Suppose a magnetic field B(t) oscillates with frequency w. A circular loop of copper lies perpendicular to the magnetic field. The radius of the circular loop is r. a. Write down an expression for the magnetic field as a function of time. Determine the induced emf E in the loop of wire and use this to calculate the current generated in the loop as a function of time. b. What is the power dissipation in the wire as a function to time? Make a sketch of this function. What is the average power Pave dissipation in the wire? Hint: what is the average value of the function you sketched? C. Recall that power is a rate of energy transfer, and that power dissipated by a resistor leads to a change in the thermal energy of the material (in this case the copper wire). We can relate a AEth change in thermal energy to a change in temperature by AT where M is the total mass and c Mc = is the specific heat capacity of the material (see page 526 for details). Find an expression for a differential change in temperature of the copper wire loop. dT dt
Suppose a magnetic field B(t) oscillates with frequency w. A circular loop of copper lies perpendicular to the magnetic field. The radius of the circular loop is r. a. Write down an expression for the magnetic field as a function of time. Determine the induced emf E in the loop of wire and use this to calculate the current generated in the loop as a function of time. b. What is the power dissipation in the wire as a function to time? Make a sketch of this function. What is the average power Pave dissipation in the wire? Hint: what is the average value of the function you sketched? C. Recall that power is a rate of energy transfer, and that power dissipated by a resistor leads to a change in the thermal energy of the material (in this case the copper wire). We can relate a AEth change in thermal energy to a change in temperature by AT where M is the total mass and c Mc = is the specific heat capacity of the material (see page 526 for details). Find an expression for a differential change in temperature of the copper wire loop. dT dt
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Transcribed Image Text:Suppose a magnetic field B(t) oscillates with frequency w. A circular loop of copper lies
perpendicular to the magnetic field. The radius of the circular loop is r.
a.
Write down an expression for the magnetic field as a function of time. Determine the
induced emf & in the loop of wire and use this to calculate the current generated in the loop as a
function of time.
b.
What is the power dissipation in the wire as a function to time? Make a sketch of this
function. What is the average power Pave dissipation in the wire? Hint: what is the average value of
the function you sketched?
C.
Recall that power is a rate of energy transfer, and that power dissipated by a resistor
leads to a change in the thermal energy of the material (in this case the copper wire). We can relate a
change in thermal energy to a change in temperature by AT
where M is the total mass and c
ΔΕth
Mc
is the specific heat capacity of the material (see page 526 for details). Find an expression for a
dT
-
differential change in temperature
of the copper wire loop.
dt
d.
Suppose the copper is initially at some temperature To. Find an expression for the
temperature of the loop as a function of the time T(t) it is exposed to the oscillating magnetic field.
Hint: Integrate.
e.
Suppose that a 10.0 mT magnetic field oscillates at 1000 Hz, and the radius of the loop
is 2.0 cm.
Assuming the initial temperature was To = 283 K, calculate the temperature of the copper
loop after 1.0 minute of exposure to the oscillating magnetic field. The mass density of copper is
Pm = 8.96 g/cm³. The resistivity of copper is found in table 27.2 and the specific heat capacity is
found on page 526. Express your answer in °C. Comment on the result. Is this a large change in
temperature? Suggest a practical application for this technology. How could this be used?
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