A circuit with a capacitor took .128s to discharge from its max charge Qmax to half of that max charge once the battery was removed. In this series circuit there also is one resistor with 100 G(ohm symbol). The voltage drop is 12V across the battery.
What is the time constant of this circuit?
I know that time constant = RC but we are given two unknowns the time constant and the capacitance. R would be 100 x 10^9 or 1 x 10^11.
1/2=e^(-t/rc)
Is this the problem statement?
Solve for c.
To find the time constant of the circuit, we need to calculate the product of the resistance and the capacitance. In this case, we are given the value of the resistance (R = 100 GΩ), but we don't have the value of the capacitance (C).
However, we can use the given information about the discharge time and voltage drop to determine the value of the capacitance.
We know that the time constant (τ) of an RC circuit is the time it takes for the capacitor to charge or discharge to approximately 63.2% of its maximum charge. In this case, the circuit is discharging, so we need to find the time it takes for the capacitor to discharge to half of its maximum charge.
Given that the discharge time is 0.128s and the voltage drop across the battery is 12V, we can use the formula for the voltage across a discharging capacitor in an RC circuit:
V(t) = V_max * e^(-t/τ)
where V(t) is the voltage at time t, V_max is the initial voltage across the capacitor, e is the base of the natural logarithm, and τ is the time constant.
In this case, we can set V_max as 12V and V(t) as (1/2)*V_max, since we want to find the time it takes for the voltage across the capacitor to reach half of the initial voltage.
Therefore, the equation becomes:
(1/2)*V_max = V_max * e^(-t/τ)
Simplifying the equation:
1/2 = e^(-t/τ)
Taking the natural logarithm (ln) of both sides:
ln(1/2) = -t/τ
ln(1/2) = -0.693 (approximation for the natural logarithm of 1/2)
Now we can solve for the time constant (τ):
-0.693 = -t/τ
By rearranging the equation:
τ = -t / (-0.693)
Using the given discharge time (t = 0.128s):
τ = -0.128s / (-0.693)
τ ≈ 0.1848s
Therefore, the time constant of the circuit is approximately 0.1848 seconds.