Question

The capacitors in each circuit are fully charged before the switch is closed. Rank, from longest to shortest, the length of time the bulbs (resistors) stay lit in each circuit.

Answer 1

Concepts and reason

The concept used to solve this problem is circuit analysis using the simplification of resistors and capacitors.

Initially, calculate the time constant for each of the circuits in figure A, figure B, figure C, figure D and figure E. Finally, rank the length of time the bulbs stay lit from longest to shortest by using the value of time constant for each circuit.

Fundamentals

The expression to calculate equivalent resistance for series combination of resistor is,

${R_s} = {R_1} + {R_2}$

Here, ${R_s}$ is the equivalent resistance in series and ${R_1}$ , ${R_2}$ are the resistances of individual resistors connected in series.

The expression to calculate equivalent resistance for parallel combination of resistor is,

${R_p} = \frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}$

Here, ${R_p}$ is the equivalent resistor in parallel.

The expression to calculate equivalent capacitance for parallel combination of capacitors is,

${C_p} = {C_1} + {C_2}$

Here, ${C_p}$ is the equivalent capacitance in parallel and ${C_1}$ , ${C_2}$ are the capacitances of individual capacitors connected in parallel.

The expression to calculate equivalent capacitance for series combination of capacitor is,

${C_s} = \frac{{{C_1}{C_2}}}{{{C_1} + {C_2}}}$

Here, ${C_s}$ is the equivalent capacitance of capacitors connected in series.

The expression for time constant in RC,

$\tau = RC$

Here, $\tau$ is the time constant, R is the resistance, and C is the capacitance.

The figure A has one resistance and one capacitor connected in series.

The expression for time constant in RC for figure A is,

$\tau = RC$

The circuit has two capacitors in series with one resistor.

The expression for series combination of capacitor is,

${C_s} = \frac{{{C_1}{C_2}}}{{{C_1} + {C_2}}}$

Substitute $C$ for ${C_1}$ and ${C_2}$ to find ${C_s}$ .

$\begin{array}{c}\\{C_s} = \frac{{CC}}{{C + C}}\\\\ = \frac{{{C^2}}}{{2C}}\\\\ = \frac{C}{2}\\\end{array}$

The expression for time constant is,

$\tau = RC$

Substitute ${\rm{C/2}}$ for C to find $\tau$ .

$\begin{array}{c}\\\tau = R\left( {\frac{C}{2}} \right)\\\\ = \frac{1}{2}RC\\\end{array}$

The figure C has two capacitors connected in parallel with one resistance in series.

The expression for parallel combination of capacitor is,

${C_p} = {C_1} + {C_2}$

Substitute $C$ for ${C_1}$ and ${C_2}$ to find ${C_p}$ .

$\begin{array}{c}\\{C_P} = C + C\\\\ = 2C\\\end{array}$

The expression for time constant is,

$\tau = RC$

Substitute $2{\rm{C}}$ for C to find $\tau$ .

$\begin{array}{c}\\\tau = R\left( {2C} \right)\\\\ = 2RC\\\end{array}$

The circuit has two capacitors in series with two resistances in parallel.

The expression for series combination of capacitor is,

${C_s} = \frac{{{C_1}{C_2}}}{{{C_1} + {C_2}}}$

Substitute $C$ for ${C_1}$ ${C_2}$ to find ${C_s}$ .

$\begin{array}{c}\\{C_s} = \frac{{CC}}{{C + C}}\\\\ = \frac{{{C^2}}}{{2C}}\\\\ = \frac{C}{2}\\\end{array}$

The circuit has two capacitors in series with one resistance in parallel.

The expression for parallel combination of resistor is,

${R_p} = \frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}$

Substitute R for ${R_1}$ , ${R_2}$ to find ${R_p}$ .

$\begin{array}{c}\\{R_p} = \frac{{RR}}{{R + R}}\\\\ = \frac{{{R^2}}}{{2R}}\\\\ = \frac{R}{2}\\\end{array}$

The expression for time constant is,

$\tau = RC$

Substitute ${\rm{C/2}}$ for C and $R/2$ for R to find $\tau$ .

$\begin{array}{c}\\\tau = \frac{R}{2}\left( {\frac{C}{2}} \right)\\\\ = \frac{{RC}}{4}\\\end{array}$

The circuit has two capacitors in parallel and two resistors in parallel.

The expression for parallel combination of capacitor is,

${C_p} = {C_1} + {C_2}$

Substitute $C$ for ${C_1}$ , ${C_2}$ to find ${C_p}$ .

$\begin{array}{c}\\{C_P} = C + C\\\\ = 2C\\\end{array}$

The expression for parallel combination of resistor is,

${R_p} = \frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}$

Here, ${R_p}$ is the equivalent resistor in parallel and ${R_1}$ ${R_2}$ is the resistance.

Substitute R for ${R_1}$ and ${R_2}$ to find ${R_p}$ .

$\begin{array}{c}\\{R_s} = \frac{{RR}}{{R + C}}\\\\ = \frac{{{R^2}}}{{2R}}\\\\ = \frac{R}{2}\\\end{array}$

The expression for time constant is,

$\tau = RC$

Substitute ${\rm{2C}}$ for C and $R/2$ for R and solve for $\tau$ .

$\begin{array}{c}\\\tau = 2R\left( {\frac{C}{2}} \right)\\\\ = RC\\\end{array}$

Ans:

Rank of the time constant of the circuits is $C > A = E > B > D$