91Ó°ÊÓ

Understanding the capacitance of a spherical capacitor is essential for analyzing its electrical properties. Capacitance is a measure of a capacitor's ability to store charge per unit of electric potential difference between its plates. In the case of a spherical capacitor, which consists of two concentric spherical conductors (an inner and an outer shell), the calculation hinges on the formula \[\begin{equation}C = \frac{4 \.pi \epsilon_0}{(1/a) - (1/b)}\end{equation}\],
where \(a\) and \(b\) are the radii of the inner and outer spheres, respectively, and \(\epsilon_0\) is the permittivity of free space. This equation takes advantage of the geometry of a sphere and the uniform electric field between the two shells. To perform this calculation, one must first convert any measurements into the SI unit of meters to ensure consistency and accuracy. For instance, the radii provided in our problem need to be converted from centimeters to meters before substituting into the equation.

The electric potential difference, often referred to as voltage \(V\), between two points is the work done per unit charge to move a charge from one point to another in an electric field. In a spherical capacitor, the potential difference between the inner and outer spheres is directly related to the charge \(Q\) on the capacitor and its capacitance \(C\) through the equation:\[\begin{equation}V = \frac{Q}{C}\end{equation}\].
This relationship indicates that for a given amount of charge, a higher capacitance will result in a lower voltage and vice versa. In the context of our problem, once the capacitance is calculated, the potential difference can be determined by rearranging the above equation and plugging in the values of the charge (converted to Coulombs) and the capacitance.

The charge \(Q\) stored in a capacitor and its capacitance \(C\) share a direct relationship expressed by the equation \(Q = C \cdot V\), where \(V\) is the potential difference across the capacitor. This equation is derived from the fundamental definition of capacitance, which is the charge per unit potential difference. In simpler terms, a capacitor with higher capacitance will hold more charge at a given voltage. For students working through textbook exercises, it's crucial to grasp that capacitance is not just a measure of how much charge a capacitor can store but also how that storage relates to the voltage across it. When dealing with problems like the one provided, always ensure to keep track of units during the calculation, as mixing units can lead to errors in computation. By understanding this fundamental relationship, solving for any one of the three variables becomes more intuitive when the other two are known.