Question

Seven identical large conducting plates are placed parallel to each other. The separation between any two consecutive plates is $a$. Area of each plate is $S$. The effective capacitance of the system between A and B is $\frac{x}{y}\frac{\epsilon_0S}{a}$. Find the value of $y$ if $x$ is a single digit integer. Then represent it as the sum of digits of $y$.

Answer 1

3

Answer 2

The system of seven parallel conducting plates connected as shown can be analyzed as a combination of parallel plate capacitors. Let the plates be numbered 1 to 7 from top to bottom. The capacitance between any two adjacent plates is $C_0 = \frac{\epsilon_0S}{a}$. Terminal A is connected to plate 1. Terminal B is connected to plate 4 and plate 7. Plates 2, 3, 5, and 6 are floating.

The capacitors $C_{12}, C_{23}, C_{34}$ are in series between plate 1 (connected to A) and plate 4 (connected to B). The equivalent capacitance of this series is $C_{series1} = \frac{1}{\frac{1}{C_0} + \frac{1}{C_0} + \frac{1}{C_0}} = \frac{C_0}{3}$. This capacitance is between A and B.

The capacitors $C_{45}, C_{56}, C_{67}$ are in series between plate 4 (connected to B) and plate 7 (connected to B). Since both ends are at the same potential B, this series combination does not contribute to the capacitance between A and B.

The effective capacitance of the system between A and B is $C_{eff} = C_{series1} = \frac{C_0}{3} = \frac{1}{3}\frac{\epsilon_0S}{a}$.

The given effective capacitance is $\frac{x}{y}\frac{\epsilon_0S}{a}$. Comparing this with the calculated capacitance, we have $\frac{x}{y} = \frac{1}{3}$.

We are given that $x$ is a single digit integer. Assuming the fraction is in simplest form, $x=1$ and $y=3$. Here $x=1$ is a single digit integer.

The value of $y$ is 3.

We need to represent the value of $y$ as the sum of its digits. The sum of digits of 3 is 3.

Answer: 3