The Electric Field Intensity

Oct 13, 2025 by pinkandtrash.com 29 views

Introduction

The electric field intensity produced by a point charge distribution is a fundamental concept in physics, particularly in the study of electromagnetism. In this article, we will explore how to find the electric field intensity produced by a point charge distribution at a given point $P(1,1,1)$ . We will consider a distribution of four identical point charges located at specific points in space.

The Electric Field Intensity

The electric field intensity, also known as the electric field strength, is a measure of the force per unit charge that a point charge would experience at a given point in space. It is a vector quantity, denoted by the symbol $\mathbf{E}$ , and is typically measured in units of newtons per coulomb (N/C).

The Point Charge Distribution

The point charge distribution consists of four identical point charges, each with a charge of $3 \, \text{nC}$ . The charges are located at the following points in space:

$P_1(1,1,0)$
$P_2(-1,1,0)$
$P_3(1,-1,0)$
$P_4(0,0,1)$

Calculating the Electric Field Intensity

To calculate the electric field intensity produced by the point charge distribution at point $P(1,1,1)$ , we can use the following formula:

\mathbf{E} = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{N} \frac{q_i \mathbf{r}_{iP}}{|\mathbf{r}_{iP}|^3}

where:

$\mathbf{E}$ is the electric field intensity at point $P$
$N$ is the number of point charges in the distribution
$q_i$ is the charge of the $i$ th point charge
$\mathbf{r}_{iP}$ is the vector from the $i$ th point charge to point $P$
$\epsilon_0$ is the electric constant (also known as the permittivity of free space)

Calculating the Electric Field Intensity at Point $P$

Using the formula above, we can calculate the electric field intensity at point $P(1,1,1)$ due to each of the four point charges.

Charge $P_1(1,1,0)$

The vector from $P_1$ to $P$ is:

\mathbf{r}_{1P} = (1-1, 1-1, 1-0) = (0, 0, 1)

The magnitude of this vector is:

|\mathbf{r}_{1P}| = \sqrt{0^2 + 0^2 + 1^2} = 1

The electric field intensity due to charge $P_1$ is:

\mathbf{E}_1 = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} \mathbf{r}_{1P}}{|\mathbf{r}_{1P}|^3} = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} (0, 0, 1)}{1^3}

Charge $P_2(-1,1,0)$

The vector from $P_2$ to $P$ is:

\mathbf{r}_{2P} = (1-(-1), 1-1, 1-0) = (2, 0, 1)

The magnitude of this vector is:

|\mathbf{r}_{2P}| = \sqrt{2^2 + 0^2 + 1^2} = \sqrt{5}

The electric field intensity due to charge $P_2$ is:

\mathbf{E}_2 = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} \mathbf{r}_{2P}}{|\mathbf{r}_{2P}|^3} = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} (2, 0, 1)}{(\sqrt{5})^3}

Charge $P_3(1,-1,0)$

The vector from $P_3$ to $P$ is:

\mathbf{r}_{3P} = (1-1, 1-(-1), 1-0) = (0, 2, 1)

The magnitude of this vector is:

|\mathbf{r}_{3P}| = \sqrt{0^2 + 2^2 + 1^2} = \sqrt{5}

The electric field intensity due to charge $P_3$ is:

\mathbf{E}_3 = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} \mathbf{r}_{3P}}{|\mathbf{r}_{3P}|^3} = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} (0, 2, 1)}{(\sqrt{5})^3}

Charge $P_4(0,0,1)$

The vector from $P_4$ to $P$ is:

\mathbf{r}_{4P} = (1-0, 1-0, 1-1) = (1, 1, 0)

The magnitude of this vector is:

|\mathbf{r}_{4P}| = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{2}

The electric field intensity due to charge $P_4$ is:

\mathbf{E}_4 = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} \mathbf{r}_{4P}}{|\mathbf{r}_{4P}|^3} = \frac{1}{4\pi\epsilon_0} \frac{3 \times 10^{-9} (1, 1, 0)}{(\sqrt{2})^3}

Combining the Electric Field Intensities

To find the total electric field intensity at point $P$ , we need to combine the electric field intensities due to each of the four point charges.

\mathbf{E} = \mathbf{E}_1 + \mathbf{E}_2 + \mathbf{E}_3 + \mathbf{E}_4

Substituting the expressions for each of the electric field intensities, we get:

\mathbf{E} = \frac{1}{4\pi\epsilon_0} \left( \frac{3 \times 10^{-9} (0, 0, 1)}{1^3} + \frac{3 \times 10^{-9} (2, 0, 1)}{(\sqrt{5})^3} + \frac{3 \times 10^{-9} (0, 2, 1)}{(\sqrt{5})^3} + \frac{3 \times 10^{-9} (1, 1, 0)}{(\sqrt{2})^3} \right)

Simplifying the expression, we get:

\mathbf{E} = \frac{1}{4\pi\epsilon_0} \left( \frac{3 \times 10^{-9} (0, 0, 1)}{1} + \frac{3 \times 10^{-9} (2, 0, 1)}{(\sqrt{5})^3} + \frac{3 \times 10^{-9} (0, 2, 1)}{(\sqrt{5})^3} + \frac{3 \times 10^{-9} (1, 1, 0)}{(\sqrt{2})^3} \right)

