equation} \]

Here $U_0$ is an arbitrary constant. If the electric field is constant in some region, and the charge particle is moved from position $\vec{r}_1$ [m] to position $\vec{r}_2$ [m], then the change in potential energy is,

\[ \begin{equation}
\large \Delta U = -q \vec{E}\cdot (\vec{r}_2-\vec{r}_1) \hspace{1cm}\text{[J]}.
\end{equation} \]

Drag force: 
The drag force is a frictional force that always points in the opposite direction as the velocity $\vec{v}$ of an object. This means that it is not a conservative force and a potential energy cannot be defined for a drag force.

Electrostatics

Electrostatics relates the charge density $\rho$ to the electric field, $\vec{E}$, and to the electrostatic potential $\varphi$. The electric field lines point from regions of positive charge towards regions of negative charge. The work needed push a charged particle $q$ from position $\vec{r}_1$ to position $\vec{r}_2$ is $q(\varphi(\vec{r}_2) - \varphi(\vec{r}_1))$. This work is independent of the path that is taken to get from $\vec{r}_1$ to $\vec{r}_2$. The difference in electrostatic potential between two positions can be determined by integrating the Coulomb force or the electric field along any path between those two positions,

\[ \begin{equation}
\large \Delta \varphi=-\frac{1}{q}\int\limits_{\vec{r}_1}^{\vec{r}_2}q\vec{E}\cdot d\vec{r}=-\int\limits_{\vec{r}_1}^{\vec{r}_2}\vec{E}\cdot d\vec{r}\text{    [V]}.
\end{equation} \]

The simplest charge density is a point charge $q$ [C] located at position $\vec{r}_0$ [m]. The electric field produced by a point charge is,

\[ \begin{equation}
\large \vec{E}(\vec{r})= \frac{q}{4\pi \epsilon_0}\frac{\vec{r}-\vec{r}_0}{|\vec{r}-\vec{r}_0|^3} \hspace{1cm}\text{[V/m]},
\end{equation} \]

and the corresponding electrostatic potential for a point charge is,
 
\[ \begin{equation}
\large \varphi(\vec{r})= \frac{q}{4\pi \epsilon_0 |\vec{r}-\vec{r}_0|} \hspace{1cm}\text{[V]}.
\end{equation} \]

If the charge density consists of a collection of point charges, the expressions for the electric field and the electrostatic potential can be added together for the different charges. For instance, an electric dipole consists of a positive charge $q$ [C] at $\vec{r}_+$ [m] and a negative charge $-q$ [C] at $\vec{r}_-$ [m] causes an electric field,

\[ \begin{equation}
\large \vec{E}(\vec{r})= \frac{q}{4\pi \epsilon_0}\frac{\vec{r}-\vec{r}_+}{ |\vec{r}-\vec{r}_+|^3} - \frac{q}{4\pi \epsilon_0}\frac{\vec{r}-\vec{r}_-} {|\vec{r}-\vec{r}_-|^3}\hspace{1cm}\text{[V/m]},
\end{equation} \]

and an electrostatic potential,

\[ \begin{equation}
\large \varphi(\vec{r})= \frac{q}{4\pi \epsilon_0 |\vec{r}-\vec{r}_+|} -\frac{q}{4\pi \epsilon_0 |\vec{r}-\vec{r}_-|}\hspace{1cm}\text{[V]}.
\end{equation} \]

<img src="formulas/dipole.png">
The electric field pattern formed by two equal but opposite charges. The electric field lines point from the positive charge to the negative charge. The electric field lines and electrostatic potential lines can be visualized with the <a href="http://public.mitx.mit.edu/gwt-teal/PCharges.html">Two point charge simulation</a>.

