Practicing Success
Four large metal plates are located a small distance apart from one another as shown in the Figure. The extreme plates are connected by means of a conductor, while a potential difference V is applied to the internal plates. Find the electric fields between the neighbouring plates. |
$E_{23}=\frac{V}{d}, E_{34}=\frac{V}{2 d}$ $E_{12}=\frac{V}{2 d}, E_{23}=E_{34}=\frac{V}{2}$ zero None of these |
$E_{23}=\frac{V}{d}, E_{34}=\frac{V}{2 d}$ |
We may take the plates to be connected as shown in the Figure. Let E23 be the electric field between plates 2 and 3. Then from the fact that electric field = rate of change of potential, $E_{22}=\frac{V}{d}$. Applying `loop rule' to the loop `A – 1 – 2 – 3 – 4 – B – A' V' – V – + V' ⇒ V' = $\frac{V}{2}$ where V' is the numerical value of the potential difference between plates 1 and 2 or 3 and 4 ∴ $E_{12}$ or $E_{34} = \frac{V}{2 d}$ $E = \frac{\sigma}{\varepsilon_0}$ ∴ $\sigma_1$ (density of charge on the plate 1) $=\varepsilon_0 E_{12}=\varepsilon_0 \frac{V}{2 d}=\sigma_4$ (numerically) The surface density of charge of the right surface of plate 2 is $\sigma'=\varepsilon_0, E_{23}=\varepsilon_0 V d$ ∴ $\sigma_2$ (surface density of charge on plate 2) $=\frac{\varepsilon_0 V}{2 d}+\frac{\varepsilon_0 V}{d}=\frac{3}{2} \frac{\varepsilon_0 V}{d}$ Thus (i) $E_{23}=\frac{V}{d} E_{12}$ or $E_{34}=\frac{V}{2 d}$ (ii) $\sigma_1=\sigma_4=\varepsilon_0 \frac{V}{2 d} \sigma_2=\varepsilon_0=\frac{3}{2} \varepsilon_0 \frac{V}{d}$. |