5.9 A circular coil of 16 turns and radius 10 cm carrying a current of 0.75 A rests with its plane normal to an external field of magnitude 5.0 × 10^–2 T. The coil is free to turn about an axis in its plane perpendicular to the field direction. When the coil is turned slightly and released, it oscillates about its stable equilibrium with a frequency of 2.0 s^–1. What is the moment of inertia of the coil about its axis of rotation?

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Payal Gupta | Contributor-Level 10

4 months ago

5.9 Number of turns, n = 16

Radius of the coil, r = 10 cm = 0.1 m

Cross-section of the coil, A = $π r^{2}$ = $π \times {0.1}^{2}$ $m^{2}$ = 0.0314 $m^{2}$

Current in the coil, I = 0.75 A

Magnetic field strength, B = 5.0 $\times 10^{- 2}$ T

Frequency of oscillation of the coil, $ν$ = 2/s

Magnetic moment, M = NIA = 16 $\times 0.75 \times 0.0314$ = 0.377 A $m^{2}$

Frequency is given by the relation, $ν$ = $\frac{1}{2 π} \sqrt{\frac{M B}{I}}$ , where I = moment of Inertia of the coil

I = $\frac{M B}{4 π^{2} ν^{2}}$ = $\frac{0.377 \times 5.0 \times 10^{- 2}}{4 π^{2} \times 2^{2}}$ = 1.19 $\times 10^{- 4}$ kg $m^{2}$

5.9 Number of turns, n = 16Radius of the coil, r = 10 cm = 0.1 mCross-section of the coil, A = <math> <mi>π</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> = <math> <mi>π</mi> <mo>×</mo> <mi> </mi> <msup> <mrow> <mrow> <mn>0.1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> <math> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> = 0.0314 <math> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> Current in the coil, I = 0.75 AMagnetic field strength, B = 5.0 <math> <mo>×</mo> <mi> </mi> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </mrow> </msup> </math> TFrequency of oscillation of the coil, <math> <mi>ν</mi> </math> = 2/sMagnetic moment, M = NIA = 16 <math> <mo>×</mo> <mn>0.75</mn> <mi> </mi> <mo>×</mo> <mn>0.0314</mn> </math> = 0.377 A <math> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> Frequency is given by the relation, <math> <mi>ν</mi> </math> = <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mrow> </mfrac> <msqrt> <mrow> <mfrac> <mrow> <mrow> <mi>M</mi> <mi>B</mi> </mrow> </mrow> <mrow> <mrow> <mi>I</mi> </mrow> </mrow> </mfrac> </mrow> </msqrt> </math> , where I = moment of Inertia of the coilI = <math> <mfrac> <mrow> <mrow> <mi>M</mi> <mi>B</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mrow> <mrow> <mi>ν</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> = <math> <mfrac> <mrow> <mrow> <mn>0.377</mn> <mi mathvariant="normal"> </mi> <mo>×</mo> <mn>5.0</mn> <mi mathvariant="normal"> </mi> <mo>×</mo> <mi> </mi> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>×</mo> <msup> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> = 1.19 <math> <mo>×</mo> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </mrow> </msup> </math> kg <math> <msup> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math>

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