(c) Find the period of oscillations if and . · OlympiadBench #1505

Physics · electromagnetism

Problem

V=\sqrt{V_{1} V_{2}}

and

R_{2}=16 R_{1}

. Given constants: A resistor is made of a material which undergoes a phase transition so that its resistance takes one of the two values,

R_{1}

if its temperature is smaller than

T_{c}

, and

R_{2}>R_{1}

if the temperature is larger than

T_{c}

. <image_1> This resistor is connected to a voltage source through an inductor of inductance

L

. It appears that if the applied voltage

V

is between two critical values,

V_{1}<V<V_{2}

, the temperature of the resistor starts oscillating. Assume that (i) the heat flux

P

from the resistor to the ambient medium is given by

P=\alpha\left(T-T_{0}\right)

, where

\alpha

is a constant,

T

denotes the temperature of the resistor, and

T_{0}

is the ambient temperature; (ii) the geometrical size of the resistor is so small that it will reach a thermal equilibrium much faster than the characteristic time

L / R_{2}

. Context question: (a) Express

V_{1}

and

V_{2}

in terms of the other parameters defined above. Context answer:

V_{1}=\sqrt{R_{1} \alpha\left(T_{c}-T_{0}\right)}

V_{2}=\sqrt{R_{2} \alpha\left(T_{c}-T_{0}\right)}

Context question: (b) Assuming that

V_{1}<V<V_{2}

, sketch qualitatively how the temperature of the resistor

T

depends on time

t

, and find the ratio

\left(T_{\max }-T_{0}\right) /\left(T_{\min }-T_{0}\right)

, where

T_{\max }

and

T_{\min }

denote the maximal and minimal values of

T

, respectively. Context answer:

\frac{R_{2}^{2}}{R_{1}^{2}}

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