In a uniform magnetic field of 0.049 T, a magnetic needle performs 20 complete oscillations in 5 seconds as shown. The moment of inertia of the needle is $9.8 \times 1 0^{- 6} kg m^{2}$ . If the magnitude of magnetic moment of the needle is $x \times 1 0^{- 5} A m^{2}$ , then the value of ' $x$ ' is :

Option 1 -

$5 π^{2}$

Option 2 -

$128 π^{2}$

Option 3 -

$50 π^{2}$

Option 4 -

$1280 π^{2}$

4 Views | Posted a month ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 4

Detailed Solution:

Time period of Oscillation, $T = 2 π \sqrt[]{\frac{I}{M B}}$

$\Rightarrow \frac{1}{4} = 2 π \sqrt[]{\frac{9.8 \times 1 0^{- 6}}{M \times 0.049}}$

$\Rightarrow \frac{1}{16} = 4 π^{2} \times \frac{9.8 \times 1 0^{- 6}}{M \times 49 \times 1 0^{- 3}}$

$\Rightarrow M = \frac{4 π^{2} \times 9.8 \times 1 0^{- 6}}{49 \times 1 0^{- 3}} \times 16$

$= \frac{4 π^{2} \times 9.8 \times 16 \times 1 0^{- 3}}{49}$

$= 12.8 π^{2} \times 1 0^{- 3} \times 1 0^{- 2} \times 1 0^{2}$

$= 1280 π^{2} \times 1 0^{- 5} A m^{2}$

Time period of Oscillation, <math> <mrow> <mi>T</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mroot> <mrow> <mfrac> <mrow> <mi>I</mi> </mrow> <mrow> <mi>M</mi> <mi>B</mi> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mroot> <mrow> <mfrac> <mrow> <mn>9</mn> <mo>.</mo> <mn>8</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <mi>M</mi> <mo>×</mo> <mn>0</mn> <mo>.</mo> <mn>0</mn> <mn>4</mn> <mn>9</mn> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mn>6</mn> </mrow> </mfrac> <mo>=</mo> <mn>4</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mfrac> <mrow> <mn>9</mn> <mo>.</mo> <mn>8</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <mi>M</mi> <mo>×</mo> <mn>4</mn> <mn>9</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </math>  <math> <mrow> <mo>⇒</mo> <mi>M</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>9</mn> <mo>.</mo> <mn>8</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mn>9</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> <mo>×</mo> <mn>1</mn> <mn>6</mn> </mrow> </math>  <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>9</mn> <mo>.</mo> <mn>8</mn> <mo>×</mo> <mn>1</mn> <mn>6</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mn>9</mn> </mrow> </mfrac> </mrow> </math>  <math> <mrow> <mo>=</mo> <mn>1</mn> <mn>2</mn> <mo>.</mo> <mn>8</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math>  <math> <mrow><mo>=</mo><mn>1</mn><mn>2</mn><mn>8</mn><mn>0</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo><mn>5</mn></mrow></msup>A<msup><mrow>m</mrow><mrow><mn>2</mn></mrow></msup></mrow> </math>

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In a uniform magnetic field of 0.049 T, a magnetic needle performs 20 complete oscillations in 5 seconds as shown. The moment of inertia of the needle is 9.8×10−6 kg m2 . If the magnitude of magnetic moment of the needle is x×10−5Am2 , then the value of ' x ' is :