Chapter 13, Problem 13.1CYU

Chek sour Understanding A closely und coil has a radius of 4.0 cm. 50 turns, and a total ieslstancc of 40 . At what rae must a magnetic field peipendicular lathe face of the coil change En euler to prodice Joule hearing in the cull a a tale of 2.0 mW

To determine

To Calculate:

The rate at which magnetic field perpendicular to the face of the coil carrying current changes in order to produce the given Joule heating in the coil.

Answer to Problem 13.1CYU

The rate change of the magnetic field is $1.1\text{ T/s}$.

Explanation of Solution

Given:

Radius of the coil, $r=4.0\text{cm = 0}\text{.04 m}$

Number of turns, $N=50$

Total resistance, $R=40\Omega$

Joule heating in the coil that is power, $P=2\text{mW}=2\times {10}^{-3}\text{W}$

Formula used:

Power loss in a circuit carrying current is, $P=I^{2}R$ ----(1).

Here $R$ is the resistance of the wire.

Emf produced in a coil is equal to negative of rate change of flux, $\epsilon =-\frac{d\phi }{dt}$ ----(2).

Here $\phi$ is magnetic flux which is given as $\phi =n\overrightarrow{B}.\overrightarrow{A}$ ----(3)

Where, n is the number of turns, $B$ is the magnetic field & $A$ is the area of the coil.

From equation(1) current in the coil is given as,

  $I=\sqrt{\frac{P}{R}}$

Substituting the value of $P$ & $R$ we get,

  $I=\sqrt{\frac{P}{R}}=\sqrt{\frac{2\times {10}^{-3}}{40}}=7.0\times {10}^{-3}=7.0\text{mA}$

And also power $P=VI$ or the emf is, $emf=V=\frac{P}{I}=\frac{2\times {10}^{-3}}{7\times {10}^{-3}}=0.28\text{V}$

From equation (2) and equation (3), the emf produced in the coil is $\epsilon =-A\frac{dB}{dt}$. Here $A$ is the area of the coil which is $A=\pi r^2$. Now

  $\epsilon =-n\pi r^2\frac{dB}{dt}$

So, the rate at which magnetic field is changing is

  $\frac{dB}{dt}=-\frac{\epsilon }{n\pi r^2}$

Substituting the value of $\epsilon$, $\pi$ & $r$, we get,

  $\frac{dB}{dt}=-\frac{0.28}{50\times (3.14){(0.04)}^2}=-1.1\text{ T/s}$

Conclusion:

Thus, the rate at which the magnetic field perpendicular to the face of the coil carrying current should change is $1.1\text{ T/s}$.