An ac generator has emf 𝒞=8m sin(ω⊥ t-π/ 4), where ℰm=25.0 V and ωd=270 rad / s. The current pr

An ac generator has emf 𝒞=8m sin(ω⊥ t-π/ 4), where ℰm=25.0...

An ac generator has emf $\mathscr{C}={8}_{m} \sin \left(\omega_{\perp} t-\pi / 4\right)$, where $\mathscr{E}_{m}=25.0 \mathrm{~V}$ and $\omega_{d}=270 \mathrm{rad} / \mathrm{s}$. The current produced in a connected circuit is $i(t)=I \sin \left(\omega_{\mu} t-3 \pi / 4\right)$, where $I=620 \mathrm{~mA}$. At what time after $t=0$ does (a) the generator emf first reach a maximum and (b) the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance, or resistance, as the case may be?

Key Concepts

Sinusoidal Waveforms

Sinusoidal waveforms are mathematical functions that describe periodic oscillations and are commonly used in AC (alternating current) analysis. They are characterized by parameters such as amplitude, angular frequency, period, and phase shift. In electrical engineering, these waveforms represent the time-dependent behavior of voltages and currents in circuits, making it possible to analyze their behavior using trigonometric relationships.

Phase Angle and Phase Shifts

A phase angle in a sinusoidal function indicates a horizontal shift from a reference time, typically representing a delay or advance in the waveform. Phase shifts are crucial when comparing two sinusoidal signals, such as the voltage from a generator and the current through a circuit element. The difference in phase between these two signals can reveal important characteristics of the circuit, such as whether it is purely resistive (in-phase), inductive (current lags voltage), or capacitive (current leads voltage).

Time of Maximum in Sinusoids

For a sinusoidal function, the maximum occurs when the argument of the sine function reaches ?/2 (or an equivalent value modulo 2?). By setting the phase in the sinusoidal equation equal to ?/2, one can solve for the time at which the maximum voltage or current occurs. This approach is widely used in analyzing AC circuits to determine key timing characteristics of alternating quantities.

AC Circuit Element Phase Relationships

In AC circuits, different circuit elements introduce characteristic phase shifts between voltage and current. A resistor produces no phase shift (voltage and current are in phase), an inductor causes the current to lag behind the voltage by 90° (or ?/2 radians), and a capacitor causes the current to lead the voltage by 90°. These phase relationships are fundamental in determining the nature of the circuit element when analyzing its behavior in response to an applied sinusoidal voltage or current.

Impedance and Reactance

Impedance generalizes resistance to AC circuits by accounting for both the resistive and reactive (capacitive or inductive) components, and is represented as a complex number. The magnitude of the impedance is related to the ratio of the voltage and current amplitudes, while the phase angle indicates if the circuit is resistive, inductive, or capacitive. In circuits with a single reactive element, the reactance (X) is given by ?L for an inductor or 1/(?C) for a capacitor, and can be determined by comparing the voltage and current phase shift and amplitudes.

Transcript

00:01 Hello student this problem belongs to electromagnetic oscillation and alternating current chapter and this problem we have given emfe is equal to emmax sign of omega t minus pi by 4 and the value of em is 30 volt and omega is 350 radiant second and the current is given i sine omega t minus 3 pi by 4 so the maximum value of current is given 620 millie ampere in part a we have to find what is the time when the emf first reach its maximum value that means emf will be equal to emex when sine 3 3 3d000 will be equal to 1 that means theta will be equal to pi by 2 here theta is equal to omega t minus pi by 4 it should be equal to pi by 2 that means omega t will be equal to pi by 4 plus pi by 2 if we solve this this will be equal to t will be equal to 3 pi y 4 omega if we put these values here omega is 350 we get the time 6 .73 into 10 to the power minus 3 second this is the solution of part a now in part b we have to find when will the current will be maximum so again here sign of omega t minus 3 pi by 4 should be equal to 1 that means omega t minus 3 pi by 4 should be equal to pi by 2 so omega t will be equal to 3 pi by 4 plus pi by 2 that means t should be equal to 5 pi by 4 omega if we again put the value of here which is 350 radian per second the value of time is 1 .12 into 10 to the power minus 2 second this is the solution of part b now in part c we have to find which is the circuit contains a single element other than generator is it a capacitor and inductor or a gesture we have to find that element if we see these two equation these two equation this is the voltage equation this is the current equation if we see this we can see here clearly that current lacks from the voltage by pi by 2 degree when current lacks current lacks the voltage that means the circuit element circuit element should be inductor because in inductor only the current legs in part d we have to find what is the value of the inductor inductance so we know that vl is equal to i into xl and here this is equal to e so xl will be equal to omega into l so l will be equal to e upon i into omega.

04:47 Here e is given 30 volt...

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