# The plane of a rectangular coil, 3.2 cm \times 9.2 cm, is perpendicular to the direction of a...

## Question:

The plane of a rectangular coil, 3.2 cm {eq}\times {/eq} 9.2 cm, is perpendicular to the direction of a uniform magnetic field B. If the coil has 98 turns and a total resistance of 7.6 {eq}\Omega {/eq}, at what rate must the magnitude of B change to induce a current of 0.09 A in the windings of the coil? Answer in units of T/s.

## Electromagnetic Induction:

The process of inducing the voltage in the coil, by changing the magnetic flux through the coil is based on electromagnetic induction. The magnitude of induced emf is given by Faraday's law.

## Answer and Explanation:

Given Data

• Rectangular coil of dimensions:
Length, {eq}L\ = 9.2\ cm\ = 9.2\times 10^{-2}\ m{/eq}
Width, {eq}W\ = 3.2\ cm\ = 3.2\times 10^{-2}\ m{/eq}
• Number of turns of the coil, N = 98
• Angle made by the magnetic field with the normal to the coil, {eq}\theta\ = 0^\circ{/eq}
• Total resistance of the coil, {eq}R\ = 7.6\ \Omega{/eq}
• Induced current, {eq}I\ = 0.09\ A{/eq}

Finding the rate of change of magnetic field ({eq}\dfrac{\Delta B}{\Delta t} {/eq})

Applying Faraday's law of electromagnetic induction:

• {eq}\text{Induced emf}\ = \text{No. of turns}\times \dfrac{\text{change of magnetic flux}}{\text{time}} {/eq}
• {eq}\text{Induced emf}\ = \text{No. of turns}\times \text{Area of coil}\times\dfrac{\Delta B}{\Delta t}\times \cos \theta {/eq}
• {eq}E\ =N\times (L\times W)\times\dfrac{\Delta B}{\Delta t}\times \cos \theta {/eq}

Applying Ohm's law, i.e. {eq}E\ = I\times R{/eq}

• {eq}I\times R\ =N\times (L\times W)\times\dfrac{\Delta B}{\Delta t}\times \cos \theta {/eq}
• {eq}0.09\times 7.6\ =98\times (9.2\times 10^{-2}\times 3.2\times 10^{-2})\times\dfrac{\Delta B}{\Delta t} \times \cos 0^\circ {/eq}
• {eq}\dfrac{\Delta B}{\Delta t} \ = 2.37\ T/s{/eq}