User:JesseSlim/Sandbox

$$\omega = \omega_0$$

$$T = \frac{\epsilon_0 \pi r^2 \gamma^2}{2}\Bigg(\frac{V}{d}\Bigg)^2$$

$$\gamma = \frac{L(1+\frac{d}{r})}{d+L}$$

$$a_1 = a_2 \frac{cosh(kL)+cos(kL)}{sin(kL)-sinh(kL)}$$

$$0 = a_2 \frac{cosh(kL)+cos(kL)}{sin(kL)-sinh(kL)} cosh(kL) + a_2 sinh(kx)+ a_2 \frac{cosh(kL)+cos(kL)}{sin(kL)-sinh(kL)} cos(kx) + a_2 sin(kx)$$

$$0 = \frac{(cosh(kL)+cos(kL))(cosh(kL)+cos(kL))}{sin(kL)-sinh(kL)} + \frac{(sin(kL)+sinh(kL))(sin(kL)-sinh(kL))}{sin(kL)-sinh(kL)}$$

$$0 = \frac{cosh^2(kL)+2 cos(kL) cosh(kL) + cos^2(kL) + sin^2(kL) - sinh^2(kL)}{sin(kL)-sinh(kL)} = \frac{2 cos(kL) cosh(kL) + 2}{sin(kL)-sinh(kL)}$$

$$0 = 2 cos(kL) cosh(kL) + 2 = cos(kL) cosh(kL) + 1\,\!$$

$$u(0) = u'(0) = u(L) = u'(L) = 0\,\!$$

$$u(x) = a_1 cosh(k x) + a_2 sinh(k x) + a_3 cos(k x) + a_4 sin(k x)\,\!$$

$$u'(x) = k a_1 sinh(k x) + k a_2 cosh(k x) - k a_3 sin(k x) + k a_4 cos(k x)\,\!$$

$$u''(x) = k^2 a_1 cosh(k x) + k^2 a_2 sinh(k x) - k^2 a_3 cos(k x) - k^2 a_4 sin(k x)\,\!$$

$$u'''(x) = k^3 a_1 sinh(k x) + k^3 a_2 cosh(k x) + k^3 a_3 sin(k x) - k^3 a_4 cos(k x)\,\!$$

$$\rho A \frac{\partial^2}{\partial t^2} Z(x,t) + EI \frac{\partial^4}{\partial x^4} Z(x,t) - T \frac{\partial^2}{\partial x^2} Z(x,t) = 0$$

$$Z(x,t) = u(x)e^{i \omega t}\,\!$$

$$\frac{\partial^2}{\partial t^2} Z(x,t) = -\omega^2 u(x) e^{i \omega t}$$

$$\frac{\partial^2}{\partial x^2} Z(x,t) = e^{i \omega t} \frac{d}{d x^2}u(x)$$

$$\frac{\partial^4}{\partial x^4} Z(x,t) = e^{i \omega t} \frac{d}{d x^4}u(x)$$

$$- \rho A \omega^2 u(x) e^{i \omega t} + EI e^{i \omega t} \frac{d}{d x^2}u(x) - T e^{i \omega t} \frac{d}{d x^2}u(x) = 0$$

$$- \rho A \omega^2 u(x) + EI \frac{d^4}{d x^4}u(x) - T \frac{d^2}{d x^2}u(x) = 0$$

$$u(x) = e^{kx}\,\!$$

$$EI k^4 e^{kx} - T k^2 e^{kx} - \rho A \omega^2 e^{kx} = 0\,\! $$

$$EI k^4 - T k^4 - \rho A \omega^2 = 0\,\!$$

$$k^2 = \frac{T+\sqrt{T^2 + 4EI \rho A \omega^2}}{2EI}\,\!$$

$$c = \sqrt{\omega}\sqrt[4]{\frac{m}{B}}$$

$$B'' = \frac{h^3 E}{12(1-\nu^2)}$$

$$m'' = \rho h\,\!$$

$$c = \sqrt{\omega}\sqrt[4]{\frac{E h^2}{12 \rho (1 - \nu^2)}}$$

$$h = \frac{c^2}{\omega} \sqrt{\frac{12 \rho (1 - \nu^2)}{E}} = \frac{c^2}{2 \pi f} \sqrt{\frac{12 \rho (1 - \nu^2)}{E}}$$

