There are two current carrying planar coil made each from identical wires of length L. C1is circular (radius R) and C2 is square (side a). They are so constructed that they have same frequency of oscillation when they are placed in the same uniform B and carry the same current. Find a in terms of R.

This is a Long Answer Type Questions as classified in NCERT ExemplarExplanation- as we know n = L/2 π R Magnetic moment of circle = m1= n1IA1= L.I π R .2/2 π R = LIR/2………(1)Magnetic moment of square = m2= n2IA2= L 4 a .I.a2= Lia/4…………….(2)Moment of inertia of circle = MR2/2…………….(3)Moment of inertia of square = Ma2/12…………….(4)Frequency of circle f1= 2 π I 1 m 1 B Frequency of square f2= 2 π I 2 m 2 B As f1=f22 π I 1 m 1 B =2 π I 2 m 2 B So m2/m1= I2/I1From eqn 1,2,3 and 4 L I a . 2 4 × L I R = M a 2 2 12 M R 2 a 2 R = a 2 6 R 2 3 R = a

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Magnetism and Matter Physics Ncert Solutions Class 12th Physics Class 12th Physics NCERT Exemplar Solutions Class 12th Chapter Five

There are two current carrying planar coil made each from identical wires of length L. C₁is circular (radius R) and C₂ is square (side a). They are so constructed that they have same frequency of oscillation when they are placed in the same uniform B and carry the same current. Find a in terms of R.

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alok kumar singh | Contributor-Level 10

4 months ago

This is a Long Answer Type Questions as classified in NCERT Exemplar

Explanation- as we know n = L/2 $π R$

Magnetic moment of circle = m₁= n₁IA₁= L.I $π R$ .²/2 $π$ R = LIR/2………(1)

Magnetic moment of square = m₂= n₂IA₂= $\frac{L}{4 a}$ .I.a²= Lia/4…………….(2)

Moment of inertia of circle = MR²/2…………….(3)

Moment of inertia of square = Ma²/12…………….(4)

Frequency of circle f₁= 2 $π \sqrt[]{\frac{I_{1}}{m_{1} B}}$

Frequency of square f₂= 2 $π \sqrt[]{\frac{I_{2}}{m_{2} B}}$

As f₁=f₂

2 $π \sqrt[]{\frac{I_{1}}{m_{1} B}}$ =2 $π \sqrt[]{\frac{I_{2}}{m_{2} B}}$

So m₂/m₁= I₂/I₁

From eqn 1,2,3 and 4

$\frac{L I a . 2}{4 \times L I R}$ = $\frac{M a^{2} 2}{12 M R^{2}}$

$\frac{a}{2 R} = \frac{a^{2}}{6 R^{2}}$

$3 R = a$

This is a Long Answer Type Questions as classified in NCERT ExemplarExplanation- as we know n = L/2 <math> <mi>π</mi> <mi>R</mi> </math> Magnetic moment of circle = m1= n1IA1= L.I <math> <mi>π</mi> <mi>R</mi> </math> .2/2 <math> <mi>π</mi> </math> R = LIR/2………(1)Magnetic moment of square = m2= n2IA2= <math> <mfrac> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mrow> </mfrac> </math> .I.a2= Lia/4…………….(2)Moment of inertia of circle = MR2/2…………….(3)Moment of inertia of square = Ma2/12…………….(4)Frequency of circle f1= 2 <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>I</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <mi>B</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> Frequency of square f2= 2 <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>I</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> <mi>B</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> As f1=f22 <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>I</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <mi>B</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> =2 <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>I</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> <mi>B</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> So m2/m1= I2/I1From eqn 1,2,3 and 4 <math> <mfrac> <mrow> <mrow> <mi>L</mi> <mi>I</mi> <mi>a</mi> <mo>.</mo> <mn>2</mn> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <mo>×</mo> <mi>L</mi> <mi>I</mi> <mi>R</mi> </mrow> </mrow> </mfrac> </math> = <math> <mfrac> <mrow> <mrow> <mi>M</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mn>2</mn> </mrow> </mrow> <mrow> <mrow> <mn>12</mn> <mi>M</mi> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> <math> <mfrac> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>6</mn> <msup> <mrow> <mrow> <mi>R</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> <math> <mn>3</mn> <mi>R</mi> <mo>=</mo> <mi>a</mi> </math>

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