6.17 A line charge l per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R. The wheel has light non-conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by,
B = – B0 k (r a; a < R)
= 0 (otherwise)
What is the angular velocity of the wheel after the field is suddenly switched off?
At a distance r, the magnetic force is balanced by the centripetal force. i.e.
BQ = , where v = linear velocity of the wheel. Then,
B =
v =
Angular velocity, =
For r
<p><strong>6.17 </strong>Line charge per unit length = <span title="Click to copy mathml"><math><mi>λ</mi></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>T</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi><mi></mi><mi>c</mi><mi>h</mi><mi>a</mi><mi>r</mi><mi>g</mi><mi>e</mi></mrow></mrow><mrow><mrow><mi>L</mi><mi>e</mi><mi>n</mi><mi>g</mi><mi>t</mi><mi>h</mi></mrow></mrow></mfrac></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>Q</mi></mrow></mrow><mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mrow></mfrac></math></span></p><p>Where r = distance of the point within the wheel</p><p>Mass of the wheel = M</p><p>Radius of the wheel = R</p><p>Magnetic field, <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>B</mi></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mo>-</mo><msub><mrow><mrow><mi>B</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mover accent="true"><mrow><mrow><mi>k</mi></mrow></mrow><mo>?</mo></mover></math></span></p><p>At a distance r, the magnetic force is balanced by the centripetal force. i.e.</p><p>BQ <span title="Click to copy mathml"><math><mi>v</mi></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>M</mi><msup><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mfrac></math></span> , where v = linear velocity of the wheel. Then,</p><p>B <span title="Click to copy mathml"><math><mo>×</mo><mi></mi><mn>2</mn><mi>λ</mi><mi>π</mi><mi>r</mi><mi></mi></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>M</mi><mi>v</mi></mrow></mrow><mrow><mrow><mi>r</mi></mrow></mrow></mfrac></math></span></p><p>v = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>2</mn><mi>B</mi><mi>λ</mi><mi>π</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi>M</mi></mrow></mrow></mfrac></math></span></p><p>Angular velocity, <span title="Click to copy mathml"><math><mi>ω</mi><mo>=</mo><mi></mi><mfrac><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>R</mi></mrow></mrow></mfrac></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>2</mn><mi>B</mi><mi>λ</mi><mi>π</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi>M</mi><mi>R</mi></mrow></mrow></mfrac></math></span></p><p>For r <span title="Click to copy mathml"><math><mo>≤</mo><mi>a</mi><mi></mi><mo>≤</mo><mi>R</mi><mo>,</mo><mi></mi><mi>w</mi><mi>e</mi><mi></mi><mi>g</mi><mi>e</mi><mi>t</mi></math></span> <span title="Click to copy mathml"><math><mi>ω</mi><mo>=</mo><mi></mi><mo>-</mo><mfrac><mrow><mrow><mn>2</mn><msub><mrow><mrow><mi>B</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub><mi>λ</mi><mi>π</mi><msup><mrow><mrow><mi>a</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mi>M</mi><mi>R</mi></mrow></mrow></mfrac><mover accent="true"><mrow><mrow><mi>k</mi></mrow></mrow><mo>?</mo></mover></math></span></p>
This website uses Cookies and related technologies for the site to function correctly and securely, improve & personalise your browsing experience, analyse traffic, and support our marketing efforts and serve the Core Purpose. By continuing to browse the site, you agree to Privacy Policy and Cookie Policy.
