Teaching Physics with the Physics Suite

Edward F. Redish

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Alternative Homework Assignment: Tuning a Radio

Lucy is driving along a highway listening to her car radio when thestation she is listening to begins to fade in and out forcing Lucy to tunein a new station. As she turns the knob on the radio, many different stationsbecome tuned in and then fade out, and Lucy begins to wonder exactly howturning the knob allows the radio to distinguish one station from another.Thinking more about her radio, Lucy begins to wonder how it works asidefrom selecting a station. In short she wonders how it turns radio wavesinto sound waves. In solving this problem, you will use a radio and yourknowledge of AC circuits to answer Lucy's questions.

Part 1:

a) When radio waves interact with a radio's antenna, they induce a currentin the antenna which results in an EMF across the radio's tuner. Each radiostation broadcasts radio waves of a particular frequency. How does thefrequency of the induced EMF compare to the frequency of the radio waves?

b) Get a radio and turn in on. The radio you have before you is receivingradio waves from many different sources. The radio waves from every localradio station, not to mention television stations, radio phones, and manyother sources of radio waves are causing the antenna on your radio to generatea complicated EMF made up of many different frequencies all correspondingto the different sources. The tuner's task is to seperate one of thesesignals out from the others and amplify it. One way of accomplishing thisis by using a series RCL circuit. To begin to see how this works, writean expression for the impedance of an RCL circuit as a function of frequency.R,C, and L will be parameters of this expression.

c) Find the frequency that minimizes the impedance in this circuit.This state of minimal impedance is called resonance.

d) Choose values of R, C, and L that are reasonable real world values,and for which the frequency that minimizes the impedance is one of yourfavorite radio station's frequency.

e) Now find the four radio stations that are most nearly adjacent tothe station you used in the question above. Record the frequencies of thesestations.

f) Calculate the impedance in the circuit you have designed at eachof these frequencies.

g) In order to easily supress the other signals, the impedance for theresonant frequency should be one tenth that of the next smallest impedanceyou calculated. If this is not the case, change your values of R, C, andL until your circuit meets this requirement.

Picture courtesy of Rick McMillion

Part 2:

a) Now consider changing the station. In order to do this you must changethe resonant frequency. Change the capicitance of your circuit so thatit tunes in a different station and check to see if the impedence at adjacentstations' frequencies is large enough to supress these stations' signals.

b) Would the effect have been different if you had changed the inductanceof the circuit rather than the capacitance? If so, how?

c) If a variable capacitor is used to change the station, what rangeof capacitance must this capacitor have in order to reach the entire A.M.dial? How about the F.M. dial?

Part 3:

a) Once a particular broadcast has been seperated from the other radiosources, the radio must read the information that is contained in the signaland translate this into a series of electrical impulses that are to sentto the speaker in order to changed to sound. The exact method for doingthis is beyond the scope of this problem, but we can look at how the speakerchanges these impulses into sound. The components of a loudspeaker areshown disassembled in the figure to the right. A pair of permanent magnetscreate a magnetic field that exerts forces on the current in the voicecoil. The voice coil is attached to the speaker cone and the magnetic forcescause the cone to vibrate in a way that matches the vibrations of currentin the voice coil. What is the net force on a current loop in a uniformmagnetic field?

b) The magnetic force on the voice coil arises because the magneticfield at the coil is not uniform. Suppose a voice coil in a loudspeakerhas 40 turns of wire and a diameter of 1.8 cm, and the current in the coilis 0.8 A. Assume the magnetic field at each point of the wire of the coilhas a constant magnitude of 0.2 T and is directed at an angle of 60ºoutward from the normal to the plane of the coil. Let the axis of the coilbe the y direction. The current in the coil is in the direction shown inthe figure below (counterclockwise as viewed from a point above the coilon the y-axis). Calculate the magnitude and direction of the resultantmagnetic force on the coil.

c) Explain how this system converts electric impulses to sound waves.

Work supported in part by NSF grant DUE-9455561

These problems written and collected by K. Vick, E. Redish, and P. Cooney. These problems may be freely used in classrooms. They may be copied and cited in published work if the Activity-Based Physics (ABP) Alternative Homework Assignments are mentioned and the source cited.

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Page last modified October 27, 2002