ODBAC is fixed rectangular conductor of negligible resistance (CO is not connected) and OP is a conductor which rotates clockwise with an angular ve1locity omega Fig. The entire system is in a uniform magnetic field B whose direction in along the normal to the surface of the rectangular conductor ABDC. The conductor OP is in electric contact with ABDC. The rotating conductor has a resistance of  per unit length. Find the current in the rotating conductor, as it rotates by 180

This is a long answer type question as classified in NCERT Exemplar

At t=o t=T/8= $\pi /4w$  the rod will make contact with side BD.at any time 0 $\pi /4w$

So magnetic flux through area ODQ is $\varnothing =B\times areaOQD=B\times \frac{1}{2QD}\times OD$

=B $\times \frac{1ltan\theta }{2}\times l$

$\varnothing =\frac{1}{2}B{L}^{2}wtan\theta =\frac{1}{2}B{L}^{2}w{sec}^{2}wt$

I=e/R

And R= $\lambda x=\frac{\lambda l}{coswt}$  so I= $\frac{1}{2}\frac{BLw}{coswt\lambda }$

Same result for other side too

<p><span data-teams="true">This is a long answer type question as classified in NCERT Exemplar</span></p><p>At t=o t=T/8=<span contenteditable="false"> <math> <mi>π</mi> <mo>/</mo> <mn>4</mn> <mi>w</mi> </math> </span> &nbsp;the rod will make contact with side BD.at any time 0&lt;t&lt;<span contenteditable="false"> <math> <mi>π</mi> <mo>/</mo> <mn>4</mn> <mi>w</mi> </math> &lt;/t&lt;</span></p><p>So magnetic flux through area ODQ is<span contenteditable="false"> <math> <mi>∅</mi> <mo>=</mo> <mi>B</mi> <mo>×</mo> <mi>a</mi> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>O</mi> <mi>Q</mi> <mi>D</mi> <mo>=</mo> <mi>B</mi> <mo>×</mo> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>Q</mi> <mi>D</mi> </mrow> </mrow> </mfrac> <mo>×</mo> <mi>O</mi> <mi>D</mi> </math> </span></p><p>=B<span contenteditable="false"> <math> <mo>×</mo> <mfrac> <mrow> <mrow> <mn>1</mn> <mi>l</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>θ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>×</mo> <mi>l</mi> </math> </span></p><div><div><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1743054632phpgCHrLr_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1743054632phpgCHrLr.jpeg" alt="" width="249" height="132"></picture></div></div></div></div><p><span contenteditable="false"> <math> <mi>∅</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mi>B</mi> <msup> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mi>w</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mi>B</mi> <msup> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mi>w</mi> <msup> <mrow> <mrow> <mi>s</mi> <mi>e</mi> <mi>c</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mi>w</mi> <mi>t</mi> <mi> </mi> </math> </span></p><p>I=e/R</p><p>And R= <span contenteditable="false"> <math> <mi>λ</mi> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>λ</mi> <mi>l</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>w</mi> <mi>t</mi> </mrow> </mrow> </mfrac> </math> </span>&nbsp;so I=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mfrac> <mrow> <mrow> <mi>B</mi> <mi>L</mi> <mi>w</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>w</mi> <mi>t</mi> <mi>λ</mi> </mrow> </mrow> </mfrac> </math> </span></p><p>Same result for other side too</p>

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