Let's see how an emf can be induced by a changing magnetic field. A circular coil of wire with $500$ turns and an average radius of $4\mathrm{.}00c{m}^2$ is placed at a $60$ angle to the uniform magnetic field between the poles of a large electromagnet. The field changes at a rate of $-0\mathrm{.}200\frac{T}{s}.$ What is the magnitude of the resulting induced emf?

Figure

$1$ of $1$

SETUP $($Figure $1)$ shows our sketch. We remember that $$ is the angle between the magnetic field and the normal to the coil and thus is $30,$ not $60.$ The area of the coil is $A=(0\mathrm{.}0400cm{)}^2=0\mathrm{.}00503m^2.$

SOLVE To find the magnitude of the emf, we use $E=N|\frac{{}_B}{}t|.$ We first need to find the rate of change of the magnetic flux, $\frac{{}_B}{}t.$ The flux at any time is given by ${}_B=$BAcos$,$ so the magnitude of the rate of change of flux is

$|\frac{{}_B}{t}|=|\frac{B}{t}|Acos$

$=(-0\mathrm{.}200\frac{T}{s})(0\mathrm{.}00503m^2)cos30$

$=0\mathrm{.}000871T*\frac{m^2}{s}=0\mathrm{.}000871W\frac{b}{s}$

We now calculate the induced emf:

$E=N|\frac{{}_B}{t}|=(500)|(-0\mathrm{.}000871W\frac{b}{s})|=0\mathrm{.}436V$

REFLECT We need the absolute value signs on the right side of Faraday's law because we have defined $E$ to be the magnitude of the emf, always a positive quantity. As we've noted, Faraday's law can be stated so that the emf is positive or negative, depending on its direction. We avoid that complexity by stating the law in terms of the magnitude of the emf.

Part A $-$ Practice Problem:

A $350-$turn coil with an average radius of $0\mathrm{.}020$ m is placed in a uniform magnetic field so that $=41.$ The field increases at a rate of $0\mathrm{.}165\frac{T}{s}.$ What is the magnitude of the resulting emf?

Express your answer with the appropriate units.

$E=\frac{,}{b}ar(V)$

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