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Physics 202
Assignment 10
Due at 11:00pm on Sunday, April 13, 2008
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The Resonance Peak
Description:
Short quantitative problem on resonance in a series RLC circuit. Based on Young/Geller Quantitative Analysis
22.5.
In this problem we consider the resonance curve for a circuit with electrical components
,
, and
and resonant frequency
.
Part A
For given values of
and
, if you double the value of
, how does the new resonance curve differ from the original one?
Hint A.1
How to approach the problem
In a series R-L-C circuit, the resonance frequency is determined by
and
, whereas the peak value of the current in the
circuit depends only on
. Since the value of
is being kept constant, the peak of the resonance curve is unchanged.
To solve this problem, use proportional reasoning to find a relation between the resonance frequency
and inductance
.
Find the simplest equation that contains these variables and other known quantities from the problem.
Write this equation twice, once to describe the original circuit and again for the circuit with greater inductance.
You then need to write each equation with all the constants on one side and the variables on the other. Since the
variable is
in this problem, you will write the equations in the form
.
To finish the problem you need to compare the two cases presented. For this question you should find the ratio of the
resonance frequency of the original circuit to that of the circuit with greater inductance.
Hint A.2
Resonance frequency
In a series R-L-C circuit, the resonance frequency
is given by
.
Part A.3
Find an expression for the new resonance frequency
Write an expression for the resonance frequency
of the circuit when the inductance
is doubled?
Express your answer in terms of some or all of the variables
,
, and
.
ANSWER:
=
ANSWER:
Both the peak height and peak frequency double.
The peak height will be half as great, and the peak frequency will double.
The peak height won't change, and the peak frequency will be
times as great.
The peak height won't change, and the peak frequency will be half as great.
In an R-L-C circuit, the resonance peak depends only on
, while the resonant frequency is determined by both
and
. In particular, for a given value of
, the resonance frequency is inversely proportional to the square root of
.
Similarly, for a given value of
, the resonance frequency is inversely proportional to the square root of
.
Part B
For given values of
and
, if you double the value of
, how does the new rms current at resonance
differ from its
original value? Assume that the voltage amplitude of the ac source is the same.
Hint B.1