A parallel plate capacitor is made of two plates of length $l$, width $w$ and separated by distance $d$. A dielectric slab ( dielectric constant $K$) that fits exactly between the plates is held near the edge of the plates. It is pulled into the capacitor by a force $F = -\frac{{\partial U}}{{\partial x}}$ where $U$ is the energy of the capacitor when dielectric is inside the capacitor up to distance $x$ (See figure). If the charge on the capacitor is $Q$ then the force on the dielectric when it is near the edge is
JEE MAIN 2014, Diffcult
Download our app for free and get started
The electric force on the slab close to the edge is
$F=-\frac{\delta U}{\delta x}$
The energy stored in the capacitor is
$U=\frac{1}{2} \frac{Q^{2}}{C}$
The capacitance in the case shown in the figure is
$C = {C_1} + {C_2} = \frac{{{\varepsilon _0}wxK}}{d} + \frac{{{\varepsilon _0}w(l - x)}}{d}$
$=\varepsilon_{\circ} w \frac{x(K-1)+l}{d}$
$U=\frac{1}{2} \frac{Q^{2}}{C}$
$F=-(\delta U) /(\delta x)=-\frac{Q^{2} d}{2 \varepsilon_{\mathrm{o}} w} \frac{\delta}{\delta x}\left(\frac{1}{x(K-1)+l}\right)$
$F=\frac{Q^{2} d}{2 \varepsilon_{\mathrm{o}} w} \cdot \frac{K-1}{(x(K-1)+l)^{2}}$
At $x=0$ (edge):
$F=\frac{Q^{2} d(K-1)}{2 \varepsilon_{0} w l^{2}}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1Two insulated metallic spheres of $3\,\mu F$ and $5\,\mu F$ capacitances are charged to $300\, V$ and $500\,V$ respectively. The energy loss, when they are connected by a wire isView Solution
- 2A solid conducting sphere of radius $R_1$ is surrounded by another concentric hollow conducting sphere of radius $R_2$. The capacitance of this assembly is proportional toView Solution
- 3View SolutionThe potential at a point due to an electric dipole will be maximum and minimum when the angles between the axis of the dipole and the line joining the point to the dipole are respectively
- 4A resistance $R$ and a capacitance $C$ are connected in series to a battery of negligible internal resistance through a key. The key is closed at $t = 0$. If after $t\, sec$ the voltage across the capacitance was seven times the voltage across $R$, the value of $t$ isView Solution
- 5Four equal charges $Q$ are placed at the four corners of a square of each side is $'a'$. Work done in removing a charge $-Q$ from its centre to infinity isView Solution
- 6A circuit shown in the figure consists of a battery of $emf$ $10$ $V$ and two capacitance $C_1$ and $C_2 $ of capacitances $1.0$ $ \mu F$ and $2.0$ $\mu F$ respectively. The potential difference $V_A - V_B$ is $5\,V$View Solution
- 7Four capacitors of each of capacity $3\,\mu F$ are connected as shown in the adjoining figure. The ratio of equivalent capacitance between $A$ and $B$ and between $A$ and $C$ will beView Solution
- 8The capacitance of a parallel plate capacitor with air as medium is $6\, \mu F$. With the introduction of a dielectric medium, the capacitance becomes $30\, \mu F$. The permittivity of the medium is..........$C ^{2} N ^{-1} m ^{-2}$View Solution
$\left(\varepsilon_{0}=8.85 \times 10^{-12} C ^{2} N ^{-1} m ^{-2}\right)$
- 9The expression for the capacity of the capacitor formed by compound dielectric placed between the plates of a parallel plate capacitor as shown in figure, will be (area of plate $ = A$)View Solution
- 10Three plates $A,\;B,\;C$each of area $50\,c{m^2}$ have separation $3\,mm$ between $A$ and $B$ and $3\,mm$ between $B$ and $C$. The energy stored when the plates are fully charged isView Solution
