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Exponential decay is a fundamental concept in understanding how the charge on a capacitor decreases over time. At its simplest, exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In this context, our charge function, \( C(t) = 100 e^{-0.2t} \) illustrates exponential decay.

The term \( e^{-0.2t} \) is what makes the function decay exponentially. Here, the base \( e \) is a constant approximately equal to 2.718, often called Euler's number. As time \( t \) increases, the exponent \( -0.2t \) becomes more negative, making the whole expression \( e^{-0.2t} \) smaller. Thus, the charge on the capacitor decreases, demonstrating exponential decay.

Understanding this process is key, as it appears in many natural phenomena where a quantity lessens quickly at first, and then slows over time. In a capacitor, it means that the charge will drop substantially initially, before leveling out more slowly.

Calculating the charge on a capacitor at a given time is a matter of evaluating the exponential function at that point. The exercise provides the function \( C(t) = 100 e^{-0.2t} \), and you simply substitute the desired time value of \( t \) into the equation.

For example, to find the charge at \( t=3 \):

• First, substitute \( t = 3 \) into the function: \( C(3) = 100 e^{-0.2 \times 3} \).
• Calculate the exponent: \( -0.2 \times 3 = -0.6 \).
• Evaluate the exponential expression: \( e^{-0.6} \approx 0.5488 \).
• Finally, calculate the charge: \( C(3) = 100 \times 0.5488 = 54.88 \).

Similarly, for \( t=5 \):

• Substitute \( t = 5 \) into the function: \( C(5) = 100 e^{-0.2 \times 5} \).
• Calculate the exponent: \( -0.2 \times 5 = -1.0 \).
• Evaluate the exponential expression: \( e^{-1.0} \approx 0.3679 \).
• Calculate the charge: \( C(5) = 100 \times 0.3679 = 36.79 \).

This straightforward process illustrates how to compute the charge at any time given the exponential decay function.

The capacitor function describes how a capacitor stores and releases electrical energy. Fundamentally, a capacitor is a device that can store energy in an electric field by separating positive and negative charges on two conductive plates.

When charged, a capacitor holds a potential difference (voltage) between its plates. The charge it holds is dependent on this voltage and the capacitance, a characteristic of the capacitor that defines how much charge it can store at a given voltage.

• Capacitance is usually measured in farads (F).
• The energy stored in a capacitor is proportional to the square of the charge and inversely proportional to the distance between the plates.

The equation \( C(t) = 100 e^{-0.2t} \) models the charge decay over time, as a capacitor naturally releases its stored charge. The exponential decay reflects the common behavior of real-world capacitors where charge dissipates gradually, impacting the device's functionality.

Understanding how a capacitor functions not only helps in calculating decay, but also in designing circuits that efficiently manage electrical energy, which is crucial in electronics.