$$\frac{c^2}{2 \pi f}$$

$$c_B=\sqrt[4]{\frac{B'\omega^2}{m''}}$$

$$\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}) = q \mathbf{E} + q (\mathbf{v} \times \mathbf{B})$$

$$\mathbf{F} = q (\mathbf{v} \times \mathbf{B}) = l(\mathbf{I} \times \mathbf{B})$$

$$\times$$

$$\mathbf{a} \times \mathbf{b} = \mathbf{c}$$

$$\mathbf{\hat{n}}$$

$$\mathbf{a}\cdot\mathbf{b} = c$$

$$\mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix}, \mathbf{b} = \begin{bmatrix} x_b\\ y_b\\ z_b\\ \end{bmatrix}, \mathbf{c} = \mathbf{a} + \mathbf{b} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix} + \begin{bmatrix} x_b\\ y_b\\ z_b\\ \end{bmatrix} = \begin{bmatrix} x_a + x_b\\ y_a + y_b\\ z_a + y_c\\ \end{bmatrix} $$

$$\mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix}, -\mathbf{a} = \begin{bmatrix} -x_a\\ -y_a\\ -z_a\\ \end{bmatrix}$$

$$\mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix}, \mathbf{b} = \begin{bmatrix} x_b\\ y_b\\ z_b\\ \end{bmatrix}, \mathbf{c} = \mathbf{a} - \mathbf{b} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix} - \begin{bmatrix} x_b\\ y_b\\ z_b\\ \end{bmatrix} = \begin{bmatrix} x_a - x_b\\ y_a - y_b\\ z_a - y_c\\ \end{bmatrix} $$

$$ \mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix}, \left\|\mathbf{a}\right\| = \sqrt{{x_a}^2 + {y_a}^2 + {z_a}^2} $$

$$\mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix}, \mathbf{b} = \begin{bmatrix} x_b\\ y_b\\ z_b\\ \end{bmatrix}, \mathbf{a}\cdot\mathbf{b} =\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta = x_a x_b + y_a y_b + z_a z_b$$

$$\mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix} = \left(x_a, y_a, z_a\right)$$

$$ \mathbf{a_x} = x_a \mathbf{\hat{x}} = \begin{bmatrix} x_a\\ 0\\ 0\\ \end{bmatrix}, \mathbf{a_y} = y_a \mathbf{\hat{y}} = \begin{bmatrix} 0\\ y_a\\ 0\\ \end{bmatrix}, \mathbf{a_z} = z_a \mathbf{\hat{z}} = \begin{bmatrix} 0\\ 0\\ z_a\\ \end{bmatrix}$$

$$\mathbf{\hat{x}} = \begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix}, \mathbf{\hat{y}} = \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix}, \mathbf{\hat{z}} = \begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix}$$

$$ c\mathbf{a} = \begin{bmatrix} c x_a\\ c y_a\\ c z_a\\ \end{bmatrix}$$

$$\mathbf{a} = \begin{bmatrix} x_a\\ y_a\\ z_a\\ \end{bmatrix},\mathbf{b} = \begin{bmatrix} x_b\\ y_b\\ z_b\\ \end{bmatrix}, \mathbf{a} \times \mathbf{b} = \begin{bmatrix} y_a z_b - y_b z_a\\ z_a x_b - z_b x_a\\ x_a y_b - x_b y_a\\ \end{bmatrix} = \mathbf{\hat{n}}\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\sin\theta$$

$$\mathbf{a} = x_a \mathbf{\hat{x}} + y_a \mathbf{\hat{y}} + z_a \mathbf{\hat{z}}$$

$$ \mathbf{B} = \begin{bmatrix} B_x\\ B_y\\ B_z\\ \end{bmatrix}, \frac{\partial \mathbf{B}}{\partial x} = \begin{bmatrix} \frac{\partial B_x}{\partial x}\\ \frac{\partial B_y}{\partial x}\\ \frac{\partial B_Z}{\partial x}\\ \end{bmatrix} $$

