Find the time constant (T) for the RC circuit with an open switch given
t = 10 ms
50% of Qmax charge
T = ?
Q = CV (1-e -t/T)
To find the time constant (T), we can use the given information and the equation for the charge (Q) in an RC circuit:
Q = CV(1 - e^(-t/T))
In this equation, Q is the charge in the capacitor, C is the capacitance, V is the voltage across the capacitor, t is the time, and T is the time constant.
From the information provided, we know that:
t = 10 ms (milliseconds)
Q = 0.5Qmax (50% of maximum charge)
We need to find T.
To solve for T, we rearrange the equation:
Q/Qmax = 1 - e^(-t/T)
Given that Q = 0.5Qmax, we substitute this value into the equation:
0.5Qmax / Qmax = 1 - e^(-t/T)
0.5 = 1 - e^(-10 ms / T)
Rearranging the equation to isolate the exponential term:
e^(-10 ms / T) = 1 - 0.5
e^(-10 ms / T) = 0.5
To solve for T, we need to take the natural logarithm (ln) of both sides:
ln(e^(-10 ms / T)) = ln(0.5)
Simplifying further:
-10 ms / T = ln(0.5)
Now, we can solve for T:
T = -10 ms / ln(0.5)
Calculating T using this formula will give you the time constant for the given RC circuit.