For more point charges, the expressions for the electric field and the electrostatic potential can be written as sums. The electric field generated by a collection of $N$ point charges $q_i$ at positions $\vec{r}_i$ is,

\[ \begin{equation}
\large \vec{E}(\vec{r})= \sum\limits_{i=1}^N\frac{q_i}{4\pi \epsilon_0}\frac{\vec{r}-\vec{r}_i} {|\vec{r}-\vec{r}_i|^3} \hspace{1cm}\text{[V/m]},
\end{equation} \]

and the corresponding electrostatic potential is,

\[ \begin{equation}
\large \varphi (\vec{r})=\sum \limits_{i=1}^{N} \frac{q_i}{4\pi \epsilon_0 |\vec{r}-\vec{r}_i|}\hspace{1cm}\text{[V]}.
\end{equation} \]

There is an app to calculate the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/coulombE.en.php">electric field and electrostatic potential produced by a collection of point charges</a>.

If there are very many point charges, it becomes more practical to use the charge density $\rho$ [C/m³] to describe the distribution of the charges. The sums are converted into integrals and the equations for the electric field and electrostatic potential become,

\[ \begin{equation}
\large \vec{E}(\vec{r})= \int\frac{\rho(\vec{r})}{4\pi \epsilon_0}\frac{\vec{r}}{ |\vec{r}|^3}d^3r \hspace{1cm}\text{[V/m]},
\end{equation} \]

and the corresponding electrostatic potential is,

\[ \begin{equation}
\large \varphi (\vec{r})=\int  \frac{\rho(\vec{r})}{4\pi \epsilon_0 |\vec{r}|}d^3r\hspace{1cm}\text{[V]}.
\end{equation} \]

There is an app that will calculate the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/linecharge.en.php">electric field and electrostatic potential if charge is distributed uniformly along a line</a> with charge density $\lambda$ [C/m]. This can be used to calculate the electric field around a uniformly charged ring or rod or spiral or any line that can be described in three dimensions by parametric equations.

The electric field around an infinitely long line parallel to the $z$-axis that is uniformly charged with a charge density $\lambda$ [C/m] is,

\[ \begin{equation}
\large \vec{E}(\vec{r})=\frac{\lambda(\vec{r}-\vec{r}_0)}{2\pi \epsilon_0|\vec{r}-\vec{r}_0|^2}\hspace{1cm}\text{[V/m]},
\end{equation} \]

where $\vec{r}_0$ [m] is a two-dimensional vector in the $x-y$ plane that specifies the position of the charged line. The corresponding electrostatic potential is,

\[ \begin{equation}
\large \varphi (\vec{r})= \frac{-\lambda}{2\pi \epsilon_0 }\ln\left(|\vec{r}-\vec{r}_0|\right)\hspace{1cm}\text{[V]}.
\end{equation} \]
 
There is an app that will calculate the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/coulombline.en.php">electric field and electrostatic potential generated by a collection of line charges</a>.

The general expressions for the relationships between the charge density $\rho$, the electric field $\vec{E}$, and to the electrostatic potential $\varphi$ are:

\[ \begin{equation}
\large \nabla\cdot\vec{E}(\vec{r})=\frac{\rho(\vec{r})}{\epsilon_r\epsilon_0}\hspace{1cm}\text{Gauss's law},
\end{equation} \]

\[ \begin{equation}
\large \vec{E}(\vec{r})=-\nabla \varphi(\vec{r}),
\end{equation} \]

\[ \begin{equation}
\large -\nabla^2\varphi(\vec{r})=\frac{\rho(\vec{r})}{\epsilon_r\epsilon_0}\hspace{1cm}\text{Poisson equation}.
\end{equation} \]

Here $\epsilon_r$ is the relative dielectric constant. If the charge density, the electric field, and the electrostatic potential are constant in the $y-$ and $z-$directions and only vary in the $x-$direction, then these equations can be written as,

$\large \varphi(x)=-\int\limits_{x_1}^{x}\left(\int\limits_{x_1}^{x'}\frac{\rho(x'')}{\epsilon_r\epsilon_0}dx''+E(x_1)\right)dx'+\varphi(x_1) = -\int\limits_{x_1}^{x} E(x')dx'+\phi(x_1), $

