Question

A 5.00-pf parallel-plate air-filled capacitor with circular plates is to be used in a circuit in which it will be subjected to potentials of up to 1.00×102v. the electric field between the plates is to be no greater than 1.00×104n/c. as a budding electrical engineer for live-wire electronics, your task is to design the capacitor by finding what its physical dimensions and separation must be.

Answer 1

The surface area of each plate of the capacitor is $\boxed{5.65 \times {{10}^{ - 3}}\,{{\text{m}}^2}}$ and the separation between the plates is $\boxed{0.01\,{\text{m}}}$ or $\boxed{10\,{\text{mm}}}$ or $\boxed{1\,{\text{cm}}}$.

Further Explanation:

A capacitor is a device which is used to store electric energy in the electric field. The capacitance of the capacitor is the ability of the capacitor to store the charge on it.

Given:

The capacitance $\left( C \right)$ of the capacitor is $5\,{\text{pF}}\,$.  

The potential difference across the plates of the capacitor $\left( V \right)$ is ${10^2}\,{\text{V}}$.  

The electric field between the plates of the capacitor $\left( E \right)$ is ${10^4}\,{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}$.

Formula and Concept used:  

The expression for electric field intensity within the capacitor is:

$E=\dfrac{V}{d}$

Simplify the above equation for $d$.  

$\boxed{d=\dfrac{V}{E}}$                                                                    …… (1)  

Here, $E$ is the electric field intensity, $V$ is the potential difference across the plates of the capacitor and $d$ is the separation between the parallel plate capacitor.

The expression for the capacitance of the parallel plate capacitor is:

$C=\dfrac{{{\varepsilon _0}A}}{d}$

Simplify the above equation for area of the plate $A$.  

$\boxed{A=\dfrac{{Cd}}{{{\varepsilon _0}}}}$                                          …… (2)  

Here, $C$ is the capacitance of the capacitor , ${\varepsilon _0}$ is the permittivity of the free space or air and $A$ is the area of the each parallel plate in the capacitor.  

Calculation:

Substitute the value of $V$ as ${10^2}\,{\text{V}}$, ${10^4}\,{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}$ for $E$ in the equation (1).  

$\begin{aligned}d&=\frac{{{{10}^2}}}{{{{10}^4}}} \hfill \\&=\frac{1}{{100}} \hfill \\&=0.01\,{\text{m}} \hfill \\ \end{aligned}$

Separation between the plates in $\text{cm}$ and $\text{mm}$.  

$\begin{aligned}d&=0.01 \times 100 \hfill \\&=1\text{ cm}\times10\\&=10\,{\text{mm}} \hfill \\ \end{aligned}$

Substitute the value of $C$ as $5\,{\text{pF}}\,$, $d$ as $0.01\,{\text{m}}$ and ${\varepsilon _0}$ as $8.85 \times {10^{ - 12}}\,{{\text{F}} \mathord{\left/ {\vphantom {{\text{F}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}$ in the equation (2).

$\begin{aligned}A&=\frac{{\left( {5 \times {{10}^{ - 12}}} \right)\left( {0.01} \right)}}{{\left( {8.85 \times {{10}^{ - 12}}} \right)}} \\&=5.65 \times {10^{ - 3}}\,{{\text{m}}^2} \\ \end{aligned}$

Thus, the surface area of each plate of the capacitor is $\boxed{5.65 \times {{10}^{ - 3}}\,{{\text{m}}^2}}$ and the separation between the plates is $\boxed{0.01\,{\text{m}}}$ or $\boxed{10\,{\text{mm}}}$ or $\boxed{1\,{\text{cm}}}$.  

Learn more:

1. The component of the lever: brainly.com/question/1073452  

2. Net force on helicopter: brainly.com/question/6125929

3. The energy density: brainly.com/question/9617400

Answer detail:

Grade: High School  

Subject: Physics  

Chapter: Current Electricity  

Keywords:

Parallel plate capacitor, electric field between the plate,separation between the plates, area of the plates, 10mm, 1cm, 0.01 m, 5.65X10^-13m^2, 5.65X10^-13m2, 5.65X10-13m^2, 5.65X10-13m2.

Answer 2

A 5 pF parallel-plate air-filled capacitor, with a potential difference equal to 100 Volts and an electric field of 10.000 Volts per meter, must have a plate separation of 10 milimeters, and plate's area equal to 0.0056 meters squared.

Further explanation

Capacitors are electrical components use to store electrical energy in the form of potential difference. This is done by putting 2 plates of a conductive material very close to each other, but without getting them into contact. The idea is that, when you charge one plate with (for example) positive charges, the other plate will "feel" this and will charge itself with negative charges (due to the attraction between charges of opposite signs).

The trick is that, even though the charges in the two plates attract each other, they can't come into contact since the 2 plates are separated by a certain distance. Now let's suppose there is only air between the capacitor's plates, we know that air is not a conductive material, however if you charge both plates just enough, you'll be able to make air conduct electricity and so charges from one plate will go to the other and equilibrium will be achieved.

This is the same thing that happens when a lightning strikes the ground. Imagine that the Earth is a gigantic capacitor, with the clouds being one plate and the ground being the other plate. When the clouds are charged enough, air is forced to conduct electricity towards the ground (which is the lightning we see), until equilibrium is achieved. This is the reason why, in Physics, we say that everything is a conductive material if you force it just enough.

Back to our problem... We can compute the separation of the capacitor's plates with the following equation:

$V = E\cdot d$

Which says that the Voltage V is equal to the product of the Electric field E and the distance between the plates d. This equation is valid if the electric field across the capacitor is constant (which for small magnitudes of d it is). So solving for the distance we get:

$d= \frac{V}{E} = \frac{100}{10000} \frac{V}{V/m} = 0.01 m = 10 mm$

Where we can see that the separation of plates is 10 millimeters. Now for the area between the plates we can use this other formula:

$C= \frac{\epsilon \cdot A}{d}$

Where $\epsilon$ is the air's permittivity which has a value of around $8.86 \cdot 10^{-12} \frac{F}{m}$. By solving for the plate's area, we can find:

$A= \frac{C \cdot d}{\epsilon} = 0.0056 m^2 =$

Which corresponds to a plate's radius of 42.4 millimeters.

Learn more

• What is electricity: brainly.com/question/9194793
• Relation between elementary particles and charge: brainly.com/question/12502147
• Another capacitor problem: brainly.com/question/3203280

Keywords

Capacitor, Electric charge, Electric field, Voltage