Physics homework help. Swing Time

Before You Begin

Today’s lab involves making a quantitative study of common circuits that have inductors. This section will

help you review inductance and inductors before you begin.

Since an electric current produces a magnetic field and a magnetic field exerts a force on an electric current

or moving electric charge, it should come as no surprise that a magnetic field can produce an electric current.

Faraday’s law of induction tells us that the emf induced in a circuit is equal to the rate of change of magnetic

flux through the circuit. Combining all these ideas, one might expect that a changing current in one circuit

ought to induce an emf and a current in a second nearby circuit and even induce an emf in itself. The first

situation is known as mutual inductance, when the changing current in one circuit induces a current in a

second circuit. Within a single coil, a changing current induces an opposing emf, so a coil has a selfinductance, L which has units of 1

volt ∙second

ampere

= 1 henry. A coil that has significant self-inductance is called

an inductor.

An inductor stores energy in the magnetic field surrounding its current-carrying wires, just as a capacitor

stores energy in the electric field between its charged plates.

Discussion Questions

While your lab report is an individual assignment, remember that part of your grade for the week is to

participate in Discussions on Canvas with your group members. You are encouraged to discuss any part of

this week’s lab/concepts. You may also want to discuss the following practice problems and questions:

1. At the instant the battery is connected into an LR circuit, like in Task #1, the emf in the inductor has

its maximum value even though the current is zero. Explain.

2. In a battery, when the current is in the same direction as the emf, the energy of the battery decreases,

whereas if the current is in the opposite direction, the energy of the battery increases (as in charging

a battery). Is this also true for an inductor?

Task #1: LR Circuit

For Task #1, you will use a simulation to observe and measure what happens to the voltage across a resistor

and an inductor when they are placed in series in a direct current (DC) circuit.

Go to http://tinyurl.com/tkz8ay8 to find the simulation. Here you should find an LR circuit with the switch

open like in the following picture.

The battery is set to 5V, the inductor is set to 0.85 H (or 850 mH), and the resistor is set to 10 Ω. On the left

side of the screen there is a button to reset the circuit, a button to start/stop the simulation, and various

sliders to change the simulation speed, current speed, and inductance and resistance values. On the bottom is

a scope view that shows both the voltage across the resistor and the voltage across the inductor.

Close the switch and allow current to flow.

A. Describe in words the behavior of the voltage across the resistor in this LR circuit.

B. Look at the scope for the voltage across the inductor. How does it compare to the voltage across the

resistor? Based on the voltage across the resistor and Kirchhoff’s rules, does this make sense? How

do you know?

C. How does the behavior of this circuit compare to an RC circuit? How are they the same and how are

they different?

D. Recall that for RC circuits you were able to define the time (the half-life time) that it took the

voltage to decay to half its original value (?1

2

⁄ = ????2). Can you define a similar time for this

circuit? What would be the value of that (half-life? twice-life?) time for this circuit? Explain how

you determined the value for this time from the voltage graphs.

E. Knowing the units of R and L and using dimensional analysis, can you predict a relation between R

and L that will result in a time value (units of seconds)? Justify your prediction. Using this

prediction, theoretically predict the half-life time for this circuit. How does it compare to the

experimental value you determined in part D?

Task #2: LRC Circuit

For Task #2, you will make a quantitative study of an LRC system. You will do this by investigating the premade circuit simulation, as shown below, at the following url: http://tinyurl.com/vyqerfc. This consists of

an inductor of 0.85 H, a resistor of 10 Ω, a capacitor of 1μF, and a battery at 5 V which can be connected and

disconnected by a switch. The setup is like the previous task. This time the scopes at the bottom are

showing the voltages across the inductor, VL, the capacitor, VC, and the resistor, VR.

A. Predict what happens to VC, VR, and VL as time increases after opening the switch.

B. Open the switch on the LRC circuit. The battery is now disconnected, but the LRC components

make their own independent loop. Describe in words the behavior of the voltages across R, L, and C

in this LRC circuit. Can you think of a mechanical system that has a similar behavior? Describe in

detail how they are similar.

C. Based on Ohm’s law, what can you say about the current in this circuit as a function of time?

Describe it as precisely as possible.

D. Using values from the scope (voltage graph), estimate the time it takes for the system to reach

equilibrium.

a. In a table, record several amplitudes and their corresponding time values over a sufficient

length of time (at least 10 data points).

b. Using Excel or your favorite spreadsheet program, graph the voltage amplitudes as a

function of time, V vs t. Be sure to add a trendline and display your equation. (Hint: it’s

not linear and don’t use a polynomial). Using this graph estimate the time it takes for the

system to reach equilibrium.

c. Estimate the half-life time (the time it takes for the amplitude of oscillations to decay to half

the initial value) for this circuit.

d. Measure the time it takes for the system to oscillate ten times. From this value, determine

the average frequency of oscillation for this circuit.

Using Kirchhoff’s laws, the LRC circuit equation can be written:

?? + ?? + ?? = 0

?

?

2?

??

2 + ?

??

?? +

1

?

? = 0

An exact solution for the charge q in the LRC circuit as a function of time can be written:

? = ?0?

−

?

2?

?

cos (?

′

? + ?)

where ?

′ = √

1

??

− (

?

2?

)

2

and ?

′ = 2??

E. Using your measured experimental values, determine the theoretically predicted oscillation

frequency for the LRC circuit. Compare this to your experimental value of the oscillation frequency

of the circuit.

F. Considering your description of the current in the circuit and knowing the relation between charge

q(t) and current I(t), does this solution seem reasonable? That is, does it make sense that the solution

has an exponential contribution and a cosine contribution? Explain why or why not.

G. Try to think back to Lab CE-01: Spring Into Action where you graphed the position of a cart/spring

system as a function of time. How do the graphs seen in this lab compare to those graphs? One

could say for the cart/spring system, that work was initially done by compressing and stretching the

springs. Once released, the energy stored in the springs was converted into kinetic energy of the

cart, which was converted back into spring potential energy. The total energy decreased in time due

to the losses to friction. Write a similar paragraph describing the energy transfers that are occurring

in the RLC circuit. How the energy is initially stored in the circuit? What does the energy oscillate

between? How is the energy lost?

H. Significantly increase the value of the resistor in your circuit (say, by a factor of five). Collect data

for the decay of this system. How did the increased resistance change the total decay time? How did

the increased resistance change the half-life time? How did the increase in resistance change the

frequency of oscillations? Does this make sense based on your previous experiences with oscillating

systems?

Wrap-Up

• Don’t forget to write your Implications section!

• Submit your individual lab report on Canvas. (Make sure to upload all relevant files.)

References

The simulations used in this lab are works by Paul Falstad and were adapted to JavaScript by Iain Sharp. The

program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public

License as published by the Free Software Foundation, either version 2 of the License, or (at your option)

any later version.