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      <text>9.3.5 Vectors and electrostatics</text>
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      <text><![CDATA[<h4><b>Vectors and electrostatics</b></h4><p>Electricity is important in several branches of engineering &ndash; not only in electrical or electronic engineering. For example, the design of the electrostatic precipitator plates for cleaning the solid fuel power stations involves both mechanical engineering (structures and mechanical rapping systems for cleaning the plates) and electrostatics (to determine the electrical forces between solid particles and plates).&nbsp;</p><p>The following example and tasks relate to the electrostatic forces between particles. Electric charge is measured in coulombs (C). Charges can be either positive or negative.</p>

<p><span><strong>The force between two charges</strong></span></p>
<p>Let <span class="math inline">\(q_1\)</span> and <span class="math inline">\(q_2\)</span> be two charges in free space located at points <span class="math inline">\(P_1\)</span> and <span class="math inline">\(P_2\)</span>. Then <span class="math inline">\(q_1\)</span> will experience a force due to the presence of <span class="math inline">\(q_2\)</span> and directed from <span class="math inline">\(P_2\)</span> towards <span class="math inline">\(P_1\)</span>.&nbsp;</p><p>This force is of magnitude <span class="math inline">\(K \dfrac{q_1q_2}{r^2}\)</span> where <span class="math inline">\(r\)</span> is the distance between <span class="math inline">\(P_1\)</span> and <span class="math inline">\(P_2\)</span> and <span class="math inline">\(K\)</span> is a constant.&nbsp;</p><p>In vector notation this coulomb force (measured in newtons) can then be expressed as <span class="math inline">\(\underline{F}=K\dfrac{q_1q_2}{r^2} \hat{\underline{ r}}\)</span> where <span class="math inline">\(\hat{\underline{ r}}\)</span> is a unit vector directed from <span class="math inline">\(P_2\)</span> towards <span class="math inline">\(P_1\)</span>.&nbsp;</p><p>The constant <span class="math inline">\(K\)</span> is known to be <span class="math inline">\(\dfrac{1}{4\pi\varepsilon_0}\)</span> where <span class="math inline">\(\varepsilon_0=8.854\times 10^{-12}\)</span> F m<span class="math inline">\(^{-1}\)</span> (farads per metre).</p>

<p><span><strong><br></strong></span></p><p><span><strong>The electric field</strong></span>&nbsp;</p><p>A unit charge located at a general point <span class="math inline">\(G\)</span> will then experience a force <span class="math inline">\(\dfrac{Kq_1}{r_1^2}\hat{\underline{&nbsp; r}}_1\)</span> (where <span class="math inline">\(\hat{\underline{ r}}_1\)</span> is the unit vector directed from <span class="math inline">\(P_1\)</span> towards <span class="math inline">\(G\))&nbsp;</span>due to a charge <span class="math inline">\(q_1\)</span> located at <span class="math inline">\(P_1\)</span>. This is the electric field \( \underline{E} \)&nbsp;newtons per coulomb (N C<span class="math inline">\(^{-1}\)</span> or alternatively V m<span class="math inline">\(^{-1}\)</span>) at <span class="math inline">\(G\)</span> due to the presence of <span class="math inline">\(q_1\)</span>.</p><p>For several point charges <span class="math inline">\(q_1\)</span> at <span class="math inline">\(P_1,\ q_2\)</span> at <span class="math inline">\(P_2\)</span> etc., the total electric field<span class="math inline"><span>&nbsp;\( \underline{E} \)</span>&nbsp;</span>&nbsp;at <span class="math inline">\(G\)</span> is given by&nbsp;</p><p><span class="math inline"><br></span></p><p><span class="math inline">\(\underline{E}= \dfrac{K q_1}{r_1^2} \hat{\underline{ r}}_1+\dfrac{Kq_2}{r_2^2}\ \hat{\underline{ r}}_2+\ldots\)</span></p><p>where <span class="math inline">\(\hat{\underline { r}}_i\)</span> is the unit vector directed from point <span class="math inline">\(P_i\)</span> towards <span class="math inline">\(G\)</span>.</p><p><br></p>

<p>From the definition of a unit vector we see that&nbsp;</p><p><span class="math inline">\(\underline{E}=\dfrac{Kq_1}{r_1^2} \dfrac{\underline{r_1}}{|\underline{r_1}|}+\dfrac{Kq_2}{r_2^2} \dfrac{\underline{r_2}}{|\underline{r_2}|}+\ldots=\dfrac{Kq_1}{|r_1|^3}\underline{r_1}+\dfrac{Kq_2}{|\underline{r_2}|^3}\underline{r_2}+\ldots\)</span></p><p><span class="math inline">\(\underline{E}=\dfrac{1}{4\pi\varepsilon_0}\left[\dfrac{q_1}{|\underline{r_1}|^3}\underline{r_1}+\dfrac{q_2}{|\underline{r_2}|^3}\underline{r_2}+\ldots\right]\)</span>&nbsp;</p><p>where <span class="math inline">\(\underline{r_i}\)</span> is the vector directed from point <span class="math inline">\(P_i\)</span> towards <span class="math inline">\(G\)</span>, so that <span class="math inline">\(\underline{r_1}=\underline{OG}-\underline{OP}_1\)</span> etc., where <span class="math inline">\(\underline{OG}\)</span> and <span class="math inline">\(\underline{OP}_1\)</span> are the position vectors of <span class="math inline">\(G\)</span> and <span class="math inline">\(P_1\)</span> (see the figure below).</p>
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<p><span class="math inline">\[\underline{OP_1}+\underline{P_1G}=\underline{OG}\qquad\qquad \underline{P_1G}=\underline{OG}-\underline{OP_1}\]</span></p>
<p><span><strong>The work done</strong></span>&nbsp;</p><p>The work done <span class="math inline">\(W\)</span> (energy expended) in moving a charge <span class="math inline">\(q\)</span> through a distance <span class="math inline">\(\text{d} S\)</span>, in a direction given by the unit vector <span class="math inline">\({{\underline  S}}/|{{\underline  S}}|\)</span>, in an electric field <span class="math inline">\({\underline  E}\)</span> is (defined by)&nbsp;</p><p><span class="math inline">\(\qquad W=-q{{\underline  E}}.\text{d} {{\underline  S}}\)</span> where <span class="math inline">\(W\)</span> is in joules.</p>]]></text>
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