Respuesta :
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 [tex]\boxed{ \ \kappa = \varepsilon_r = 2.6 \ }[/tex]
- Dielectric strength is [tex]\boxed{ \ 2.0 \times 10^7 \ V/m \ }[/tex]
- The electric field between the plates is 75 % of the dielectric strength.
Question:
The energy density [tex]\boxed{ \ u \ } \ in \ Joule/m^3[/tex]
The Process:
The expression for the energy density inside a dielectric is
[tex]\boxed{\boxed{ \ u = \frac{1}{2} \varepsilon E^2 \ }}[/tex]
Permittivity, i.e., [tex]\boxed{ \ \varepsilon = \varepsilon_r \varepsilon_o \ }[/tex] with vacuum permittivity [tex]\boxed{ \ \varepsilon_o = 8.85 \times 10^{-12} \ Fm^{-1} \ }[/tex]
Let's calculate the permittivity and the electric field.
- [tex] \boxed{ \varepsilon = 2.6 \times 8.85 \times 10^{-12}} \rightarrow \boxed{ \ \varepsilon = 2.301 \times 10^{-11} \ } [/tex]
- [tex] \boxed{ \ E = 75 \% \times 2.0 \times 10^7 \ V/m \ } \rightarrow \boxed{ \ E = 1.5 \times 10^7 \ V/m \ } [/tex]
Both data are substituted into the formula to calculate the energy density of stored energy.
[tex]\boxed{ \ u = \frac{1}{2} (2.301 \times 10^{-11}) (1.5 \times 10^7)^2 \ }[/tex]
Hence, we get [tex]\boxed{\boxed{ \ u = 2,589 \ J.m^{-3} \ }}[/tex]
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Keywords: polystyrene, dielectric constant, strength, a parallel-plate capacitor, electric field, energy density, the stored energy, relative permittivity, vacuum
The energy density of stored energy within the capacitor is [tex]\boxed{2588.625{\text{ }}{{\text{J}} \mathord{\left/ {\vphantom {{\text{J}} {{{\text{m}}^3}}}} \right. \kern-\nulldelimiterspace} {{{\text{m}}^3}}}}[/tex].
Further explanation:
The energy density is defined as the energy stored in the capacitor per unit volume.
Given:
The dielectric constant of polystyrene is [tex]2.6[/tex].
The dielectric strength of polystyrene is [tex]2 \times {10^7}{\text{ }}{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}[/tex].
The electric field between the plates of capacitor is [tex]75\%[/tex] of the dielectric strength of polystyrene.
Formula and concept used:
The expression for the relative permittivity of a medium is:
[tex]K=\dfrac{\varepsilon }{{{\varepsilon _0}}}[/tex]
By simplifying the above equation we can calculate the permittivity of polystyrene.
The permittivity of the polystyrene is:
[tex]\boxed{\varepsilon=K{\varepsilon _0}}[/tex] …… (1)
Here, [tex]K[/tex] is the dielectric strength of polystyrene , [tex]\varepsilon[/tex] is the electric permittivity of polystyrene and [tex]{\varepsilon _0}[/tex] 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:
[tex]u=\dfrac{1}{2}\varepsilon {E^2}[/tex]
Substitute the value of [tex]\varepsilon[/tex] from equation (1) in the above equation.
[tex]\boxed{u=\frac{1}{2}K{\varepsilon _0}{E^2}}[/tex] …… (2)
Calculation:
The electric field is the [tex]75\%[/tex] of the dielectric strength.
The value of electric field is:
[tex]\boxed{E=1.5\times {{10}^7}{\text{ }}{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}}[/tex]
Substitute the value of [tex]K[/tex] as [tex]2.6[/tex], value of [tex]{\varepsilon _0}[/tex] as [tex]8.85 \times {10^{ - 12}}{{{\text{ F}}} \mathord{\left/ {\vphantom {{{\text{ F}}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}[/tex] and value of [tex]E[/tex] as [tex]1.5 \times {10^7}{{\text{V}} \mathord{\left/ {\vphantom {{\text{V}} {\text{m}}}} \right. \kern-\nulldelimiterspace} {\text{m}}}[/tex] in equation (2).
[tex]\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}[/tex]
Thus, the energy density of stored energy is [tex]\boxed{2588.625{\text{ }}{{\text{J}} \mathord{\left/ {\vphantom {{\text{J}} {{{\text{m}}^3}}}} \right. \kern-\nulldelimiterspace} {{{\text{m}}^3}}}}[/tex].
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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.
