# 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}

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}

{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}