$\large E(x)=\int\limits_{x_1}^{x}\frac{\rho(x')}{\epsilon_r\epsilon_0}dx'+E(x_1) = - \frac{d\varphi(x)}{dx}, $

$\large \rho(x) = \epsilon_r\epsilon_0\frac{dE(x)}{dx} = -\epsilon_r\epsilon_0\frac{d^2\varphi(x)}{dx^2},$

$\large F(x) = qE(x)= q\int\limits_{x_1}^{x}\frac{\rho(x')}{\epsilon_r\epsilon_0}dx'+qE(x_1) = - q\frac{d\varphi(x)}{dx}.$

Calculations of this sort can be checked with the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/numerical_integration_x.en.php">Numerical integration and differentiation app</a>.

For an infinite sheet uniformly-charged with a positive charge, the electric field is perpendicular to the sheet and it points away from the sheet. For a negatively charged sheet, the electric field is also perpendicular to the sheet but points towards the sheet. The magnitude of the electric field for a uniformly-charged sheet at position $x_0$ is,

\[ \begin{equation}
\large E_x = \frac{\text{sgn}(x-x_0)\sigma}{2\epsilon_r\epsilon_0}\hspace{0.5cm}\text{[V/m]},
\end{equation} \]

where $\sigma$ is the charge density in C/m² and $\text{sgn}(x)$ is the sign function: $\text{sgn}(x<0) = -1$, $\text{sgn}(x>0) = 1$.

Consider two electrically uncharged metal conductors. If some charge $q$ is taken from one and added to the other then the difference in electrostatic potential between these two conductors is related to the charge by the capacitance,

\[ \begin{equation}
\large \Delta\varphi = V = \frac{q}{C}\hspace{0.5cm}\text{[V/m]}.
\end{equation} \]

Here $C$ is the capacitance measured in Farads. If the two conductors are two flat metal plates then the electric field between them is constant, $E = -\frac{\sigma}{\epsilon_r\epsilon_0}$ and the voltage between them is $V=-\int E dx = \frac{\sigma d}{\epsilon_r\epsilon_0}$ where $d$ is the distance between the plates. The charge is the sheet charge times the area $q=\sigma A$. Solving for the capacitance yields,

\[ \begin{equation}
\large C =  \frac{\epsilon_r \epsilon_0 A}{d}\hspace{0.5cm}\text{[F]}.
\end{equation} \]

If a capacitor has a voltage across it, the work need to bring a small quantity of charge $dq$ across the capacitor is $Vdq$. This small charge then changes the voltage on the capacitor by $dq/C$. The total work needed to charge a capacitor to charge $Q$ is,

\[ \begin{equation}
\large W =  \int \limits_{0}^Q \frac{q}{C}dq=\frac{Q^2}{2C}=\frac{CV^2}{C}\hspace{0.5cm}\text{[J]}.
\end{equation} \]

The total current flowing onto a conductor is the change of charge on the conductor,

\[ \begin{equation}
\large \frac{dq}{dt} =  I_{\text{total}}\hspace{0.5cm}\text{[A]}.
\end{equation} \]

Magnetostatics

Magnetic forces are used in electrical motors. An electrical current produces a magnetic field. This field results in a force on nearby wires that carry a current due to the Lorentz force $q\vec{v}\times\vec{B}$.

The magnetic field produced by a current $I$ [A] flowing through a wire can be determined by breaking the current path into short segments and adding all of the contributions of those segments together. The contribution to the magnetic field at position $\vec{r}$ due to a short segment of length $d\vec{r}_{wire}$ at position $\vec{r}_{wire}$ is,

\[ \begin{equation}
\large d\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\frac{I d\vec{r}_{wire} \times (\vec{r}-\vec{r}_{wire})}{|\vec{r}-\vec{r}_{wire}|^3}\hspace{1cm}\text{[T]}\hspace{1cm}\text{Biot-Savart law}.
\end{equation} \]

Here $d\vec{r}_{wire}$ points in the direction that the current is flowing. The constant $\mu_0 = 4\pi \times 10^{-7}$ T m/A is the permeability constant.

