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Next: Worked Example 2: Lenz's Up: Magnetic Induction Previous: Worked Examples

Example 9.1: Faraday's law

Question: A plane circular loop of conducting wire of radius $r=10$cm which possesses $N=15$ turns is placed in a uniform magnetic field. The direction of the magnetic field makes an angle of $30^\circ$ with respect to the normal direction to the loop. The magnetic field-strength $B$ is increased at a constant rate from $B_1=1$T to $B_2=5$T in a time interval of ${\mit\Delta}t=10$s. What is the emf generated around the loop? If the electrical resistance of the loop is $R=15\,\Omega$, what current flows around the loop as the magnetic field is increased?
 
Answer: The area of the loop is

\begin{displaymath}
A = \pi\,r^2 = \pi\,(0.1)^2 = 0.0314\,{\rm m}^2.
\end{displaymath}

The component of the magnetic field perpendicular to the loop is

\begin{displaymath}
B_\perp = B\,\cos\theta = B\,\cos 30^\circ = 0.8660\,B,
\end{displaymath}

where $B$ is the magnetic field-strength. Thus, the initial magnetic flux linking the loop is

\begin{displaymath}
{\mit\Phi}_{B\,1} = N\,A\,B_1\,\cos\theta = (15)\,(0.0314)\,(1)\,(0.8660) = 0.408\,{\rm Wb}.
\end{displaymath}

Likewise, the final flux linking the loop is

\begin{displaymath}
{\mit\Phi}_{B\,2} = N\,A\,B_2\,\cos\theta =
(15)\,(0.0314)\,(5)\,(0.8660) = 2.039\,{\rm Wb}.
\end{displaymath}

The time rate of change of the flux is

\begin{displaymath}
\frac{d{\mit\Phi}_B}{dt} = \frac{{\mit\Phi}_{B\,2}-
{\mit\P...
...Delta}t} = \frac{(2.039-0.408)}{(10)}=0.163\,{\rm Wb\,s}^{-1}.
\end{displaymath}

Thus, the emf generated around the loop is

\begin{displaymath}
{\cal E} = \frac{d{\mit\Phi}_B}{dt} = 0.163\,{\rm V}.
\end{displaymath}

Note, incidentally, that one weber per second is equivalent to one volt.

According to Ohm's law, the current which flows around the loop in response to the emf is

\begin{displaymath}
I = \frac{{\cal E}}{R} = \frac{(0.163)}{(15)} = 0.011\,{\rm A}.
\end{displaymath}


next up previous
Next: Worked Example 2: Lenz's Up: Magnetic Induction Previous: Worked Examples
Richard Fitzpatrick 2007-07-14