Evelyn wears glasses, whose wire frames are shaped like to circles, each with an area of...

Question:

Evelyn wears glasses, whose wire frames are shaped like to circles, each with an area of 7.0x10{eq}^{-4} {/eq}m{eq}^2 {/eq}. The horizontal component of the Earth's magnetic field, in Evelyn's hometown, is 4.9x10{eq}^{-5} {/eq}T. If Evelyn turns her head back and forth, rotating it through 90 degrees every 0.5s, what is the induced voltage in the wire frame of one eyepiece?

Faraday's law of electromagnetic induction:

Faraday's law of electromagnetic induction states that when a conducting loop is placed in a magnetic field an emf is induced in the loop until there is a change in the magnetic flux through the loop. This emf is called induced emf and the current thus produced is called induced current.

The magnetic flux through a small area {eq}\vec{ds} {/eq} due to a magnetic field {eq}\vec{B} {/eq} is given by

{eq}d\phi = \vec{B} \cdot \vec{ds} {/eq}

From the Faraday's law of electromagnetic induction, the emf induced in the closed loop is;

{eq}\varepsilon = -\dfrac{d\phi}{dt} =-\dfrac{d}{dt} (\oint_{s}\vec{B}\cdot\vec{ds}) {/eq}

Answer and Explanation:

Given:

  • Area of the wire frame is {eq}A = 7\times 10^{-4}\ m^2 {/eq}
  • The horizontal component of the earth magnetic field is {eq}B = 4.9\times 10^{-5}\ T {/eq}
  • The angle of rotation is {eq}\theta = 90^o {/eq}

Change in the magnetic flux through each frame in time {eq}\Delta t = 0.5\ s {/eq} is;

{eq}\begin{align} \Delta \phi &= BA\cos0^o - BA\cos 90^o\\ &= BA\\ \end{align} {/eq}

So, the induced voltage in the wire of one eyepiece is;

{eq}\begin{align} V &=|\dfrac{-\Delta \phi}{\Delta t}|\\ &= \dfrac{BA}{\Delta t}\\ &= \dfrac{4.9\times 10^{-5}\times 7\times 10^{-4}}{0.5}\\ &=6.86\times 10^{-8}\ V\\ \end{align} {/eq}


Learn more about this topic:

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Faraday's Law of Electromagnetic Induction: Equation and Application

from High School Physics: Help and Review

Chapter 13 / Lesson 10
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