There is an app that calculates the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/biotsavart.en.php">magnetic field produced by a current flowing through a wire</a>. The shape of the wire can be any form that can be described by parametric equations.

The magnetic field around an infinitely-long straight wire is,

\[ \begin{equation}
\large |\vec{B}(\vec{r})|=\frac{\mu_0I}{2\pi r}\hspace{1cm}\text{[T]},\hspace{2cm}
\end{equation} \]

</td><td><img src="formulas/bwire.png"></dt></tr></table>

where $I$ [A] is the current flowing through the wire and $r$ [m] is the shortest distance from the wire to the point where the magnetic field is measured. The direction of the magnetic field is given by the right-hand rule. When the thumb of a right hand points in the direction that the current is flowing, the fingers point in the direction of the magnetic field.

The magnetic field inside a long solenoid is,

\[ \begin{equation}
\large |\vec{B}|=\mu_0 n I \hspace{1cm}\text{[T]},
\end{equation} \]

where $I$ is the current flowing through the solenoid and $n$ is the number of turns per meter that the wire is wrapped around the solenoid. The direction of the magnetic field in the solenoid is given by the right-hand rule. The thumb of a right hand points in the direction of the magnetic field when the fingers are pointing in the direction that the current is flowing.

The total force on a wire with a current $I$ flowing through it in a magnetic field is,

\[ \begin{equation}
\large \vec{F}= I \int\limits_0^Ld\vec{l}\times\vec{B}\hspace{1cm}\text{[N]},
\end{equation} \]

where $L$ is the length of the wire. If the wire is straight and the magnetic field is constant everywhere along the wire then this simplifies to,

\[ \begin{equation}
\large \vec{F}= I \vec{r}\times\vec{B}\hspace{1cm}\text{[N]},
\end{equation} \]

where $\vec{r}$ is a vector pointing along the wire in the direction that the current is flowing. The length of $\vec{r}$ is the length of the wire.

Ampère's law relates the magnetic field to the current density,

\[ \begin{equation}
\large \nabla\times\vec{B}=\mu_0 \vec{J},
\end{equation} \]

where $\vec{J}$ is the current density. Using <a href="http://en.wikipedia.org/wiki/Stokes%27_theorem">Stokes's theorem</a>, this can be transformed to a form where the line integral of the magnetic field once around some loop is related to the total current $I_{enc}$ passing through the loop,

\[ \begin{equation}
\large \oint\limits_{C}\vec{B}\cdot d\vec{l}=\mu_0 I_{enc}.
\end{equation} \]

This <a href="http://public.mitx.mit.edu/gwt-teal/AmperesLaw.html">simulation illustrates the integral form of Ampère's law</a>.

Oscillations

In the section on oscillations we take a closer look at a damped mass-spring system that can be described by the differential equation,

\[ \begin{equation}
\large m\frac{d^2x}{dt^2}+b\frac{dx}{dt} + kx = F_0\cos (\omega t).
\end{equation} \]

Here $m$ [kg] is the mass, $b$ [N s/m] is the damping constant, $k$ [N/m] is the spring constant, $F_0$ [N] is the amplitude of the driving force, and $\omega$ [rad/s] is the frequency of the driving force.

The solution to the differential equation in the undriven case, $F_0=0$, can be found using the app: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/2nd_ode_an.en.php">Analytic solutions to second order linear differential equations</a>. The oscillations of the undriven system occur at an angular frequency of,

\[ \begin{equation}
\large \omega=\frac{\sqrt{4mk-b^2}}{2m}\hspace{1cm}\left[\frac{\text{rad}}{\text{s}}\right].
\end{equation} \]

If $b=0$, $\omega=\sqrt{\frac{k}{m}}$. The response of the mass-spring system is divided into three cases: the overdamped case $(4mk-b^2)<0$, the critically damped case $(4mk-b^2)=0$, and the underdamped case $(4mk-b^2)>0$.

