Understanding Time Constants and the Mathematical Equation for Current Decay

[transgalactic]

i understood every thing except at the last 30 seconds

he says "after 5 time constants t=rc" current will be 0
i can't understand by what mathematical equation he got this number 5
??

 

[transgalactic]

intuitively i know that after the capacitor will be filled with charge
it will stop being a conductor

but how he came up with the "5"
??

 

[Hao]

The 5 he choose is an example of a 'magic number'.

We know that current decays as $$I = I_{initial} e^{-\frac{t}{R C}}$$.

Hence, as time passes, I gets smaller and smaller.

After 1 time constant,
$$\frac{I}{I_{initial}} = e^{-1} = 0.368$$
After 2 time constants,
$$\frac{I}{I_{initial}} = e^{-2} = 0.135$$
After 3 time constants,
$$\frac{I}{I_{initial}} = e^{-3} = 0.050$$
After 4 time constants,
$$\frac{I}{I_{initial}} = e^{-4} = 0.018$$
After 5 time constants,
$$\frac{I}{I_{initial}} = e^{-5} = 0.007$$

So a more accurate statement is, "After 5 time constants, the current is approximately 0."

He could have easily said 4 or 6 or 100. They key point is that after about 3-5 time constants, the current is effectively 0.

Edit: Formatting error, sorry,

 

[transgalactic]

The 5 he choose is an example of a 'magic number'.

We know that current decays as $$I = I_{initial} e^{-\frac{t}{R C}}$$.

Hence, as time passes, I gets smaller and smaller.

After 1 time constant, $$\frac{I}{I_{initial} = e^{-1} = 0.368}$$
After 2 time constant, $$\frac{I}{I_{initial} = e^{-2} = 0.135}$$
After 3 time constant, $$\frac{I}{I_{initial} = e^{-3} = 0.050}$$
After 4 time constant, $$\frac{I}{I_{initial} = e^{-4} = 0.018}$$
After 5 time constant, $$\frac{I}{I_{initial} = e^{-5} = 0.007}$$

So a more accurate statement is, "After 5 time constants, the current is approximately 0."

He could have easily said 4 or 6 or 100. They key point is that after about 3-5 time constants, the current is effectively 0.

xyzt

 

[transgalactic]

thanks