# Derivation of energy in capacitor

Electrical Engineering Asked on January 4, 2022

While searching for something totally unrelated to this, I came a cross a website that derived it in this fashion:
the instanteous power in a capacitor is given by $$p_c= v_c(t)cdot i_c$$

since $$i_c(t) = Cfrac{dv_c}{dt}$$, this becomes $$p_c = v_c(t)cdot Cfrac{dv_c}{dt}$$

No issues so far….but, he then proceeds to write:
$$frac{dw_c(t)}{dt}=frac{d}{dt}[frac{1}{2}Cv_c^2(t)]$$.

power is the derivative of energy, so I get the left hand side of the equation. However, how does $$Cfrac{dv_c}{dt}cdot v_c(t)=frac{d}{dt}[frac{1}{2}Cv_c^2(t)]$$ on the right hand side of the equation?

Just use the chain rule of calculus:

$$frac{d}{dt}v_c^2(t) = 2v_c(t)frac{dv_c(t)}{dt}$$

therefore $$v_c(t)frac{dv_c(t)}{dt} = frac{1}{2}frac{d[v_c^2(t)]}{dt}$$

Answered by rpm2718 on January 4, 2022

## Related Questions

### Predicting decay time of a resonant bandpass filter?

3  Asked on December 13, 2020

### Using 2 laptop power supplies to make plus and minus supply

0  Asked on December 13, 2020 by freddie

### Help me to understand VOOC fast charging for lithium ion batteries?

1  Asked on December 13, 2020 by ridham

### OpAmp VS Voltage Divider VS Buck Converter (Current Sensor Output)

3  Asked on December 12, 2020 by marcelo-queirs

### op-amp outputs supply voltage on GND Signal

2  Asked on December 12, 2020 by rijik

### Xilinx Vivado: How are inputs/outputs handled that are not in the constraints file?

1  Asked on December 11, 2020 by tschaboo

### Can eddy current sensors measure through a thin metallic sheet?

2  Asked on December 11, 2020 by tarti

### How exactly should I build a smart wall light switch for a smart home

2  Asked on December 10, 2020 by mostafanfs

### Low Overhead Constant Current LED Driver

2  Asked on December 10, 2020 by easyohm

### Band-pass filter vs. Serial Low-pass+High-pass filter – Is there a difference?

2  Asked on December 10, 2020

### Equivalent DC circuit of 3 phase induction motor

1  Asked on December 9, 2020 by steelixb

### Performing loadflow on islanded microgrid

0  Asked on December 8, 2020

### Using the frame of an RV trailer for the ground

1  Asked on December 8, 2020 by user158568

### Electrical frequency of motor with different stator and rotor poles

2  Asked on December 8, 2020 by jrvinayak

### How to know component’s offset when making custom footprint

1  Asked on December 7, 2020 by sim-son

### Can anyone tell me why nothing is displayed on the scope?

0  Asked on December 7, 2020 by mmm

### Is a special charger required for Li-ION battery packs with a BMS?

1  Asked on December 7, 2020 by cpuguru

### What is the purpose of this diode?

1  Asked on December 7, 2020 by st4rgut