<table align="center"><tr><td>Overdamped oscillations <img src="formulas/overdamped.png"></td><td> </td>
<td>Underdamped oscillations <img src="formulas/underdamped.png"></td></tr></table>

The quality factor $Q=\frac{\sqrt{mk}}{b}$ describes how underdamped the system is. Higher $Q$ implies more oscillations before the oscillations decay away: $Q=\frac{\pi\tau}{T}$, where $T$ is the period of the oscillations and $\tau$ describes the exponential decay.

The motion of a driven mass-spring system can be calculated using the app: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/ode2nx.en.php">Numerical solutions to second order linear differential equations</a>. An <a href="http://lampz.tugraz.at/~hadley/physikm/apps/resonance.en.php">analytic treatment of the driven mass-spring system</a> typically uses complex numbers. This description uses Euler's formula,

\[ \begin{equation}
\large e^{i\omega t}=\cos(\omega t)+i\sin(\omega t).
\end{equation} \]

<a href="http://lampz.tugraz.at/~hadley/physikm/apps/imaginary/circular.en.php">A sinusoidal oscillation can be though of as circular motion in the complex plane</a>. In the complex differential equation,

$\large m\frac{d^2z}{dt^2}+b\frac{dz}{dt} + kz = F_0 e^{i\omega t}$,

the driving function $F_0 e^{i\omega t}$ describes a point that moves in the complex plane in a circle of radius $F_0$ at a constant angular velocity of $\omega$. The real part of this motion is the original driving force, $F_0\cos (\omega t)$. The response $z$ also describes a point moving in the complex plane but with a different amplitude $A$ and phase $\theta$ as the driving term, $z=|A|e^{i\omega t-\theta}$. <a href="http://lampz.tugraz.at/~hadley/physikm/apps/resonance.en.php">The amplitude and phase of the response can be calculated to be:</a>

<table align="center"><tr><td>
\[ \begin{equation}
\large \frac{|A|}{F_0} = \frac{1}{\sqrt{(k-m\omega^2)^2+\omega^2b^2}}.
\end{equation} \]

\[ \begin{equation}
\tan\theta = \frac{\omega b}{k-m\omega^2}
\end{equation} \]

</td><td> </td><td><img src="formulas/resonance.png"></td></tr></table>

For $Q>\frac{1}{2}$, there is a resonance (a peak in the amplitude of the response) at the frequency where the undriven system oscillates.

Waves

The wave equation is a <a href="http://en.wikipedia.org/wiki/Partial_differential_equation">partial differential equation</a> that describes the motion of waves. In one dimension it is,

\[ \begin{equation}
\large \frac{\partial ^2 u}{\partial t^2} = c^2\frac{\partial ^2 u}{\partial x^2},
\end{equation} \]

where $c$ is the speed of the waves. In three dimensions the wave equation is,

\[ \begin{equation}
\large \frac{\partial ^2 u}{\partial t^2} = c^2\nabla ^2 u = c^2\left( \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}\right).
\end{equation} \]

Solutions to the wave equation are often described as a superposition of harmonic waves. When more sources are present, the resulting wave is a superposition of the simple forms. The resulting wave pattern is called an interference pattern if there are a few sources and it is called a diffraction pattern if there are many sources.

The form of a harmonic wave in one dimension is,

\[ \begin{equation}
\large u=A\cos(kx-\omega t).
\end{equation} \]

This wave is traveling in the positive $x-$direction with a velocity $c=\frac{\omega}{k}$ [m/s]. Here $A$ [m] is the amplitude, $k=\frac{2\pi}{\lambda}$ [m-1] is the wave number, $\omega=\frac{2\pi}{T}$ [rad/s] is the angular frequency, $\lambda=$ [m] is the wavelength, and $T$ [s] is the period. If $\frac{\omega}{k}<0$, the wave is traveling in the negative $x-$direction.

