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:## Step 1The 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.## Step 2The energy stored in a capacitor is given by the formula:### where is the energy, is the capacitance, and is the voltage.## Step 3The capacitance of a parallel plate capacitor is given by the formula:### where is the area of the plates, is the distance between the plates, and is the permittivity of free space.## Step 4The voltage across the capacitor is given by the formula:### ## Step 5The work done to move the plates to a separation distance of 2d is given by the formula:### ## Step 6Substituting the values of and from steps 3 and 4 into the formula from step 5, we get:### \(W = \frac{1}{2} \cdot \frac{\varepsilon_0A}{d} \cdot \left(\frac{E}{\frac{\varepsilon_0A}{d}}\right)^2\)## Step 7Simplifying the above expression, we get:### ## Step 8Further simplifying, we get:### ## Step 9Finally, simplifying the above expression, we get:### \(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