\mathbf{E} = \frac{1}{4\pi\epsilon_0} \left( 3 \times 10^{-9} (0, 0, 1) + \frac{3 \times 10^{-9} (2, 0, 1)}{(\sqrt{5})^3} + \frac{3 \times 10^{-9} (0, 2, 1)}{(\sqrt{5})^3} + \frac{3 \times 10^{-9} (1, 1, 0)}{(\sqrt{2})^3} \right)

\mathbf{E} = \frac{1}{4\pi\epsilon_0} \left( 3 \times 10^{-9} (0, 0, 1) + \frac{3 \times 10^{-9} (2, 0, 1)}{5\sqrt{5}} + \frac{3 \times 10^{-9} (0, 2, 1)}{5\sqrt{5}} + \frac{3 \times 10^{-9} (1<br/> # Find the Electric Field Intensity Produced by a Point Charge Distribution at $P(1,1,1)$: Q&A

Introduction

In our previous article, we explored how to find the electric field intensity produced by a point charge distribution at a given point $P(1,1,1)$ . We considered a distribution of four identical point charges located at specific points in space. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the electric field intensity?

A: The electric field intensity, also known as the electric field strength, is a measure of the force per unit charge that a point charge would experience at a given point in space. It is a vector quantity, denoted by the symbol $\mathbf{E}$ , and is typically measured in units of newtons per coulomb (N/C).

Q: How do I calculate the electric field intensity?

A: To calculate the electric field intensity produced by a point charge distribution, you can use the following formula:

\mathbf{E} = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{N} \frac{q_i \mathbf{r}_{iP}}{|\mathbf{r}_{iP}|^3} </span></p> <p>where:</p> <ul> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">E</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">E</span></span></span></span> is the electric field intensity at point <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> is the number of point charges in the distribution</li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the charge of the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>th point charge</li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mrow><mi>i</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{r}_{iP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5944em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the vector from the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>th point charge to point <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ϵ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\epsilon_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">ϵ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the electric constant (also known as the permittivity of free space)</li> </ul> <h2>Q: What is the significance of the electric field intensity?</h2> <p>A: The electric field intensity is a fundamental concept in physics, particularly in the study of electromagnetism. It is used to describe the force that a point charge would experience at a given point in space. The electric field intensity is also used to calculate the force on a charged particle, such as an electron or a proton.</p> <h2>Q: How do I apply the formula for electric field intensity?</h2> <p>A: To apply the formula for electric field intensity, you need to follow these steps:</p> <ol> <li>Identify the point charges in the distribution and their locations.</li> <li>Calculate the vector from each point charge to the point of interest.</li> <li>Calculate the magnitude of each vector.</li> <li>Substitute the values into the formula for electric field intensity.</li> <li>Simplify the expression to obtain the final answer.</li> </ol> <h2>Q: What are some common applications of electric field intensity?</h2> <p>A: Electric field intensity has many practical applications in fields such as:</p> <ul> <li>Electrical engineering: Electric field intensity is used to design and optimize electrical systems, such as power transmission lines and electrical circuits.</li> <li>Physics: Electric field intensity is used to study the behavior of charged particles, such as electrons and protons.</li> <li>Materials science: Electric field intensity is used to study the properties of materials, such as their electrical conductivity and dielectric properties.</li> </ul> <h2>Q: How do I troubleshoot common errors in electric field intensity calculations?</h2> <p>A: Some common errors in electric field intensity calculations include:</p> <ul> <li>Incorrectly calculating the vector from a point charge to the point of interest.</li> <li>Incorrectly calculating the magnitude of a vector.</li> <li>Incorrectly substituting values into the formula for electric field intensity.</li> <li>Failing to simplify the expression to obtain the final answer.</li> </ul> <p>To troubleshoot these errors, you can:</p> <ul> <li>Double-check your calculations for each step.</li> <li>Use a calculator or computer program to verify your calculations.</li> <li>Consult a reference book or online resource for guidance.</li> <li>Practice calculating electric field intensity for different scenarios to build your skills and confidence.</li> </ul> <h2>Conclusion</h2> <p>In this article, we have answered some of the most frequently asked questions related to electric field intensity. We have also provided a step-by-step guide on how to calculate electric field intensity and troubleshoot common errors. By following these guidelines, you can confidently apply the formula for electric field intensity and solve problems in physics and electrical engineering.</p> <h2>Additional Resources</h2> <p>For further reading and practice, we recommend the following resources:</p> <ul> <li>&quot;Electric Field Intensity&quot; by Wikipedia</li> <li>&quot;Electric Field&quot; by HyperPhysics</li> <li>&quot;Electric Field Intensity&quot; by Physics Classroom</li> <li>&quot;Electric Field&quot; by Khan Academy</li> </ul> <p>We hope this article has been helpful in your understanding of electric field intensity. If you have any further questions or need additional guidance, please don't hesitate to ask.</p>