7.6 Applications of electrostatics  (Page 3/12)

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Large electrostatic precipitators    are used industrially to remove over $99%$ of the particles from stack gas emissions associated with the burning of coal and oil. Home precipitators, often in conjunction with the home heating and air conditioning system, are very effective in removing polluting particles, irritants, and allergens.

Summary

• Electrostatics is the study of electric fields in static equilibrium.
• In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink jet printers, and electrostatic air filters.

Key equations

 Potential energy of a two-charge system $U\left(r\right)=k\frac{qQ}{r}$ Work done to assemble a system of charges ${W}_{12\cdots N}=\frac{k}{2}\sum _{i}^{N}\sum _{j}^{N}\frac{{q}_{i}{q}_{j}}{{r}_{ij}}\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}i\ne j$ Potential difference $\text{Δ}V=\frac{\text{Δ}U}{q}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\text{Δ}U=q\text{Δ}V$ Electric potential $V=\frac{U}{q}=-{\int }_{R}^{P}\stackrel{\to }{E}\cdot d\stackrel{\to }{l}$ Potential difference between two points $\text{Δ}{V}_{AB}={V}_{B}-{V}_{A}=\text{−}{\int }_{A}^{B}\stackrel{\to }{\text{E}}·d\stackrel{\to }{\text{l}}$ Electric potential of a point charge $V=\frac{kq}{r}$ Electric potential of a system of point charges ${V}_{P}=k\sum _{1}^{N}\frac{{q}_{i}}{{r}_{i}}$ Electric dipole moment $\stackrel{\to }{\text{p}}=q\stackrel{\to }{\text{d}}$ Electric potential due to a dipole ${V}_{P}=k\frac{\stackrel{\to }{\text{p}}·\stackrel{^}{\text{r}}}{{r}^{2}}$ Electric potential of a continuous charge distribution ${V}_{P}=k\int \frac{dq}{r}$ Electric field components ${E}_{x}=-\frac{\partial V}{\partial x},\phantom{\rule{0.2em}{0ex}}{E}_{y}=-\frac{\partial V}{\partial y},\phantom{\rule{0.2em}{0ex}}{E}_{z}=-\frac{\partial V}{\partial z}$ Del operator in Cartesian coordinates $\stackrel{\to }{\nabla }=\stackrel{^}{\text{i}}\frac{\partial }{\partial x}+\stackrel{^}{\text{j}}\frac{\partial }{\partial y}+\stackrel{^}{\text{k}}\frac{\partial }{\partial z}$ Electric field as gradient of potential $\stackrel{\to }{\text{E}}=\text{−}\stackrel{\to }{\nabla }V$ Del operator in cylindrical coordinates $\stackrel{\to }{\nabla }=\stackrel{^}{\text{r}}\frac{\partial }{\partial r}+\stackrel{^}{\mathit{\text{φ}}}\frac{1}{r}\phantom{\rule{0.2em}{0ex}}\frac{\partial }{\partial \phi }+\stackrel{^}{\text{z}}\frac{\partial }{\partial z}$ Del operator in spherical coordinates $\stackrel{\to }{\nabla }=\stackrel{^}{\text{r}}\frac{\partial }{\partial r}+\stackrel{^}{\mathit{\text{θ}}}\frac{1}{r}\phantom{\rule{0.2em}{0ex}}\frac{\partial }{\partial \theta }+\stackrel{^}{\mathit{\text{φ}}}\frac{1}{r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta }\phantom{\rule{0.2em}{0ex}}\frac{\partial }{\partial \phi }$

Conceptual questions

Why are the metal support rods for satellite network dishes generally grounded?

So that lightning striking them goes into the ground instead of the television equipment.

(a) Why are fish reasonably safe in an electrical storm? (b) Why are swimmers nonetheless ordered to get out of the water in the same circumstance?

What are the similarities and differences between the processes in a photocopier and an electrostatic precipitator?

They both make use of static electricity to stick small particles to another surface. However, the precipitator has to charge a wide variety of particles, and is not designed to make sure they land in a particular place.

About what magnitude of potential is used to charge the drum of a photocopy machine? A web search for “xerography” may be of use.

Problems

(a) What is the electric field 5.00 m from the center of the terminal of a Van de Graaff with a 3.00-mC charge, noting that the field is equivalent to that of a point charge at the center of the terminal? (b) At this distance, what force does the field exert on a $2.00\text{-}\mu \text{C}$ charge on the Van de Graaff’s belt?

(a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces.

a. $F=5.58\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}\phantom{\rule{0.2em}{0ex}}\text{N/C}$ ;

The electric field is towards the surface of Earth. b. The coulomb force is much stronger than gravity.

Maxwell's stress tensor is
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