5.6 KiB
ECE 106: Electricity and Magnetism
MATH 117 review
!!! definition A definite integral is composed of:
- the **upper limit**, $b$,
- the **lower limit**, $a$,
- the **integrand**, $f(x)$, and
- the **differential element**, $dx$.\[\int^b_a f(x)\ dx\]
The original function cannot be recovered from the result of a definite integral unless it is known that \(f(x)\) is a constant.
N-dimensional integrals
Much like how \(dx\) represents an infinitely small line, \(dx\cdot dy\) represents an infinitely small rectangle. This means that the surface area of an object can be expressed as:
\[dS=dx\cdot dy\]
Therefore, the area of a function can be expressed as:
\[S=\int^x_0\int^y_0 dy\ dx\]
where \(y\) is usually equal to \(f(x)\), changing on each iteration.
!!! example The area of a circle can be expressed as \(y=\pm\sqrt{r^2-x^2}\). This can be reduced to \(y=2\sqrt{r^2-x^2}\) because of the symmetry of the equation.
$$
\begin{align*}
A&=\int^r_0\int^{\sqrt{r^2-x^2}}_0 dy\ dx \\
&=\int^r_0\sqrt{r^2-x^2}\ dx
\end{align*}
$$!!! warning Similar to parentheses, the correct integral squiggly must be paired with the correct differential element.
These rules also apply for a system in three dimensions:
| Vector | Length | Area | Volume |
|---|---|---|---|
| \(x\) | \(dx\) | \(dx\cdot dy\) | \(dx\cdot dy\cdot dz\) |
| \(y\) | \(dy\) | \(dy\cdot dz\) | |
| \(z\) | \(dz\) | \(dx\cdot dz\) |
Although differential elements can be blindly used inside and outside an object (e.g., area), the rules break down as the boundary of an object is approached (e.g., perimeter). Applying these rules to determine an object’s perimeter will result in the incorrect deduction that \(\pi=4\).
Therefore, further approximations can be made using the Pythagorean theorem to represent the perimeter.
\[dl=\sqrt{(dx^2) + (dy)^2}\]
Polar coordinates
Please see MATH 115: Linear Algebra#Polar form for more information.
In polar form, the difference in each “rectangle” side length is slightly different.
| Vector | Length difference |
|---|---|
| \(\hat r\) | \(dr\) |
| \(\hat\phi\) | \(rd\phi\) |
Therefore, the change in surface area can be approximated to be a rectangle and is equal to:
\[dS=(dr)(rd\phi)\]
!!! example The area of a circle can be expressed as \(A=\int^{2\pi}_0\int^R_0 r\ dr\ d\phi\).
$$
\begin{align*}
A&=\int^{2\pi}_0\frac{1}{2}R^2\ d\phi \\
&=\pi R^2
\end{align*}
$$If \(r\) does not depend on \(d\phi\), part of the integral can be pre-evaluated:
\[ \begin{align*} dS&=\int^{2\pi}_{\phi=0} r\ dr\ d\phi \\ dS^\text{ring}&=2\pi r\ dr \end{align*} \]
So long as the variables are independent of each other, their order does not matter. Otherwise, the dependent variable must be calculated first.
!!! tip There is a shortcut for integrals of cosine and sine squared, so long as \(a=0\) and \(b\) is a multiple of \(\frac\pi 2\):
$$
\int^b_a\cos^2\phi=\frac{b-a}{2} \\
\int^b_a\sin^2\phi=\frac{b-a}{2}
$$The side length of a curve is as follows:
\[dl=\sqrt{(dr^2+(rd\phi)^2}\]
!!! example The side length of the curve \(r=e^\phi\) (Archimedes’ spiral) from \(0\) to \(2\pi\):
\begin{align*}
dl &=d\phi\sqrt{\left(\frac{dr}{d\phi}\right)^2 + r^2} \\
\tag{$\frac{dr}{d\phi}=e^\phi$}&=d\phi\sqrt{e^{2\phi}+r^2} \\
&=????????
\end{align*}Polar volume is the same as Cartesian volume:
\[dV=A\ dr\]
!!! example For a cylinder of radius \(R\) and height \(h\):
$$
\begin{align*}
dV&=\pi R^2\ dr \\
V&=\int^h_0 \pi R^2\ dr \\
&=\pi R^2 h
\end{align*}
$$Moment of inertia
The mass distribution of an object varies depending on its surface density \(\rho_s\). In objects with uniformly distributed mass, the surface density is equal to the total mass over the total area.
\[dm=\rho_s\ dS\]
The formula for the moment of inertia of an object is as follows, where \(r_\perp\) is the distance from the axis of rotation:
\[dI=(r_\perp)^2dm\]
If the axis of rotation is perpendicular to the plane of the object, \(r_\perp=r\). If the axis is parallel, \(r_\perp\) is the shortest distance to the axis. Setting an axis along the axis of rotation is easier.
!!! example In a uniformly distributed disk rotating about the origin like a CD with mass \(M\) and radius \(R\):
$$
\begin{align*}
\rho_s &= \frac{M}{\pi R^2} \\
dm &= \rho_s\ r\ dr\ d\phi \\
dI &=r^2\ dm \\
&= r^2\rho_s r\ dr\ d\phi \\
&= \rho_s r^3dr\ d\phi \\
I &=\rho_s\int^{2\pi}_{\phi=0}\int^R_{r=0} r^3dr\ d\phi \\
&= \rho_s\int^{2\pi}_{\phi=0}\frac{1}{4}R^4d\phi \\
&= \rho_s\frac{1}{2}\pi R^4 \\
&= \frac 1 2 MR^2
\end{align*}
$$Electrostatics
!!! definition - The polarity of a particle is whether it is positive or negative.
The law of conservation of charge states that electrons and charges cannot be created nor destroyed, such that the net charge in a closed system stays the same.
The law of charge quantisation states that charge is discrete — electrons have the lowest possible quantity.
Please see SL Physics 1#Charge for more information.
Coulomb’s law states that for point charges \(Q_1, Q_2\) with distance from the first to the second \(\vec R_{12}\):
\[\vec F_{12}=k\frac{Q_1Q_2}{||R_{12}||^2}\hat{R_{12}}\]
Dipoles
An electric dipole is composed of two equal but opposite charges \(Q\) separated by a distance \(d\). The dipole moment is the product of the two, \(Qd\).
The charge experienced by a positive test charge along the dipole line can be reduced to: \[\vec F_q=\hat x\frac{2kQdq}{||\vec x||^3}\]