23.A parallel plate capacitor is connected to a battery for a long time and stores 5 J of energy. The separation distance between the plates is d. The battery is then disconnected from the capacitor.Which of the following is equal to the work required to move the plates to separation distance 2d ?

# Explanation:<br />## Step 1<br />The problem involves a parallel plate capacitor that is initially connected to a battery for a long time, storing 5 J of energy. The battery is then disconnected from the capacitor. We are asked to find the work required to move the plates to a separation distance of 2d.<br /><br />## Step 2<br />The energy stored in a capacitor is given by the formula:<br />### \(E = \frac{1}{2}Cv^2\)<br />where \(E\) is the energy, \(C\) is the capacitance, and \(v\) is the voltage.<br /><br />## Step 3<br />The capacitance \(C\) of a parallel plate capacitor is given by the formula:<br />### \(C = \frac{\varepsilon_0A}{d}\)<br />where \(A\) is the area of the plates, \(d\) is the distance between the plates, and \(\varepsilon_0\) is the permittivity of free space.<br /><br />## Step 4<br />The voltage \(v\) across the capacitor is given by the formula:<br />### \(v = \frac{E}{C}\)<br /><br />## Step 5<br />The work done to move the plates to a separation distance of 2d is given by the formula:<br />### \(W = \frac{1}{2}Cv^2\)<br /><br />## Step 6<br />Substituting the values of \(C\) and \(v\) from steps 3 and 4 into the formula from step 5, we get:<br />### \(W = \frac{1}{2} \cdot \frac{\varepsilon_0A}{d} \cdot \left(\frac{E}{\frac{\varepsilon_0A}{d}}\right)^2\)<br /><br />## Step 7<br />Simplifying the above expression, we get:<br />### \(W = \frac{1}{2} \cdot \frac{\varepsilon_0A}{d} \cdot \frac{E^2}{\varepsilon_0^2A^2/d^2}\)<br /><br />## Step 8<br />Further simplifying, we get:<br />### \(W = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d}\)<br /><br />## Step 9<br />Finally, simplifying the above expression, we get:<br />### \(W = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2}{\varepsilon_0A/d} = \frac{1}{2} \cdot \frac{E^2