Show that the magnitude of the charge on a capacitor is given by Q = (Qo(1-e^-(t/T))) and Q = Qoe^-(t/T) for charging and discharging, respectively.
To derive the given formulas for charging and discharging of a capacitor, we can use the concept of exponential decay and growth.
Let's start with the charging case:
When a capacitor is connected to a voltage source, it starts to charge. The charge on the capacitor, Q, increases with time until it reaches a maximum value, denoted as Qo. The time it takes for the charge to reach this maximum value is determined by the time constant, T, of the circuit.
To derive the formula for charging, we know that the rate of change of charge on a capacitor is proportional to the voltage across it. Mathematically, this can be expressed as:
dQ/dt = V/R (1)
Here, V is the applied voltage across the capacitor, and R is the resistance in the circuit.
We can rearrange equation (1) to separate variables and integrate both sides:
(1/Qo) dQ = (1/RT) dt
Integrating both sides:
∫(1/Qo) dQ = ∫(1/RT) dt
ln(Q/Qo) = -t/RT + C
Using the properties of logarithms, we can rewrite the above equation as:
ln(Q/Qo) = -t/RT + ln(C')
Where C' is the constant of integration.
Next, we can use the property of logarithms e^ln(x) = x, which allows us to rewrite the equation as:
Q/Qo = e^(-t/RT) * e^(ln(C'))
As C' is an arbitrary constant, we can combine it with Qo, leading to:
Q = Qo * e^(-t/RT)
This equation represents the charge on a capacitor as it charges over time, which matches the formula Q = Qo * (1 - e^(-t/T)), given a time constant T = RC.
For the discharging case, the formula Q = Qo * e^(-t/RT) can be derived similarly. However, instead of applying a voltage, we start with an initial charge Qo on the capacitor. As time progresses, the charge on the capacitor decreases exponentially until it reaches zero.
Therefore, the formula for the discharging case is:
Q = Qo * e^(-t/RT)
This matches the given formula Q = Qo * e^(-t/T) for discharging.
In summary, the magnitude of the charge on a capacitor during charging is given by Q = Qo * (1 - e^(-t/T)), and during discharging, it is given by Q = Qo * e^(-t/T). These formulas can be derived by considering the rate of change of charge on a capacitor and integrating the resulting differential equation.