Question

Make a figure analogous to Fig. 21.4 for an ideal inductor in an ac circuit. Start by assuming that the voltage across an ideal inductor is \(v_{\mathrm{L}}(t)=V_{\mathrm{L}}\) sin \(\omega t .\) Make a graph showing one cycle of \(v_{\mathrm{L}}(t)\) and \(i(t)\) on the same axes. Then, at each of the times \(t=0, \frac{1}{8} T, \frac{2}{8} T, \ldots, T,\) indicate the direction of the current (or that it is zero), whether the current is increasing, decreasing, or (instantaneously) not changing, and the direction of the induced emf in the inductor (or that it is zero).

Short answer

Short Answer: For an ideal inductor in an AC circuit, the current and voltage have sinusoidal behaviors and can be given by the equations \(i(t)=\frac{V_L}{\omega L} (1 - \cos(\omega t))\) and \(v_L(t)=V_L \sin(\omega t)\). Throughout one cycle of the input voltage, there are different time intervals at which the current and induced emf exhibit specific behaviors. By analyzing these behaviors, we can observe the following: 1. At t=0, the current is zero, not changing, and the induced emf is negative. 2. At t=\(\frac{1}{8}T\), the current is positive and increasing, and the induced emf is also positive. 3. At t=\(\frac{2}{8}T\), the current is positive and instantaneously not changing, and the induced emf is zero. 4. At t=\(\frac{3}{8}T\), the current is positive and decreasing, and the induced emf is negative. 5. At t=\(\frac{4}{8}T\), the current is zero, not changing, and the induced emf is negative. 6. At t=\(\frac{5}{8}T\), the current is negative and increasing, and the induced emf is positive. 7. At t=\(\frac{6}{8}T\), the current is negative and instantaneously not changing, and the induced emf is zero. 8. At t=\(\frac{7}{8}T\), the current is negative and decreasing, and the induced emf is negative. 9. At t=T, the current is zero, not changing, and the induced emf is negative. This analysis highlights the alternating behavior of the current and induced emf in an inductor in an AC circuit throughout one cycle of the input voltage.

Step-by-step solution

Derive the equation for current through the inductor

Recall that the voltage across an ideal inductor is given by \(v_L(t)=L \frac{di}{dt}\), where L is the inductance and i is the current. We are given the input voltage as \(v_L(t)=V_L \sin(\omega t)\). Now, we need to find the current through the inductor at any time t. For this, we have to solve the following differential equation: $$L \frac{di(t)}{dt}=V_L \sin(\omega t)$$ First, we integrate both sides with respect to t: $$\int{L di(t)} = \int{V_L \sin(\omega t) dt}$$ Let's solve the right side of the equation, which results in: $$Li(t)= -\frac{V_L}{\omega} \cos(\omega t) + C$$ where C is the integration constant. To find C, we can use the initial condition \(i(0)=0\): $$0= -\frac{V_L}{\omega} \cos(\omega 0) + C \implies C=\frac{V_L}{\omega}$$ So the general equation for the current in the inductor is: $$i(t)=\frac{V_L}{\omega L} (1 - \cos(\omega t))$$

Plot the graphs of voltage and current

Now we have the equations for \(v_L(t)\) and \(i(t)\), we can plot them on the same graph for one complete cycle of the input voltage (\(0 \le t \le T\)), where \(T=\frac{2 \pi}{\omega }\) is the period. Voltage and current on the graph will have a sinusoidal shape, and the current will have a higher frequency than the voltage.

Analyze current and induced emf behavior

Next, we need to analyze the behavior of current, its rate of change, and induced emf in the inductor at the following time intervals: $$t=0, \frac{1}{8}T, \frac{2}{8}T, \ldots, T$$ At each time, we need to determine: 1. The direction of the current (or if it's zero) 2. Whether the current is increasing or decreasing (or instantaneously not changing) 3. The direction of the induced emf in the inductor (or if it's zero) Use the current equation to find the current value and its rate of change: $$i(t)=\frac{V_L}{\omega L} (1 - \cos(\omega t))$$ $$\frac{di(t)}{dt}=\frac{V_L}{L} \sin(\omega t)$$ For each interval of t, plug in the value and determine the behavior of current and induced emf accordingly.