AC circuit and standing wave questions

In summary, the first question asked for the maximum value of energy stored in an inductor connected to a 120 V rms, 60 Hz ac power supply with a 5.0 H inductance. The correct answer is 0.40 J. The second question asked for the distance between consecutive nodes in a standing wave represented by the equation y = A sin (kx) cos (ωt). The correct answer is 2Π/k.
  • #1
Violagirl
114
0

Homework Statement



1. A 5.0 H inductor is connected to a 120 V rms, 60 Hz ac power supply. Periodically, the energy stored in the inductor reaches a maximum value. What is this maximum value?

A) 0.010 J B) 0.020 J C) 0.40 J D) 0.80 J E) 10.0 J

2. A standing wave on a string is represented by the equation y = A sin (kx) cos (ωt). The distance between consecutive nodes is:

A) ∏/k B) ∏/ω C) 2∏/k D) k/∏ E) ω/∏


The Attempt at a Solution


These were two questions from an exam I had today and I wanted to check to see if my thinking was correct? For the first question, I put C.

I first found the period, 1/f, and got 1/60 = .0167 sec.

Next, I divided the 120 V rms by 5 H and got an answer of 24 V/H. Finally, I multiplied the frequency into this value and got an answer of 0.40 J. Was this correct?

For number 2, I rationalized that a node corresponds to half of a wavelength, which is equivalent to pi. And I know that k is equal the number of wave peaks in a wave. Therefore, out of the choices possible, I thought that A made the most sense. If anyone could confirm whether or not that is correct, I would appreciate it.
 
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  • #2
Violagirl said:
For the first question, I put C.

I first found the period, 1/f, and got 1/60 = .0167 sec.

Next, I divided the 120 V rms by 5 H and got an answer of 24 V/H. Finally, I multiplied the frequency into this value and got an answer of 0.40 J. Was this correct?
Hi Violagirl. It sounds like you evaluated the expression V/(f L) ?

That looks suspiciously close to V/(2 Pi f L) which would be the formula for current. You also need a formula relating energy in an inductor to current in it.

For number 2, I rationalized that a node corresponds to half of a wavelength, which is equivalent to pi. And I know that k is equal the number of wave peaks in a wave.
I don't follow that reasoning about k. Take another look at this. First, sketch what the outline (envelope) of the standing wave looks like.
 
  • #3
Thanks for clarifying number 1, it makes more sense now.

For 2, well I know from the equation that k = 2pi/λ. For a standing wave, λ corresponds to 1/2L. And I know it's half a wavelength, it should be equivalent to pi. I guess from there though, I wasn't sure how to understand number 2...
 
  • #4
Violagirl said:
For 2, well I know from the equation that k = 2pi/λ.
Do we? You're talking about the term sin (kx) ?

We know it has a node where kx = 0
 
  • #5
Oooh, true. I guess I was thinking of it in general terms like 0, pi, 2pi, etc. And I guess from a standing wave, since we see one full peak from a standing wave, that I thought that k would correspond to pi since we see it travel from 0 to pi in the distance of a peak. This question was definitely tricky for me to think about.
 
  • #6
Violagirl said:
Oooh, true. I guess I was thinking of it in general terms like 0, pi, 2pi, etc. And I guess from a standing wave, since we see one full peak from a standing wave, that I thought that k would correspond to pi since we see it travel from 0 to pi in the distance of a peak. This question was definitely tricky for me to think about.
I think you mean the product kx would correspond to Pi, 2Pi, 3Pi, etc., don't you?
 
  • #7
Yes, that is true, kx is what would correspond to pi, 2pi, etc.
 
  • #8
Violagirl said:
Yes, that is true, kx is what would correspond to pi, 2pi, etc.

So what is your answer for (2)?
 

1. What is an AC circuit?

An AC circuit is a type of electrical circuit that uses alternating current (AC) to power devices. In an AC circuit, the direction of the current changes periodically, usually at a frequency of 50 or 60 Hertz.

2. How is an AC circuit different from a DC circuit?

The main difference between an AC and DC circuit is the direction of the current flow. In DC circuits, the current flows in only one direction, while in AC circuits, the current changes direction periodically.

3. What is a standing wave in an AC circuit?

A standing wave is a type of wave that forms when two identical waves moving in opposite directions interfere with each other. In an AC circuit, standing waves can occur when the incoming and reflected waves have the same frequency and amplitude, causing a stationary pattern of energy.

4. How do you calculate the wavelength of a standing wave in an AC circuit?

The wavelength of a standing wave in an AC circuit can be calculated using the formula λ=2L/n, where L is the length of the circuit and n is the number of nodes (points of zero amplitude) in the standing wave.

5. What are some practical applications of standing waves in AC circuits?

Standing waves in AC circuits have various practical applications, such as in antennas for wireless communication, in musical instruments, and in microwave ovens. They can also be used to measure the impedance of a circuit and to detect faults in transmission lines.

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