In two dimensions, circular waves moving away from a point $\vec{r}_j$ are described by,

\[ \begin{equation}
\large u_j=\frac{A_j\cos (k|\vec{r}-\vec{r}_j|-\omega t +\phi_j)}{\sqrt{|\vec{r}-\vec{r}_j|}},
\end{equation} \]

</td><td> </td><td><img src="formulas/2dwaves.png"></td></tr></table>

where $A_j$ describes the amplitude of the wave at point $j$ and $\phi_j$ describes the phase. The complex scalar field that describes the same wave is,

\[ \begin{equation}
\large \mathcal{U}=\frac{Ae^{i(k|r-r_j|-\omega t +\phi)}}{\sqrt{|r-r_j|}}.
\end{equation} \]

In three dimensions, spherical waves moving away from a point source are described by,

\[ \begin{equation}
\large u=\frac{A_j\cos (k|\vec{r}-\vec{r}_j|-\omega t +\phi_j)}{|\vec{r}-\vec{r}_j|},
\end{equation} \]

The complex form is,

\[ \begin{equation}
\large \mathcal{U}=\frac{Ae^{i(k|r-r_j|-\omega t +\phi)}}{|r-r_j|}.
\end{equation} \]

Any wave motion or wave pulse can be written as a sum over these basic waves. There is an app that describes <a href="http://lampz.tugraz.at/~hadley/physikm/apps/interference.en.php">the interference of two waves in one dimension</a> and one that describes <a href="http://lampz.tugraz.at/~hadley/physikm/apps/interference2d.en.php">the interference of two waves in two dimensions</a>. When more waves are involved, it becomes mathematically convenient to use the complex forms. If you focus on a point in space, the wave executes a simple sinusoidal oscillation. You can describe this sinusoidal motion as circular motion in the complex plane. The superposition of two-dimensional waves from $N$ point sources is,

\[ \begin{equation}
\large \mathcal{A}=\sum\limits_{j=1}^N \frac{A_j}{\sqrt{|\vec{r}-\vec{r}_j|}} e^{ i(k|\vec{r}-\vec{r}_j|-\omega t +\phi_j)},
\end{equation} \]
Here $\mathcal{A}=|\mathcal{A}|e^{i\theta}$ is a complex number for every position in space. $|\mathcal{A}|$ is the amplitude of the oscillations at that position and $\theta$ is the phase. The oscillations at this position are the real part of $\mathcal{A}$ which is $|\mathcal{A}|\cos(\omega t +\theta)$.

There is a similar formula for waves in three dimensions,

\[ \begin{equation}
\large \mathcal{A}=\sum\limits_{j=1}^N \frac{A_j}{|\vec{r}-\vec{r}_j|} e^{ i(k|\vec{r}-\vec{r}_j|-\omega t +\phi_j)}.
\end{equation} \]

The intensity of the pattern is proportional to the square of the amplitude $I\propto\mathcal{A}^*\mathcal{A}$. Here $\mathcal{A}^*\mathcal{A}$ means that $\mathcal{A}$ is multiplied by its complex conjugate $\mathcal{A}^*$. There are apps to calculate <a href="http://lampz.tugraz.at/~hadley/physikm/apps/interferenceintensity2d.en.php">the intensity pattern of two interfering surface waves</a> and <a href="http://lampz.tugraz.at/~hadley/physikm/apps/huygensintensity.en.php">the intensity pattern of many interfering surface waves</a>. The interference patterns are divided into the near field and the far field. In the near field, the distance from all sources to the observer is not large compared to the wavelength. The interference in this regime can usually can only be calculated numerically. The interference pattern from two point sources in shown below.

