Question

The earth's magnetic field strength is 5.0x10^-5 T. How fast would you have to drive your car to create a 4.0 motional emf along your 1.0 -long radio antenna? Assume that the motion of the antenna is perpendicular to $\vec {B}$ .

Answer 1

Concepts and reason

The concepts used to solve this question is the magnetic force, Lenz’s law and the induced emf.

Initially, explain the Lenz’s law and later derive the expression for the electromotive force. Later use the given values to determine the speed of the car to create the motional emf.

Fundamentals

Lenz’s law:

“Lenz’s law states that the change in magnetic flux generates an emf called as induced emf and the polarity of the induced emf opposes the change which produces it”

The emf induced in the coil is,

$\varepsilon = - N\frac{{d\phi }}{{dt}}$

Here, $\varepsilon$ is the induced emf, $\frac{{d\phi }}{{dt}}$ is the rate of change in magnetic flux, $N$ is the number of turns in the coil.

Magnetic flux:

The magnetic flux is defined as the number of magnetic field lines passing through a certain area. The strength of the magnetic flux depends on the number of lines passing through a certain area.

The magnitude of the magnetic flux is,

$\phi = BA\cos \theta$

Here, $B$ is the magnetic field strength, $A$ is the area of the surface and $\theta$ is the angle between magnetic field lines and normal to the area.

The magnetic flux in the loop when the angle between field line and a vector normal to the area is $0^\circ$

$\begin{array}{c}\\\phi = BA\cos 0^\circ \\\\ = Blx\\\end{array}$

Here, $x$ is the thickness.

The induced emf is,

$\varepsilon = - \frac{{d\phi }}{{dt}}$

Substitute $Blx$ for $\phi$

$\begin{array}{c}\\\varepsilon = - \frac{{d\phi }}{{dt}}\\\\ = - \frac{{d\left( {BLx} \right)}}{{dt}}\\\\ = - BLv\\\end{array}$

Here, the negative sign indicates that the current flows in the opposite direction of change in flux.

The motional emf is,

$\varepsilon = Bvl$

The above equation is modified as,

$v = \frac{\varepsilon }{{Bl}}$

Substitute $4{\rm{ V}}$ for $\varepsilon$ , $5 \times {10^{ - 5}}{\rm{ T}}$ for $B$ and $1{\rm{ m}}$ for $l$

$\begin{array}{c}\\v = \frac{\varepsilon }{{Bl}}\\\\ = \frac{{4{\rm{ V}}}}{{\left( {5 \times {{10}^{ - 5}}{\rm{ T}}} \right)\left( {1{\rm{ m}}} \right)}}\\\\ = 8 \times {10^4}{\rm{ m/s}}\\\end{array}$

Ans:

The speed of the car to create the motion emf of $4{\rm{ V}}$ is $8 \times {10^4}{\rm{ m/s}}$ .