Phase Of Oscillations & Charge-Current Amplitude Relation

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Hey guys! Let's dive into the fascinating world of oscillations and tackle some physics problems. We'll break down how to calculate the phase of an oscillating charge and explore the relationship between charge and current amplitudes in a discharging capacitor. Let's get started!

Calculating the Phase of Oscillations

So, the first question asks us to figure out the phase of oscillations of a charge at a specific time. The magic formula describing these oscillations is given as q = 3.5 x 10⁻⁵ cos(4πt), where 'q' represents the charge in Coulombs and 't' is the time in seconds. We need to find the phase 5 seconds after the oscillations begin.

Let's first understand what phase means in the context of oscillations. The phase of an oscillation, often represented by the Greek letter phi (φ), tells us the state of the oscillation at a particular point in time. Think of it like the position of a swing at a certain moment – is it at the top of its arc, swinging downwards, or at the bottom? The phase captures this information.

In our equation, q = 3.5 x 10⁻⁵ cos(4πt), the term inside the cosine function, 4πt, represents the phase of the oscillation. This is because the cosine function oscillates between -1 and 1 as its argument (the phase) changes. To find the phase at t = 5 seconds, we simply substitute this value into the expression: Phase (φ) = 4π * 5 = 20π radians.

Now, let's think about what this result means. The phase is expressed in radians, which is a standard unit for measuring angles. One complete oscillation corresponds to a phase change of 2π radians. So, 20π radians represents 10 complete oscillations (20π / 2π = 10). This means that after 5 seconds, the charge has gone through 10 complete cycles of oscillation. Importantly, the fractional part of the phase (beyond the whole multiples of 2π) would tell us the exact position within the cycle, but since we have a whole multiple of 2π, the charge is at the same point in its cycle as it was at the beginning – the starting point of a cosine wave, which is the maximum amplitude.

In summary, to find the phase of oscillations, you identify the argument of the trigonometric function (sine or cosine) in the equation describing the oscillation. Substitute the given time into this expression, and the result is the phase in radians. Remembering that 2π radians corresponds to a full cycle is key to interpreting the phase value.

Charge and Current Amplitude Relationship in a Discharging Capacitor

Now, let's tackle the second question: How are the amplitudes of charge and current oscillations related during the discharge of a capacitor? This dives into the heart of how capacitors store and release energy.

Imagine a capacitor, initially charged, connected in a circuit. When the circuit is closed, the capacitor starts to discharge, meaning it releases the stored charge. This discharge process isn't instantaneous; the charge flows out gradually, creating an electric current in the circuit. The key is that both the charge on the capacitor and the current in the circuit oscillate as the capacitor discharges, if there is inductance in the circuit.

To understand the relationship between their amplitudes, we need to bring in a bit of calculus and the fundamental equations governing capacitor behavior. The charge (q) on a capacitor is related to the voltage (V) across it by the equation: q = CV, where C is the capacitance of the capacitor. The current (i) flowing through the circuit is defined as the rate of change of charge with respect to time: i = dq/dt. This is a crucial equation!

Let's assume the charge on the capacitor oscillates according to a sinusoidal function, something like: q(t) = Q₀ cos(ωt), where Q₀ is the amplitude of the charge oscillations and ω is the angular frequency of the oscillations. This is a reasonable assumption because, in an ideal LC circuit (a circuit with an inductor and a capacitor), the charge and current oscillate sinusoidally.

Now, let's differentiate this equation with respect to time to find the current: i(t) = dq/dt = d/dt [Q₀ cos(ωt)] = -Q₀ω sin(ωt). This equation tells us a lot! First, it confirms that the current also oscillates sinusoidally, but with a sine function instead of a cosine. This means the current and charge oscillations are out of phase by 90 degrees (π/2 radians).

More importantly, let's look at the amplitude of the current oscillations. From the equation i(t) = -Q₀ω sin(ωt), the amplitude of the current (I₀) is given by I₀ = Q₀ω. This is the crucial relationship we were looking for! It says that the amplitude of the current oscillations is directly proportional to the amplitude of the charge oscillations and the angular frequency of the oscillations.

In simpler terms, a larger initial charge (Q₀) on the capacitor will lead to a larger peak current (I₀) during discharge. Also, a higher oscillation frequency (ω) will result in a higher peak current for the same initial charge. Think of it like this: if the charge sloshes back and forth more quickly (higher frequency), the current will also be larger. The equation I₀ = Q₀ω beautifully encapsulates this connection.

Key Takeaways: The amplitude of the current oscillations during capacitor discharge is directly proportional to the amplitude of the charge oscillations and the angular frequency. This relationship highlights the fundamental connection between charge flow and current in capacitive circuits.

Conclusion

So, there you have it! We've successfully calculated the phase of oscillating charge and uncovered the relationship between charge and current amplitudes during capacitor discharge. These are fundamental concepts in understanding oscillatory circuits and electromagnetic phenomena. Keep exploring, guys, and the world of physics will continue to amaze you!