question_answer

Three identical capacitors are given a charge Q each and they are then allowed to discharge through resistance R1, R2 and R3. Their charges, as a function of time shown in the graph below. The smallest of the three resistance is 

A)            \[{{R}_{\text{3}}}\]           

B)            \[{{R}_{\text{2}}}\]

C)            \[{{R}_{\text{1}}}\]

D)            Cannot be predicted

Correct Answer: C

Solution :

           During the discharge of a capacitor through a resistance charge at any instant \[Q={{Q}_{0}}{{e}^{-t/CR}}\] Þ \[\frac{{{Q}_{0}}}{Q}={{e}^{t/CR}}\] Þ \[t=CR{{\log }_{e}}\frac{{{Q}_{0}}}{Q}\] If Q ® constant, then t µ R Now, draw a line parallel to the time axis as shown. Suppose this line cut the graphs at points 1, 2 and 3. Corresponding time are \[{{t}_{\text{1}}}\], \[{{t}_{\text{2}}}\] and \[{{t}_{\text{3}}}\] respectively. Hence from graph \[{{t}_{\text{1}}}<{{t}_{\text{2}}}<{{t}_{\text{3}}}\] Þ \[{{R}_{\text{1}}}<{{R}_{\text{2}}}<{{R}_{\text{3}}}\]