How long will it take the voltage across the capacitor in an RC circuit to discharge 75% of its initial charged voltage if the resistor has a resistance of 450 kilo-Ohms and the capacitor has a capacitance of 49 micro-farads?
How do I solve this problem?
Thanks!
C = q/V
q = C V
i = -dq/dt = -C dV/dt
so
dV/dt = -i/C
V = i R = 4.5 * 10^5 i
so
i = .222 * 10^-5 V = 2.22*10^-6 V
then combine
dV/dt = -(1/c)(2.22*10^-6 V)
dV/V = -(2.22/49) dt
ln V = -.0453 t + constant
V = Vi e^-.0453 t
. 75 = e^-.0453 t
ln .75 = - .288 = - .0453 t
t = 6.35 s
CHECK ARITHMETIC !!!
R = 450k Ohms.
C = 49 uF
100%-75%=25% of the voltage remaining.
x = t/RC = t/(450*49) = 4.54*10^-5t
1/e^x = 0.25
e^x = 4
x*Ln e = Ln 4
X = 1.386
x = 4.54*10^-5t = 1.386
t = 1.386/4.54*10^-5=30535 Milliseconds = 30.54 Seconds.
To solve this problem, we can use the equation for the voltage across a charging or discharging capacitor in an RC circuit:
V(t) = V0 * e^(-t / RC),
where V(t) is the voltage across the capacitor at time t, V0 is the initial voltage across the capacitor, e is the base of the natural logarithm (approximately 2.71828), t is the time in seconds, R is the resistance in ohms, and C is the capacitance in farads.
In this case, we want to find the time it takes for the voltage to discharge 75% of its initial charged voltage. Thus, we can set V(t) equal to 0.75 * V0 and solve for t.
0.75 * V0 = V0 * e^(-t / (450 * 10^3 * 49 * 10^-6)).
Dividing both sides by V0:
0.75 = e^(-t / (450 * 10^3 * 49 * 10^-6)).
To isolate the variable t, we can take the natural logarithm of both sides:
ln(0.75) = -t / (450 * 10^3 * 49 * 10^-6).
Simplifying:
t = -ln(0.75) * (450 * 10^3 * 49 * 10^-6).
Using a scientific calculator or computer software, you can calculate the value of t.
To solve this problem, you can use the formula for the voltage across a charging or discharging capacitor in an RC circuit:
V(t) = V0 * e^(-t / RC)
Where:
- V(t) is the voltage across the capacitor at time t
- V0 is the initial charged voltage across the capacitor
- e is the mathematical constant approximately equal to 2.71828
- t is the time in seconds
- R is the resistance in ohms
- C is the capacitance in farads
In this case, you want to find the time it takes for the voltage across the capacitor to discharge 75% of its initial charged voltage. So, you can set V(t) equal to 0.75 * V0 and solve for t.
0.75 * V0 = V0 * e^(-t / RC)
Divide both sides of the equation by V0:
0.75 = e^(-t / RC)
To isolate t, you can take the natural logarithm (ln) of both sides of the equation:
ln(0.75) = -t / RC
Multiply both sides of the equation by -RC:
-t = ln(0.75) * RC
Divide both sides of the equation by -RC:
t = -ln(0.75) * RC
Now, you can substitute the given values into the equation:
R = 450 kilo-Ohms = 450,000 Ohms
C = 49 micro-farads = 49 * 10^-6 farads
t = -ln(0.75) * (450,000 * 49 * 10^-6)
Calculating this expression will give you the time it takes for the capacitor to discharge 75% of its initial charged voltage in seconds.