Paragraph: Consider a simple RCRC circuit as shown in Figure 11. Process...
Paragraph:
Consider a simple RCRC circuit as shown in Figure 11.
Process 1: In the circuit the switch SS is closed at t=0t=0 and the capacitor is fully charged to voltage V0V0 (i.e., charging continues for time T>>RCT>>RC ). In the process some dissipation (ED)(ED) occurs across the resistance RR. The amount of energy finally stored in the fully charged capacitor is ECEC.
Process 2: In a different process the voltage is first set to V03V03 and maintained for a charging time T>>RCT>>RC. Then the voltage is raised to 2V032V03 without discharging the capacitor and again maintained for a time T>>RCT>>RC. The process is repeated one more time by raising the voltage to V0V0 and the capacitor is charged to the same final voltage V0V0 as in Process 1.
These two processes are depicted in Figure 2.2.


Question:
In Process 2, total energy dissipated across the resistance EDED is:
Consider a simple RCRC circuit as shown in Figure 11.
Process 1: In the circuit the switch SS is closed at t=0t=0 and the capacitor is fully charged to voltage V0V0 (i.e., charging continues for time T>>RCT>>RC ). In the process some dissipation (ED)(ED) occurs across the resistance RR. The amount of energy finally stored in the fully charged capacitor is ECEC.
Process 2: In a different process the voltage is first set to V03V03 and maintained for a charging time T>>RCT>>RC. Then the voltage is raised to 2V032V03 without discharging the capacitor and again maintained for a time T>>RCT>>RC. The process is repeated one more time by raising the voltage to V0V0 and the capacitor is charged to the same final voltage V0V0 as in Process 1.
These two processes are depicted in Figure 2.2.


Question:
In Process 2, total energy dissipated across the resistance EDED is:
Solution:
For process (i)
Charge on capacitor =CV0 3=CV0 3
Energy stored in capacitor =12CV209=CV2018=12CV209=CV2018
Work done by battery =CV03×V3=CV209=CV03×V3=CV209
Heat loss =CV209-CV2018=CV2018=CV209−CV2018=CV2018
For process (ii)
Charge on capacitor =2CV03=2CV03
Extra charge flow through battery =CV03=CV03
Work done by battery: CV03 .2V03=2CV209CV03 .2V03=2CV209
Final energy store in capacitor: 12C(2V03)2=4CV201812C(2V03)2=4CV2018
Energy store in process 2: 4CV2018-CV2018=3CV20184CV2018−CV2018=3CV2018
Heat loss in process (ii) = work done by battery in process (ii) – energy store in capacitor process (ii)
=2CV209-3CV2018=CV2018=2CV209−3CV2018=CV2018
For process (iii)
Charge on capacitor =CV0=CV0
Extra charge flow through battery: CV0-2CV03=CV03CV0−2CV03=CV03
Work done by battery in this process: (CV03) (V0)=CV203(CV03) (V0)=CV203
Find energy store in capacitor: 12CV2012CV20
Energy stored in this process: 12CV20-4CV2018=5CV201812CV20−4CV2018=5CV2018
Heat loss in process (iii): CV203-5CV2018=CV2018CV203−5CV2018=CV2018
Now total heat loss (ED): CV2018+CV2018+CV2018=CV206(ED): CV2018+CV2018+CV2018=CV206
Final energy store in capacitor: 12CV2012CV20
So we can say that ED=13(12CV20)ED=13(12CV20)
Charge on capacitor =CV0 3=CV0 3
Energy stored in capacitor =12CV209=CV2018=12CV209=CV2018
Work done by battery =CV03×V3=CV209=CV03×V3=CV209
Heat loss =CV209-CV2018=CV2018=CV209−CV2018=CV2018
For process (ii)
Charge on capacitor =2CV03=2CV03
Extra charge flow through battery =CV03=CV03
Work done by battery: CV03 .2V03=2CV209CV03 .2V03=2CV209
Final energy store in capacitor: 12C(2V03)2=4CV201812C(2V03)2=4CV2018
Energy store in process 2: 4CV2018-CV2018=3CV20184CV2018−CV2018=3CV2018
Heat loss in process (ii) = work done by battery in process (ii) – energy store in capacitor process (ii)
=2CV209-3CV2018=CV2018=2CV209−3CV2018=CV2018
For process (iii)
Charge on capacitor =CV0=CV0
Extra charge flow through battery: CV0-2CV03=CV03CV0−2CV03=CV03
Work done by battery in this process: (CV03) (V0)=CV203(CV03) (V0)=CV203
Find energy store in capacitor: 12CV2012CV20
Energy stored in this process: 12CV20-4CV2018=5CV201812CV20−4CV2018=5CV2018
Heat loss in process (iii): CV203-5CV2018=CV2018CV203−5CV2018=CV2018
Now total heat loss (ED): CV2018+CV2018+CV2018=CV206(ED): CV2018+CV2018+CV2018=CV206
Final energy store in capacitor: 12CV2012CV20
So we can say that ED=13(12CV20)ED=13(12CV20)
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