1. Given t1/2 to be 0.0137 seconds, how long should it take to reach 87.5% of maximum charge?
2. Let's say you measure t1/2 to be 0.0137 seconds and the resistance in the circuit is 99 Ohms. Then what is the capacitance in micro Farads?
3.q = qmax(1 - e -t / τ)
Using this equation one can see that when t = τ*ln2 that q = 0.5 qmax. Or, that q equals 50% of qmax.
What is the percentage of qmax when t = 4 * tau * ln2?
4.The following equation is for the next question.
t½ = τ ln2 = RC ln. Given a resistance of 99 Ohms and a capacitance of 200micro Farads,
what is t½?
What is t1/2?
1. To find the time it takes to reach 87.5% of maximum charge, we can use the concept of half-life. The half-life (t1/2) is the time it takes for a system to reduce its quantity by half. In this case, t1/2 is given as 0.0137 seconds.
To find the time it takes to reach 87.5% of maximum charge, we need to solve for t in the equation:
q = qmax(1 - e^(-t/τ))
where q is the charge at time t, qmax is the maximum charge, and τ is the time constant of the system.
To find τ, we can use the equation τ = t1/2 / ln(2), as the time constant is equal to half-life divided by the natural logarithm of 2.
Substituting the given t1/2 value into the equation, we get:
τ = 0.0137 seconds / ln(2)
Now, we can substitute the obtained τ value and the desired charge percentage (87.5% = 0.875) into the equation:
0.875 = qmax(1 - e^(-t/τ))
Solving for t, we have:
t = -τ * ln(1 - 0.875)
Note: The negative sign indicates that we are looking for the time it takes for the charge to decrease to 87.5%.
Calculate the value of -τ * ln(1 - 0.875), and you will have the time it takes to reach 87.5% of maximum charge.
2. To find the capacitance in microfarads, we can use the formula:
t1/2 = τ * ln(2) = RC * ln(2)
Given that t1/2 is 0.0137 seconds and the resistance is 99 ohms, we can rearrange the formula to solve for capacitance (C):
C = t1/2 / (R * ln(2))
Substituting the values, we have:
C = 0.0137 seconds / (99 ohms * ln(2))
Calculate the value of 0.0137 / (99 * ln(2)), and you will have the capacitance in microfarads.
3. The equation q = qmax(1 - e^(-t/τ)) represents the charge (q) at time t in a system with a time constant τ. If we substitute t = τ * ln(2) into the equation, we can find the percentage of qmax.
For t = τ * ln(2), the equation becomes:
q = qmax(1 - e^(-τ * ln(2) / τ))
= qmax(1 - e^(-ln(2)))
= qmax(1 - 1/2)
= qmax * (1/2)
= 0.5 * qmax
Therefore, when t = 4 * τ * ln(2), the equation q = 0.5 * qmax still holds true, which means q is 50% of qmax.
4. The equation t1/2 = τ * ln(2) = RC * ln(2) relates the half-life (t1/2), time constant (τ), resistance (R), and capacitance (C) in a circuit.
Given a resistance of 99 ohms and a capacitance of 200 microfarads, we can rearrange the equation to solve for t1/2:
t1/2 = (RC * ln(2)) / ln(2)
Substituting the given values, we have:
t1/2 = (99 ohms * 200 microfarads * ln(2)) / ln(2)
Calculate the value of (99 * 200 * ln(2)) / ln(2), and you will have the value of t1/2.