A beam of neutrons performs circular motion of radius, $r=1 \,m$. Under the influence of an inhomogeneous magnetic field with inhomogeneity extending over $\Delta r=0.01 \,m$. The speed of the neutrons is $54 \,m / s$. The mass and magnetic moment of the neutrons respectively are $1.67 \times 10^{-27} \,kg$ and $9.67 \times 10^{-27} \,J / T$. The average variation of the magnetic field over $\Delta r$ is approximately ....... $T$
KVPY 2020, Advanced
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$(c)$ Given, $r=1 \,m , \Delta r=0.01 \,m$, $v=54 \,ms ^{-1}, M=9.67 \times 10^{-27} \,JT ^{-1}$ and $m=1.67 \times 10^{-27} \,kg$
The magnetic force experienced by the neutron,
$F=M \frac{\delta B}{\delta r}=\frac{m v^2}{r}$
$\Delta B=\frac{m v^2}{r}$
$\Delta B=\frac{m v^2}{M r} \Delta r$
$=\frac{1.67 \times 10^{-27} \times 54^2 \times 0.01}{9.67 \times 10^{-27} \times 1}=5.04\; T$
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