Calculate the capa atance of two capaotos consivitin of two co.adions cylinder of inner iadius acin and puter radius 4 mathrm(~cm) . The spaco writhin the Ginder is filled with a substance of electrio. constand of 2. and the longth of the cylinder

To calculate the capacitance of the given system, we need to find the capacitance of each capacitor and then add them together.<br /><br />The capacitance of a capacitor is given by the formula:<br /><br />\[ C = \frac{\varepsilon \cdot A}{d} \]<br /><br />where:<br />- \( C \) is the capacitance,<br />- \( \varepsilon \) is the permittivity of the substance,<br />- \( A \) is the area of the plates,<br />- \( d \) is the distance between the plates.<br /><br />In this case, we have two capacitors. The first capacitor is a solid cylinder with an inner radius of \( a \) cm and a outer radius of \( b \) cm. The second capacitor is a hollow cylinder with an inner radius of \( a \) cm and an outer radius of \( b \) cm.<br /><br />For the solid cylinder, the area of the plates is given by \( A = 2 \pi a h \), where \( h \) is the height of the cylinder. The distance between the plates is equal to the height of the cylinder, \( h \).<br /><br />For the hollow cylinder, the area of the plates is given by \( A = 2 \pi (b^2 - a^2) h \). The distance between the plates is equal to the height of the cylinder, \( h \).<br /><br />Substituting the given values into the formula, we can calculate the capacitance of each capacitor:<br /><br />For the solid cylinder:<br />\[ C_1 = \frac{\varepsilon \cdot 2 \pi a h}{h} = 2 \pi a \varepsilon \]<br /><br />For the hollow cylinder:<br />\[ C_2 = \frac{\varepsilon \cdot 2 \pi (b^2 - a^2) h}{h} = 2 \pi (b^2 - a^2) \varepsilon \]<br /><br />Finally, we add the capacitances of the two capacitors to get the total capacitance:<br /><br />\[ C_{\text{total}} = C_1 + C_2 = 2 \pi a \varepsilon + 2 \pi (b^2 - a^2) \varepsilon \]<br /><br />Simplifying the expression, we get:<br /><br />\[ C_{\text{total}} = 2 \pi \varepsilon (a + b^2 - a^2) \]<br /><br />Therefore, the capacitance of the given system is \( 2 \pi \varepsilon (a + b^2 - a^2) \).