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A circular loop in the plane of the paper lies in a 0.40T magnetic field pointing into the paper....

Question:

A circular loop in the plane of the paper lies in a {eq}0.40T {/eq} magnetic field pointing into the paper. The loop's diameter changes from {eq}22.8cm {/eq} to {eq}6.4cm {/eq} in {eq}0.55s {/eq}.

a) What is the magnitude of the average induced emf?

b) If the coil resistance is {eq}2.1\Omega {/eq}, what is the average induced current?

Electromagnetic induction

Electromagnetic induction was discovered by Michael Faraday in the first half of the 19th century. This phenomenon suggests that the variation in time of the magnetic field flux ({eq}\Phi_B {/eq}) that crosses a conductive surface will generate an electromotive force (potential difference) ({eq}\varepsilon {/eq}). The mathematical expression of this law is known by the Faraday-Lenz Law, since Heinrich Lenz discovered that the effects generated in the material opposed the change of flow. This is reflected in the minus sign of the equation:

{eq}\varepsilon =-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} {/eq}

Answer and Explanation:

a) It is possible to demonstrate that for magnetic fields constant in time, the magnetic field flux that crosses a given surface can be calculated by the expression:

{eq}\Phi_B=\vec{B}\cdot\vec{S}=BS\cos\theta {/eq}

In this case the magnetic field is perpendicular to the surface enclosed by the loop therefore:

{eq}\cos\theta=\cos 0=1\Rightarrow \Phi_B=BS {/eq}

Applying now the Faraday-Lenz's law:

{eq}\varepsilon =-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}=-B\frac{\mathrm{d}S}{\mathrm{d}t}=-B\frac{\Delta S}{\Delta t}=-2\pi B\left (\frac{r_f^2-r_0^2}{\Delta t} \right )\\ \therefore \varepsilon =-2\pi (0.40\,\mathrm{T})\left (\frac{(6.4\cdot10^{-2}\,\mathrm{T})^2-(22.8\cdot10^{-2}\,\mathrm{T})^2}{0.55\,\mathrm{s}} \right )=0.22\,\mathrm{V} {/eq}

b) To calculate the current flowing through the loop, simply use Ohm's law, therefore:

{eq}I=\frac{\varepsilon }{R}=\frac{0.22\,\mathrm{V}}{2.1\,\mathrm{\Omega}}=0.10\,\mathrm{A} {/eq}


Learn more about this topic:

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Lenz's Law, Magnetic Flux and Motional EMF

from GACE Physics (530): Practice & Study Guide

Chapter 23 / Lesson 9
7.6K

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