In the far field, the distance from all sources to the observer is much greater that the wavelength. In this limit it is often possible to calculate a formula for the interference pattern. <a href="http://lampz.tugraz.at/~hadley/physikm/apps/double_slit.en.php">The far-field interference pattern caused by two narrow slits is,</a>,

<table align="center"><tr><td>
\[ \begin{equation}
\large I \propto \cos^2\left( \frac{kdy}{2L}\right)
\end{equation} \]

$d$ is the distance between the slits 
$L$ is the distance to the screen 
$y$ is position on the screen 
$k=\frac{2\pi}{\lambda}$ is the wavenumber 
far field: $L >> d,\lambda,y$

</td><td> </td><td><img src="formulas/doubleslit.png"></td></tr></table>

<a href="http://lampz.tugraz.at/~hadley/physikm/apps/single_slit.en.php">The interference pattern of a single slit</a>:

<table align="center"><tr><td>
\[ \begin{equation}
\large I \propto \frac{\sin^2\left( \frac{\beta}{2}\right)}{\left( \frac{\beta}{2}\right)^2}
\end{equation} \]

$\beta=2\pi a\sin (y/L)/\lambda$ 
$a$ is the width of the slit 
$L$ is the distance to the screen 
$y$ is position on the screen 
$\lambda$ is the wavelength 
far field: $L >> a,\lambda,y$

</td><td> </td><td><img src="formulas/singleslit.png"></td></tr></table>

Waves that pass through a slit of width $a$ spread out with an angle that is about $\theta \approx \frac{\lambda}{a}$.
<img src="formulas/divergenceslit.png">

<a href="http://lampz.tugraz.at/~hadley/physikm/apps/2single_slit.en.php">The interference pattern of two finite-width slits</a>:

<table align="center"><tr><td>
\[ \begin{equation}
\large I \propto \frac{\sin^2\left( \frac{\beta}{2}\right)}{\left( \frac{\beta}{2}\right)^2}\cos^2\left( \frac{\delta}{2}\right)
\end{equation} \]

$\beta=2\pi a\sin (y/L)/\lambda$ 
$\delta=2\pi d\sin (y/L)/\lambda$ 
$d$ is the distance between the slits 
$a$ is the width of a slit 
$L$ is the distance to the screen 
$y$ is position on the screen 
$\lambda$ is the wavelength 
far field: $L >> d,a,\lambda,y$

</td><td> </td><td><img src="formulas/doubleslit_2.png"></td></tr></table>

Moving wave source A wave source emits waves that travel at a speed $c$. If the wave source is traveling at a velocity $v < c$, and observer detects a higher frequency if the source is moving towards the observer and a lower frequency when the source is moving away from the observer. This is the Doppler effect. If $\vec{r}_s(t)$ describes the time dependence of the position vector of the wave source and $\vec{r}_o(t)$ describes the time dependence of the position vector of the observer then the following equations determine the frequency $\tilde{f}$ heard by the observer at time $t_1$.

$\large |\vec{r}_o(t_1) - \vec{r}_s(t_0)|=c(t_1 -t_0)$,
$\large |\vec{r}_o(t_2) - \vec{r}_s(t_0+T)|=c(t_2-t_0-T)$,
$\large \tilde{f}=\frac{1}{t_2-t_1}$.

Here $c$ is the speed of the waves. Given $t_1$, solve the first equation for $t_0$. Then solve the second equation for $t_2$. Finally solve the third equation for $\tilde{f}$.

Apps: <a href="http://lampz.tugraz.at/~hadley/physikm/apps/shockwave.en.php">Moving wave source</a>, <a href="http://lampz.tugraz.at/~hadley/physikm/apps/doppler.en.php">Doppler effect</a>, <a href="http://lampz.tugraz.at/~hadley/physikm/apps/graphsol.en.php">Graphical solutions</a>.

If the wave source moves at a velocity faster than the wave velocity $(\frac{v}{c} < 1)$, a conical <a href="http://en.wikipedia.org/wiki/Shock_wave">shock wave</a> is produced. The cone forms an angle $\theta=\sin^{-1}(\frac{c}{v})$ with the velocity vector of the source. The wave fronts add constructively on the cone so there is a loud noise.

