Question

The charge \(q\) on a capacitor in a simple a-c circuit varies with time according to the equation \(q=3 \sin (120 \pi t+\pi / 4) .\) Find the amplitude, period, and frequency of this oscillation. By definition, the current flowing in the circuit at time \(t\) is \(I=d q / d t\) Show that \(I\) is also a sinusoidal function of \(t,\) and find its amplitude, period, and frequency.

Short answer

Amplitude of charge: 3; period: \( \frac{1}{60} \) s; frequency: 60 Hz. Amplitude of current: 360 \( \pi \); same period and frequency as the charge.

Step-by-step solution

Identify the Form of the Equation

The charge on the capacitor is given by the equation: \( q = 3 \sin (120 \pi t + \frac{\pi}{4}) \).

Determine the Amplitude

The amplitude of the sinusoidal function \( q \) is the coefficient of the sine function. Thus, the amplitude is 3.

Determine the Angular Frequency

The equation \( q = 3 \sin(120 \pi t + \frac{\pi}{4}) \) has the form \( q = A \sin( \omega t + \phi ) \), where \( \omega \) is the angular frequency. From the given equation, \( \omega = 120 \pi \).

Calculate the Period

The period \( T \) of the oscillation is given by the formula \( T = \frac{2 \pi}{\omega} \). Substituting \( \omega = 120 \pi \), we get \( T = \frac{2 \pi}{120 \pi} = \frac{1}{60} \) seconds.

Calculate the Frequency

The frequency \( f \) is the reciprocal of the period: \( f = \frac{1}{T} \). Given that \( T = \frac{1}{60} \) seconds, \( f = 60 \) Hz.

Differentiate to Find the Current

The current \( I \) is the time derivative of the charge \( q \):\( I = \frac{dq}{dt} \).Differentiating \( q = 3 \sin(120 \pi t + \frac{\pi}{4}) \) with respect to time gives\( I = 3 \cdot 120 \pi \cos(120 \pi t + \frac{\pi}{4}) = 360 \pi \cos(120 \pi t + \frac{\pi}{4}) \).

Identify the Amplitude of the Current

The amplitude of the sinusoidal current \( I \) is the coefficient of the cosine function. Thus, the amplitude of \( I \) is \( 360 \pi \).

Determine the Period and Frequency of the Current

Since the current is a cosine function with the same angular frequency \( \omega \) as the charge, the period \( ( T ) \) and frequency \( ( f ) \) of the current will be the same as those of the charge: \( T = \frac{1}{60} \) seconds and \( f = 60 \) Hz.

Key concepts

sinusoidal oscillations

Sinusoidal oscillations describe the manner in which quantities like charge, current, and voltage vary over time in AC circuits. In an AC circuit, these quantities usually change in a smooth, repetitive wave-like pattern called a sine wave. The mathematical representation involves trigonometric functions such as sine or cosine. The function includes parameters like amplitude, angular frequency, and phase, which together help define the wave's characteristics.

amplitude calculation

Amplitude is a measure of the maximum value attained by the oscillating quantity. For a function of the form \(\text{q} = \text{A} \text{sin}( \text{ω t} + \text{φ} ) \), the amplitude is represented by the constant \(\text{A}\). In our specific example, the charge function is given by \(\text{q} = 3 \text{sin}(120 \text{π t} + \frac{\text{π}}{4}) \). Here, the amplitude, \(\text{A}\), is 3, indicating that the maximum charge on the capacitor will be 3 units.

period and frequency

The period (T) and frequency (f) are crucial aspects of sinusoidal functions in AC circuits. The period (\text{T}) is the time required for one complete cycle of the oscillation. It can be calculated using the angular frequency (\text{ω}), which is already determined from the given function. The formula to find the period is \[ T = \frac{2π}{ω} \]. With an angular frequency of \(\text{120 π}\), the period becomes \[ T = \frac{2π}{120 π} = \frac{1}{60} \text{ seconds} \]. The frequency (\text{f}) is the number of cycles per second, reciprocal of the period: \[ f = \frac{1}{T} = 60 \text{ Hz} \]. This means there are 60 cycles each second.

differentiation

Differentiation is a mathematical process used to determine the rate at which a quantity changes. In AC circuit analysis, to find the current (\text{I}) from the charge (\text{q}), we differentiate the charge function with respect to time (\text{t}). For our given function \(\text{q} = 3 \text{sin}(120 \text{π t} + \frac{\text{π}}{4}) \), differentiation yields \[ I = \frac{dq}{dt} = 3 \times 120 π \text{cos}(120 π t + \frac{π}{4}) = 360 π \text{cos}(120 π t + \frac{π}{4}) \]. This implies that the current is also a sinusoidal function, but with a cosine term.

current and charge relationship

The current (\text{I}) in an AC circuit is directly related to the charge (\text{q}) by differentiation. From the equation \[ q = 3 \text{sin}(120 \text{π t} + \frac{π}{4}) \], differentiating gives us the current function \(\text{I} = 360 π \text{cos}(120 π t + \frac{π}{4}) \). The amplitude of the current here is \(\text{360 π}\), with the same period and frequency as the charge due to sharing the same angular frequency (\text{ω}). This demonstrates that variations in charge directly influence current, as seen by their relationship through differentiation.