# A 137-turn circular coil of radius 2.07 cm is immersed in a uniform magnetic field that is...

## Question:

A 137-turn circular coil of radius 2.07 cm is immersed in a uniform magnetic field that is perpendicular to the plane of the coil. During 0.149 s the magnetic field strength increases from 57.5 mT to 94.7 mT. Find the magnitude of the average EMF, in millivolts, that is induced in the coil during this time interval.

## Faraday's law of Electromagnetic Induction:

When a conducting loop is placed in a magnetic field an emf is induced in the loop until there is a change in the magnetic flux through the loop. This emf is called induced emf and the current thus produced is called induced current.

The magnetic flux of a field {eq}B {/eq} through an area {eq}A {/eq} is given by:

{eq}\begin{align*} \Phi_B &= \int \vec B \cdot d\vec A \\ &= BA & \text{[when the magnetic field is perpendicular to the cross-section of the area]} \end{align*} {/eq}

Given:

• Number of turns in a circular coil is {eq}N = 137 \ \text {turns} {/eq}
• The radius of a coil is {eq}r = 2.07 \ cm = 2.07 \times 10^{-2} \ m {/eq}
• The change in magnetic field is {eq}dB = B_2 - B_1 = 94.7 - 57.5 = 37.2 \ mT = 37.2 \times 10^{-3} \ T {/eq}
• The angle between the magnetic field lines and the normal (perpendicular) to A is {eq}\theta = 0^\circ {/eq}
• The time duration is {eq}dt = 0.149 \ s {/eq}

Let

• The magnitude of the average EMF induced in the coil is {eq}\varepsilon {/eq}.

After plugging the given values in the formula of average induced emf, we get the magnitude of induced emf in the coil.

The induced emf can be calculated using the formula;

{eq}\begin{align} \varepsilon & = - N \dfrac { d\phi} {dt} \\ & = - N \dfrac { d } {dt} (BA\cos\theta ) \\ & = - N A \cos\theta \dfrac { dB } {dt} \\ & = - N A \cos\theta \dfrac { (B_2 -B_1 ) } {dt} \\ & = - (137) \pi (2.07 \times 10^{-2})^2 \ cos0^\circ \left \{ \dfrac { 37.2 \times 10^{-3} } { 0.149 } \right \} \\ & = -( 1.84 \times 10^{-1}) ( 249.66 \times 10^{-3} ) \\ \implies \varepsilon & = -45.9 \ mV \\ \end{align} {/eq}

Hence, the magnitude of the average EMF induced in the coil is {eq}45.9 \ mV {/eq}.