Physics Electric Charge and Field

 $E=\frac{\lambda }{4\pi {\epsilon }_{0}a}\left[sin\alpha +sin\alpha \right]=\frac{\lambda sin\alpha }{2\pi {\epsilon }_{0}a}$

$\Rightarrow E=\frac{Q}{L2\pi {\epsilon }_0*\left(\frac{\sqrt[]{3}L}{2}\right)}*\frac{1}{2}=\frac{Q}{2\sqrt[]{3}\pi {\epsilon }_0{L}^2}$

$tan\alpha =\frac{\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sqrt[]{3}L/2}\Rightarrow \alpha =\frac{\pi }{6}\Rightarrow sin\alpha =\frac{1}{2}$

The best conductor of electricity is Silver (Ag) at room temperature. It is not used in normal wiring even though being the best conductor because of its high cost.

• Electrical Conductivity ($\sigma$): $6.3 \times 10^ {7} \ \text {S/m}$

• Electrical Resistivity ($\rho$): $1.59 \times 10^ {-8} \ \Omega \cdot \text {m}$ (the lowest among all metals)

Flux through the 6 sides of square (i.e. cube)

${?}_T=\frac{q_{in}}{{\epsilon }_0}=\frac{12*10^{-6}C}{{\epsilon }_0}$

$\therefore$ Flux through a square

$?=\frac{{?}_T}{6}=\frac{12*10^{-6}}{{\epsilon }_0*6}=\frac{2*10^{-6}}{{\epsilon }_0}$  

$\Rightarrow ?=225.98*10^{3}N{m}^2/C?226N{m}^2/C$

$\overrightarrow{AB}+\overrightarrow{AH}=\overrightarrow{AO}$

$\overrightarrow{AC}+\overrightarrow{AG}=2\overrightarrow{AO}$

$\overrightarrow{AD}+\overrightarrow{AF}=3\overrightarrow{AO}$

$\overrightarrow{AE}=2\overrightarrow{AO}$

Adding All,  

$\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}+\overrightarrow{AG}+\overrightarrow{AH}=8\overrightarrow{AO}=16\widehat{i}+24\widehat{j}-32\widehat{k}$

Since force will act in horizontal direction so vertical component of speed will be same

v₁ cos α = v₂ cos β

$\Rightarrow \frac{K_1}{K_2}={\left(\frac{v_1}{v_2}\right)}^2=\frac{co{s}^2\beta }{co{s}^2\alpha }$

When two or more individual charges are present in a system, the total charge will be an algebraic sum of all individual charges and not the vector sum. Therefore, an electric charge is considered as a scalar quantity.

This is an interesting question! Even though a proton has a positive charge, the net positive charge in a conducting material is always due to the removal of free electrons. 

This happens due to the availability of only free electrons in all conductors. Since protons are present in the nucleus, they can not roam freely in the conductor, as electrons travel freely in the conductor due to their presence in the outer shell.

Using gauss law in differential form

$\overrightarrow{\nabla }.\overrightarrow{E}=\frac{\rho }{{\epsilon }_0}$

$\overrightarrow{\nabla }.\left({\epsilon }_0\overrightarrow{E}\right)=\rho$

$\Rightarrow \overrightarrow{\nabla }.\overrightarrow{D}=\rho$

$\Rightarrow \left(\frac{\partial }{\partial x}\widehat{i}+\frac{\partial }{\partial y}\widehat{j}+\frac{\partial }{\partial z}\widehat{k}\right).\overrightarrow{D}=\rho$

$\begin{array}{l}\Rightarrow \rho =-e^{-x}siny+e^{-x}siny+2=2\\ \Rightarrow Q=2\left(Volume\right)=4*10^{-9}c=4nc\end{array}$