# 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

### Is there such a thing as an automatic variable resistor

2  Asked on December 15, 2021

### Constant current source with bypass option

2  Asked on December 15, 2021 by david-cutting

### Automation – Stealing a signal from a circuit

1  Asked on December 15, 2021 by craig-monyhan

### Clock usage in the always block and in the event

2  Asked on December 15, 2021 by animesh-srivastava

### Determining AXI4-stream Data FIFO size, understanding Packet Mode?

1  Asked on December 15, 2021 by comc-cmoc

### Standards Compliant High DC Load Safety Interlock System

0  Asked on December 15, 2021 by buttercup

### Button short press and long press without using microcontroller

2  Asked on December 15, 2021 by spockshr

### read negative voltage with mcp3008

1  Asked on December 15, 2021

### How to learn PID Control?

6  Asked on December 12, 2021

### Using car charger for small sealed-type lead-acid batteries

6  Asked on December 12, 2021

### Why does only the fermi level shift in a semiconductor under an external electric field?

1  Asked on December 12, 2021

### Variation of Fermi level in doped semiconductor with applied voltage

2  Asked on December 12, 2021 by tvv

### Solving circuit design problem with two approaches

1  Asked on December 12, 2021

### L-C-L vs Pi network

1  Asked on December 12, 2021

### What is the “DanTherm” model in the SOATherm library in LTSpice?

0  Asked on December 12, 2021 by spice_rack

### Is there any limitation in how quickly/often I can change a 6V DC motor direction of rotation?

1  Asked on December 12, 2021

### How to light LEDs with Arduino without having any resistors

6  Asked on December 12, 2021

### How do I connect this crane machine motherboard to my power supply? What connectors do I need for the power supply header pins?

1  Asked on December 12, 2021 by sam-jacka

### Magnetic vector potential and displacement current

2  Asked on December 12, 2021 by pr871