If ratio of magnetic induction on the axial line of a long magnet at distance 20 cm and 30 cm is 128 : 27. Find length of the magnet.

10 c m

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20 c m

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30 c m

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40 c m

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Solution

The correct option is B $20 c m$
$B_{a x i a l} = \frac{μ_{0}}{4 π} \frac{2 M r}{(r^{2} - l^{2})^{2}}$
$B_{20} : B_{30} = 128 : 27$
$\frac{20}{(20^{2} - l^{2})^{2}} \times \frac{(30^{2} - l^{2})^{2}}{30} = \frac{128}{127}$
$2 (30^{2} - l^{2})^{2} (27) = 3 (20^{2} - l^{2})^{2} 128$
$\sqrt{54} (900 - l^{2}) = \sqrt{384} (400 - l^{2})$
$900 \sqrt{54} - \sqrt{54} l^{2} = \sqrt{384} \times 400 - \sqrt{384} l^{2}$
$l^{2} (12.2474) = 12224. 75$
$l^{2} = 100; l = \sqrt{100} - 10 c m$
$∴$ Magnetic length $= 2 l = 20 c m$

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If ratio of magnetic induction on the axial line of a long magnet at distance 20 cm and 30 cm is 128 : 27. Find length of the magnet.

The correct option is B 20cmBaxial=μ04π2Mr(r2−l2)2B20:B30=128:2720(202−l2)2×(302−l2)230=1281272(302−l2)2(27)=3(202−l2)2128√54(900−l2)=√384(400−l2)900√54−√54l2=√384×400−√384l2l2(12.2474)=12224.75l2=100;l=√100−10cm∴ Magnetic length =2l=20cm