Optics

Optics is the study of light waves and these waves interfere and diffract as was described in the waves section. If all of the components in an optical system (such as lenses) are much bigger than the wavelength of light, then there is a limit called geometric optics where diffraction effects are ignored and light is described by rays that travel in straight lines. A central formula in geometric optics is Snell's law which states how light rays are refracted at an interface between materials with different indices of refraction. The derivation uses the concepts of the waves section and matches the plane waves at an interface where the speed of the waves is different on the two sides of the interface.

\[ \begin{equation}
\large n_1\sin\theta_1 = n_2\sin\theta_2.
\end{equation} \]

</td><td>   </td><td><img src="formulas/snell.png"> <a href="http://lampz.tugraz.at/~hadley/physikm/script/waves/snell.en.php">Refraction app</a></td></tr></table>

If $n_1 > n_2$, there are no solutions for $\theta_2$ for $\sin(\theta_1) > n_2/n_1$. This condition leads to total internal reflection of the light ray. Using Snell's law and a little geometry it is possible to calculate how a light ray will get bent through a system of lenses arranged along an optical axis. If a light ray only makes a small angle $\phi_i$ to the optical axis, then when it reaches a curved interface between two materials it will bent to a new angle,

\[ \begin{equation}
\large \phi_{i+1} = \frac{(n_i-n_{i+1})y_i}{R_{i}n_{i+1}}+\frac{n_i}{n_{i+1}}\phi_i.
\end{equation} \]

Here $\phi_i$ [rad] is the angle to the left before the ray strikes the interface, $\phi_{i+1}i$ [rad] is the angle after it strikes the interface, $n_i$ is the index of refraction of the material the ray moves through before it strikes the interface, $n_{i+1}$ is the index of refraction of the material the ray moves through after it strikes the interface, $y_i$ [m] is the distance from the optical axis that the ray strikes the interface, and $R_i$ [m] is the radius of the curved interface. For $R_i < 0$ the center of curvature on the side of the interface before the ray strikes the interface and for $R_i > 0$ the center of curvature is on the side of the interface after the ray strikes the interface.

Between the interfaces the rays travel in straight lines,

\[ \begin{equation}
\large y_{i+1} = \phi_{i+1}(x_{i+1}-x_i)+y_i.
\end{equation} \]

If two spherical interfaces are close to each other so that they form a thin lens, the focal length of the lens $f$ is,

\[ \begin{equation}
\large \frac{1}{f} = \frac{n_2-n_1}{n_1}\left(\frac{1}{R_1}-\frac{1}{R_2}\right).
\end{equation} \]

Here $n_2$ is the index of refraction of the material that the lens is made of and $n_1$ is the index of refraction of the surrounding medium; $n_1=1$ if the lens is in air. The condition for a thin lens is that the distance between the two surfaces of the lens is much smaller than the focal length. Parallel light rays striking the lens are focused at a focal length from the lens. If a light ray only makes a small angle $\phi_i$ to the optical axis, then when it reaches a thin lens with focal length $f$, it will bent to a new angle,

\[ \begin{equation}
\large \phi_{i+1} = -\frac{y_i}{f}+\phi_i.
\end{equation} \]

Between the interfaces the rays travel in straight lines,

\[ \begin{equation}
\large y_{i+1} = \phi_{i+1}(x_{i+1}-x_i)+y_i.
\end{equation} \]

The two equations above can be used to calculate how light rays travel through <a href="http://lampz.tugraz.at/~hadley/physikm/apps/2lens2.en.php">optical instruments</a> such as microscopes and telescopes or more complicated <a href="http://lampz.tugraz.at/~hadley/physikm/apps/ray.en.php">optical systems</a>.

Light emitted by an object at a position $x_o <0$ [m], will be focused by a lens at $x=0$ to an image at position $x_i= fx_o/(f+x_o)$. This formula can be rearranged to yield the <a href="http://lampz.tugraz.at/~hadley/physikm/apps/thinlens.en.php">thin lens equation</a>,

\[ \begin{equation}
\Large -\frac{1}{x_o}+\frac{1}{x_i}= \frac{1}{f}.
\end{equation} \]

The magnification of a thin lens is $m=y_i/y_o= f/(f+x_o)$.






	Physik M 513.805 (511.015)