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Pre-Lecture Homework
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3.1 Pre-Lecture Homework
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3.1.2 Some mathy stuffs
Graph
y=A\sin(kx+\omega t+\phi)
on Desmos and investigate how k, \omega,\phi and A affect the shape of the wave. Also identify which variable relates to the wavelength, \lambda and frequency, f of the wave.
Google Euler's formula and complete the equation below
e^{i(kx-\omega t)}=Read Ervin's article on Fourier Representation.
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3.1.2 Force and potential
Force is a vector quantity. Sometimes it is tricky to find the force (both magnitude and direction) at all positions. Fortunately there is a scalar quantity known as Potential Energy V that is closely related to force \mathbf{F} via
\mathbf{F}=\left(\begin{array}{c}
F_{x}\\
F_{y}\\
F_{z}
\end{array}\right)=-\left(\begin{array}{c}
{\displaystyle \frac{\partial V}{\partial x}}\\
{\displaystyle \frac{\partial V}{\partial y}}\\
{\displaystyle \frac{\partial V}{\partial z}}
\end{array}\right)Consider a force
\mathbf{F}_{E}=\frac{1}{4\pi\epsilon_{0}}\frac{qq_{2}}{\left(x^{2}+y^{2}+z^{2}\right)^{3/2}}\left(\begin{array}{c}
x\\
y\\
z
\end{array}\right)Verify that
V=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{\left(x^{2}+y^{2}+z^{2}\right)^{1/2}}is the potential energy for \mathbf{F}_{E}.
(Do feel free to use
Wolfram|Alpha
to help with
the differentiation.)
The mechanics of a system of particles can be equivalently studied using either forces or energy. Often the latter is preferred as the expression for potential energy is simpler. Applying the conservation of energy also allows us intuitively understand the behaviour of a closed system.