# 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}

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}