Final answer:
To increase the separation between the plates of a parallel-plate capacitor by a factor of 3, you can use the formula W = (1/2)CV'^2 - (1/2)CV^2, where W is the work done, C is the capacitance, V' is the new voltage, and V is the original voltage. This formula is derived from the relationship between capacitance, voltage, and charge on the plates.
Step-by-step explanation:
A parallel-plate capacitor is charged to a voltage V with a certain plate separation d. To increase the separation between the plates to a factor of 3, we need to find the new separation d' and the new voltage V'. The capacitance of the capacitor, denoted by C, is given by C = υA/d, where υ is the permittivity of free space, A is the area of the plates, and d is the original separation. Since the capacitance is defined as the charge on each plate divided by the voltage, we have C = Q/V. Rearranging the equation, we get Q = CV. In this case, the charge Q is given by Q = CV and the capacitance C is given by C = υA/d. We can now rearrange the equations to solve for the new separation d': d' = υA/(Q/V'). Substituting the known values, we get d' = υA/(CV'). Therefore, the minimum work required to increase the separation between the plates by a factor of 3 can be calculated using the equation W = (1/2)CV'^2 - (1/2)CV^2.