Chapter 21, Problem 1RQ

Review Question 21.1 Your friend thinks that relative motion of a coil and a magnet is absolutely necessary to induce current in a coil that is not connected to a battery Support your friend’s point of view with a physics argument. Then provide a counterargument and describe an experiment you could perform to disprove your friend's idea.

To determine

A physics argument that relative motion between a magnet and a coil is absolutely necessary to induce current in the coil which is not connected to a battery. Also, provide a counterargument and disprove the above statement with the help of an experiment.

Answer to Problem 1RQ

Solution:

“Whenever the magnetic field flux through an area which is enclosed by a closed conducting loop (coil) changes, a current is induced in the loop (coil)”. So, in order to change the flux, a relative motion between the coil and the magnet must be provided.

A counterstatement for the above statement is, ‘in order to induce current in the coil, the magnetic flux can be changed in a number of ways and not just by providing a relative motion between the magnet and coil.’ The flux can be changed by manipulating the magnitude of the magnetic field $\overrightarrow{B}$, the area of the coil or the angle between area vector, $d\overrightarrow{S}$, and the magnetic field, $\overrightarrow{B}$.

Experiment- Take a magnet with a controllable magnetic field, and fix its position relative to the coil. Now increase the magnitude of the magnetic field. As the field changes, it produces a change in the magnetic flux, thus inducing an EMF and hence, a current is induced in the coil without relative motion between the magnet and the coil.

Explanation of Solution

Introduction:

Whenever the magnetic field flux through an area which is enclosed by a closed conducting loop (coil) changes, an EMF (Electromotive force) is produced in the loop.

The equation for Faraday’s Law of Electromagnetic Induction is:

$\epsilon =\frac{\Delta \varphi }{\Delta t}$

Here, $\epsilon$ is the induced EMF, $\Delta \varphi$ is the change in magnetic flux and $\Delta t$ is the change in time.

Explanation:

According to Faraday’s Law, whenever there is a change in the magnetic flux of the coil, an EMF is induced in the coil. The induced EMF results in an induced current inside the coil. The most common way to produce this EMF is to move the magnet or magnetic field relative to the coil or vice versa. By doing so, the flux across the coil changes and current is induced.

But contradictory to the general belief, the magnetic flux can be changed by other ways also. One way to change the magnetic flux is to change the magnitude of the magnetic field at the site of the loop.

Consider a coil of area $\left(A\right)$ $0.1{\text{ m}}^2$, number of turns $\left(N\right)$ $5$, resistance $\left(R\right)$ $1\Omega$ and a magnetic field $\left(\overrightarrow{B}\right)$ $0.3\text{ T}$ produced from a magnet passing through the coil. The plane of the coil is perpendicular to the magnetic field. Now increase the magnetic field from $0.3\text{ T}$ to $0.8\text{ T}$ in $1\text{ sec}$.

Now, write the expression for magnetic flux.

${\Phi }_0=B_{0}A$

And,

${\Phi }_F=B_{F}A$

Here, ${\Phi }_0$ and ${\Phi }_F$ are the initial and final magnetic fluxes, respectively, $B_0$ and $B_F$ are the initial and final magnetic fields, respectively and $A$ is the area of the coil.

Substitute $0.3\text{ T}$ for $B_0$, and $0.1{\text{ m}}^2$ for $A$ in the above equation of initial magnetic flux.

$\begin{array}{c}{\Phi }_0=\left(0.3\text{ T}\right)\left(0.1{\text{ m}}^2\right)\\ =0.03\text{ T}\cdot {\text{m}}^2\end{array}$

Similarly, substitute $0.8\text{ T}$ for $B_F$, and $0.1{\text{ m}}^2$ for $A$ in the above equation of final magnetic flux.

$\begin{array}{c}{\Phi }_F=\left(0.8\text{ T}\right)\left(0.1{\text{ m}}^2\right)\\ =0.08\text{ T}\cdot {\text{m}}^2\end{array}$

Now, the equation of induced EMF ($\epsilon$) is,

$\begin{array}{c}\epsilon =-N\frac{\Delta \varphi }{\Delta t}\\ =\frac{-N\left({\varphi }_F-{\varphi }_0\right)}{\Delta t}\end{array}$

Here, $N$ is the number of turns, $\Delta \varphi$ is the change in flux, ${\varphi }_F$ is the final magnetic flux, ${\varphi }_0$ is the initial magnetic flux, and $\Delta t$ is the time interval.

Substitute $5$ for $N$, $1\text{ sec}$ for $\Delta t$, $0.08\text{ T}\cdot {\text{m}}^2$ for ${\varphi }_F$ and $0.03\text{ T}\cdot {\text{m}}^2$ for ${\varphi }_0$ in the above equation.

$\begin{array}{c}\epsilon =\frac{-5\left(0.08\text{ T}\cdot {\text{m}}^2-0.03\text{ T}\cdot {\text{m}}^2\right)}{1\text{ sec}}\\ =-0.25\text{ V}\end{array}$

Induced current $\left(i\right)$ in the coil is expressed as:

$i=\frac{\epsilon }{R}$

Here, $R$ is the resistance and $\epsilon$ is the induced EMF.

Substitute $-0.25\text{ V}$ for $\epsilon$ and $1\Omega$ for $R$ in the above expression.

$\begin{array}{c}i=\frac{-0.25\text{ V}}{1\Omega }\\ =-0.25\text{ A}\end{array}$

Here, the negative sign denotes the direction of EMF. $$

Hence, an induced current has been achieved without the relative motion of the coil and magnet.

As discussed in the experiment, a change in the magnitude of magnetic field results in an induced current.

Another way is to change the area of the loop. The magnetic flux ($\Phi$) is an integral function of magnetic field ($\overrightarrow{B}$) and area of the coil or loop ($d\overrightarrow{S}$).

$\Phi ={\int }^{\text{}}\overrightarrow{B}\cdot d\overrightarrow{S}$

So, a change in area causes a change in magnetic flux, thereby inducing a current in the coil.

Also, a change in the direction of area vector and magnetic field vector can cause an induced current to develop in the coil.

Conclusion:

The statement that a relative motion between the coil and the magnet is absolutely necessary, has been subjected to a physics argument. A counterstatement has also been provided, which is proven with the help of an experiment that current in a coil can also be induced by changing the magnetic field, as given by Faraday’s law.