The mean electrical energy density between plates of a charged air capacitor is (where $\mathrm{q}=$ charge…

The mean electrical energy density between plates of a charged air capacitor is (where $\mathrm{q}=$ charge on capacitor, $\mathrm{A}=$ Area of capacitor plate)
  1. $\frac{\mathrm{q}^2}{2 \varepsilon_0 \mathrm{~A}^2}$
  2. $\frac{\mathrm{q}}{2 \varepsilon_0 \mathrm{~A}^2}$
  3. $\frac{\mathrm{q}^2}{2 \varepsilon_0 \mathrm{~A}}$
  4. $\frac{\varepsilon_0 A}{q^2}$

Solution

For a parallel plate capacitor, the energy density $=\frac{1}{2} E^2 \varepsilon_0$ But $\mathrm{E}=\frac{\sigma}{\varepsilon_0}$ $\begin{aligned} \therefore \quad \text { Energy density } & =\frac{1}{2} \frac{\sigma^2}{\varepsilon_0^2} \times \varepsilon_0 \\ & =\frac{\sigma^2}{2 \varepsilon_0} \\ & =\frac{\left(\frac{q}{A}\right)^2}{2 \varepsilon_0} \quad \ldots(\because \sigma=q / A) \\ & =\frac{q^2}{2 A^2 \cdot \varepsilon_0} \end{aligned}$ ~

Asked in: MHT CET 2023 (10 May Shift 1)

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