Electric field

<p>There is a small green magnifying glass initially at the origin that indicates where the electric field and the electrostatic potential are measured. The measurement point can be moved in three dimensions by adjusting the components of its position vector $\vec{r}$. </p>

<form name="calc">
<p style="text-align:center">$\vec{r} = $ <input type="button" value="-" onclick="calc.rx.value = calc.rx.value -0.01*L; field();"><input type='text' size='10' name='rx' value="0"><input type="button" value="+" onclick="calc.rx.value = parseFloat(calc.rx.value) + 0.01*L; field();"> $\hat{x} + $ <input type="button" value="-" onclick="calc.ry.value = calc.ry.value -0.01*L; field();"><input type='text' size='10' name='ry' value="0"><input type="button" value="+" onclick="calc.ry.value = parseFloat(calc.ry.value) + 0.01*L; field();"> $\hat{y} + $ <input type="button" value="-" onclick="calc.rz.value = calc.rz.value -0.01*L; field();"><input type='text' size='10' name='rz'  value="0"><input type="button" value="+" onclick="calc.rz.value = parseFloat(calc.rz.value) + 0.01*L; field();"> $\hat{z}$ [m]</p>
<p style="text-align:center">$\vec{E}(\vec{r}) = $ <input type='text' size='10' name='Ex' readonly> $\hat{x} + $ <input type='text' size='10' name='Ey' readonly> $\hat{y} + $ <input type='text' size='10' name='Ez' readonly> $\hat{z}$ [V/m] $\varphi(\vec{r}) = $ <input type='text' size='10' name='V' readonly>  [V]</p>
<p style="text-align:center"><input type="button" value="Measure φ and E at position r (green magnifying glass)" onclick="field();"></p>
</form>

<p>The voltage is measured between the positive (red) contact and the negative (black) contact. The positions of the contacts are indicated by a red circle and a black circle. The positions $\vec{r}^{(+)}$ and $\vec{r}^{(-)}$ of the two contacts can be adjusted in three dimensions. The voltage is the difference in electrostatic potential, $V = \phi(\vec{r}^{(+)}) - \phi(\vec{r}^{(-)})$. </p>

<p>A ring of charge is modelled by $N=120$ point charges in a circle. The electric field exactly in the middle of the ring is zero since the charges are arranged symmetrically around this point. The ring is in the $x-y$ plane. We can also see that because of symmetry there can be no $x-$ or $y-$components of the electric field along the $z-$axis. We can check this by pressing the measurement button and moving along the $z-$axis. There is a small numerical error but is it clear that the $z-$component is much larger than the $x-$ or $y-$components. It is possible to show that the electric field along the axis is,</p>

$$\vec{E} = \frac{Qz}{4\pi\epsilon_0\left(z^2+R^2\right)^{3/2}}\hat{z}$$

<p>where $Q$ is the total charge on the ring and $R$ is the radius of the ring. For distances $z >> R$, an observer would see the ring as a point and in this limit the formula becomes the formula for a point charge,</p>

$$\vec{E} = \frac{Q}{4\pi\epsilon_0 z^2}\hat{z}.$$

<p>A charged line segment is modelled by $N =100$ point charges in a line. A charge $Q$ is evenly distributed along the line that starts at point $\vec{r}_1$ and ends at point $\vec{r}_2$.</p>

<p>Far from the line segment, the electric field approaches the electric field of a point charge. Close to the line segment but away from the ends, the electrical field approaches that of an infinite line charge.</p>

<p>The electric field produced by two line charges.</p>
<form name="parallel_form">
<p style="text-align:center"><input type="button"  value="both positively charged" onclick="parallel_form.charges.value=0; parallelf(parallel_form.charges.value, parallel_form.view.value);"> <input type="button"  value="both negatively charged" onclick="parallel_form.charges.value=1; parallelf(parallel_form.charges.value, parallel_form.view.value);"> <input type="button"  value="equal and oppositely charged" onclick="parallel_form.charges.value=2; parallelf(parallel_form.charges.value, parallel_form.view.value);"><br>
<input type="button"  value="view along line" onclick="parallel_form.view.value=0; parallelf(parallel_form.charges.value, parallel_form.view.value);"> <input type="button"  value="view perpendicular to the lines" onclick="parallel_form.view.value=1; parallelf(parallel_form.charges.value, parallel_form.view.value);"> 
<input type="hidden" name="charges" value="0">
<input type="hidden" name="view" value="0">
</p>
</form>

<p>The image below illustrates a three dimensional electric field created by a collection of point charges, line charges, and sheet charges. The magnitude of the electric field at every point is given in opacity of the vectors. The black vectors have the largest magnitude and the transparent vectors have the smallest magnitude. The arrows show the components of the electric field in the $x−y$ plane. If the arrow is short and black, that vector is pointing mostly out of the $x−y$ plane. If the vector is pointing out of the screen, $\odot$ is displayed. If the vector is pointing into the screen, $\otimes$ is displayed. The positive charges are red and the negative charges are blue. The projection of the line charges into the $x-y$ plane are shown in light blue or light red. If the line is parallel to the $z$-axis, only the point is visible. The intersection of the charge sheets with the $x-y$ plane is shown as dark red or dark blue lines. It is possible to adjust the $z-$plane that is displayed by pressing the <input type="button" value="z+"> and <input type="button" value="z-"> buttons.</p>

<p>An arbitrary number of point charges, line charges, and sheet charges can be added with the <input type="button" value="+"> buttons and the last one can be removed with the <input type="button" value="-"> button. By pressing on the numbered buttons, you can change the position and the charge of an item. If a charge is added outside the field of view, press the <input type="button" value="rescale"> button. </p>

<p>When the <input type="button" value="Measure φ and E at position r"> button is pressed, a small green magnifying glass is displayed that can be moved around to measure the electric field and electrostatic potential at any position. The <input type="button" value="Voltmeter"> button displays the two contacts of a voltmeter that are represented as red and black circles. The voltmeter displays the difference in electrostatic potential between the red and black contacts. </p>

<p>The equipotentials will be drawn under the vectors when the <input type="button" value="show"> button is pressed and hidden when the <input type="button" value="hide"> button is pressed. There is a algorithm that tries to choose the right scaling for the opacity of the vectors and the electrostatic potential range for the equipotentials. However, the potentials for the point charges and line charges diverge so sometimes this algorithm fails and you have to adjust the opacity and the range manually.</p>

<p>There is a pulldown menu that will load various charge configurations.</p>






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