Question

Polystyrene has dielectric constant 2.6 and dielectric strength 2.0 × 107 v/m. a piece of polystyrene is used as a dielectric in a parallel-plate capacitor, filling the volume between the plates. part a when the electric field between the plates is 75 % of the dielectric strength, what is the energy density of the stored energy?

Answer 1

2,589 J/m³

Further explanation

This is a matter of calculating the energy density of stored energy in a parallel-plate capacitor.

Given:

• Dielectric constant (or relative permittivity) of polystyrene is $\boxed{ \ \kappa = \varepsilon_r = 2.6 \ }$
• Dielectric strength is $\boxed{ \ 2.0 \times 10^7 \ V/m \ }$
• The electric field between the plates is 75 % of the dielectric strength.

Question:

The energy density $\boxed{ \ u \ } \ in \ Joule/m^3$

The Process:

The expression for the energy density inside a dielectric  is

$\boxed{\boxed{ \ u = \frac{1}{2} \varepsilon E^2 \ }}$

Permittivity, i.e., $\boxed{ \ \varepsilon = \varepsilon_r \varepsilon_o \ }$ with vacuum permittivity $\boxed{ \ \varepsilon_o = 8.85 \times 10^{-12} \ Fm^{-1} \ }$

Let's calculate the permittivity and the electric field.

• $\boxed{ \varepsilon = 2.6 \times 8.85 \times 10^{-12}} \rightarrow \boxed{ \ \varepsilon = 2.301 \times 10^{-11} \ }$
• $\boxed{ \ E = 75 \% \times 2.0 \times 10^7 \ V/m \ } \rightarrow \boxed{ \ E = 1.5 \times 10^7 \ V/m \ }$

Both data are substituted into the formula to calculate the energy density of stored energy.

$\boxed{ \ u = \frac{1}{2} (2.301 \times 10^{-11}) (1.5 \times 10^7)^2 \ }$

Hence, we get $\boxed{\boxed{ \ u = 2,589 \ J.m^{-3} \ }}$

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Keywords: polystyrene, dielectric constant, strength, a parallel-plate capacitor, electric field, energy density, the stored energy, relative permittivity, vacuum

Answer 2

The energy density of stored energy within the capacitor is $\boxed{2588.625{\text{ }}{{\text{J}} \mathord{\left/ {\vphantom {{\text{J}} {{{\text{m}}^3}}}} \right. \kern-\nulldelimiterspace} {{{\text{m}}^3}}}}$.

Further explanation:

The energy density is defined as the energy stored in the capacitor per unit volume.

Given:  

The dielectric constant of polystyrene is $2.6$.  

The dielectric strength of polystyrene is $2 \times {10^7}{\text{ }}{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}$.  

The electric field between the plates of capacitor is $75\%$ of the dielectric strength of polystyrene.

Formula and concept used:

The expression for the relative permittivity of a medium is:

$K=\dfrac{\varepsilon }{{{\varepsilon _0}}}$

By simplifying the above equation we can calculate the permittivity of polystyrene.

The permittivity of the polystyrene is:

$\boxed{\varepsilon=K{\varepsilon _0}}$ …… (1)  

Here, $K$ is the dielectric strength of polystyrene , $\varepsilon$ is the electric permittivity of polystyrene and ${\varepsilon _0}$ is the electric permittivity of free space or air.

The energy density of the stored energy in the parallel plate capacitor can be expressed as:

$u=\dfrac{1}{2}\varepsilon {E^2}$

Substitute the value of $\varepsilon$ from equation (1) in the above equation.  

$\boxed{u=\frac{1}{2}K{\varepsilon _0}{E^2}}$                              …… (2)

Calculation:

The electric field is the $75\%$ of the dielectric strength.

The value of electric field is:

$\boxed{E=1.5\times {{10}^7}{\text{ }}{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}}$

Substitute the value of $K$ as $2.6$, value of ${\varepsilon _0}$ as $8.85 \times {10^{ - 12}}{{{\text{ F}}} \mathord{\left/ {\vphantom {{{\text{ F}}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}$ and value of $E$ as $1.5 \times {10^7}{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}$ in equation (2).  

$\begin{aligned}u&=\frac{1}{2}\left( {2.6} \right)\left( {8.85 \times {{10}^{ - 12}}} \right){\left( {1.5 \times {{10}^7}} \right)^2} \\&=2588.625{\text{ }}{{\text{J}} \mathord{\left/ {\vphantom {{\text{J}} {{{\text{m}}^3}}}} \right. \kern-\nulldelimiterspace} {{{\text{m}}^3}}} \\ \end{aligned}$

Thus, the energy density of stored energy is $\boxed{2588.625{\text{ }}{{\text{J}} \mathord{\left/ {\vphantom {{\text{J}} {{{\text{m}}^3}}}} \right. \kern-\nulldelimiterspace} {{{\text{m}}^3}}}}$.

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1. Change in momentum due to its collision: brainly.com/question/9484203  

2. Type of mirror used by dentist: brainly.com/question/997618  

3. Projectile motion: brainly.com/question/161035

Answer detail:  

Grade: College  

Subject: Physics  

Chapter: Current Electricity  

Keywords:  

Polystyrene, dielectric strength, parallel plate capacitor, energy density, electric field between the plates , relative permitivitty, volume, energy, 2588.625 J/m3, 2589 J/m3, 2588.625 J/m^3, 2589 J/m^3.