$$   \int_{AB} \mathbf{f}\ \cdot d\mathbf{s} $$

$$   \oint_C \mathbf{f}\ \cdot d\mathbf{s} $$

$$\iiint\limits_V\mathbf{F}\ dV$$

$$\iint\limits_{S}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf{F}\ \cdot d\mathbf{S} $$

$$\iint\limits_{S} \mathbf{F}\ \cdot d\mathbf{S} $$

$$ \nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) $$

$$\vec F$$

$$y_F\,\!$$

$$F_y\,\!$$

$$   \operatorname{div} F=\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}+ \frac{\partial F_z}{\partial z} $$

$$   \operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} = \begin{bmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}\\ \end{bmatrix} \cdot \begin{bmatrix} F_x\\ F_y\\ F_z\\ \end{bmatrix} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}+ \frac{\partial F_z}{\partial z} $$

$$ \operatorname{curl} \ \mathbf{F}=  \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{\hat{x}} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{\hat{y}} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{\hat{z}} $$

$$ \operatorname{curl} \ \mathbf{F} = \nabla \times \mathbf{F} = \begin{bmatrix} \frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}\\ \end{bmatrix} \times \begin{bmatrix} F_x\\ F_y\\ F_z\\ \end{bmatrix} = $$$$ \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \mathbf{\hat{x}} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{\hat{y}} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{\hat{z}} $$

$$   \iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_S\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf{F}\cdot d\mathbf{S} $$

$$\iint\limits_S \left(\nabla\times\mathbf{F}\right) dS = \oint\limits_s \mathbf{F}\ ds$$

$$\operatorname{grad}\ f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$

$$\operatorname{grad}\ f = \nabla f = \left( \frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \right)

f =

\left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$

$$\mathbf{B} \cdot d\boldsymbol{\ell} = Bd\ell$$

$$   \oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \oint_C Bd\ell = 2 \pi r B = \mu_0 I_{\mathrm{enc}} $$

$$ B = \frac{\mu_0 I_{\mathrm{enc}} }{2 \pi r} $$

$$\mathbf{N} = \mathbf{m} \times \mathbf{B}$$

$$\mathbf{m} = I\mathbf{a}$$

$$F = 2 \pi r I B_r\,\!$$

$$2\pi r B_r \Delta z\,\!$$

$$\pi r^2 ( - B_z(z) + B_z(z + \Delta z) = \pi r^2(\frac{\partial B_z}{\partial z}) \Delta z$$

$$2\pi r B_r \Delta z + \pi r^2(\frac{\partial B_z}{\partial z}) \Delta z = 0$$

$$B_r = - \frac{r}{2} \frac{\partial B_z}{\partial z}$$

$$F = 2 \pi r I \frac{r}{2} \frac{\partial B_z}{\partial z} = \pi r^2 I \frac{\partial B_z}{\partial z}$$

$$F = m \frac{\partial B_z}{\partial z}$$

$$F_x = \mathbf{m} \cdot \operatorname{grad}\ B_x$$

$$m = I\ A = M\ da\ dz $$

$$   \oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = B_{as} L_{pad} = \mu_0 I_{\mathrm{enc}} $$

$$B_{as} L_{pad} = \mu_0 I_{\mathrm{enc}} = \mu_0 N_{pad} I_{spoel} \,\!$$

$$n = \frac{N_{spoel}}{L_{spoel}} $$

$$N_{pad} = \frac{N_{spoel}}{L_{spoel}} L_{pad} = n L_{pad} $$

$$B_{as} L_{pad} = \mu_0 N_{pad} I_{spoel} = \mu_0 n L_{pad} I_{spoel} \,\!$$

$$B_{as} = \mu n I_{spoel} = \mu \frac{N_{spoel}}{L_{spoel}} I_{spoel}$$

$$P = I^2 R\,\!$$

$$I = \frac{q}{t}$$

$$\Phi_{B,S} = \iint\limits_{S} \mathbf{B}\ \cdot d\mathbf{S}$$

$$F_x = m_z \frac{\partial B_x}{\partial z}$$

$$\frac{\partial B_x}{\partial z}$$