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A tightly wound long solenoid carries a current of $$1.5\,A.$$ An electron is executing uniform circular motion inside the solenoid with a time period of $$75\,ns.$$ The number of turns per metre in the solenoid is $$\underline{\hspace{1cm}}.$$ [Take mass of electron $$m_e=9\times10^{-31}\,kg,$$ change of electron $$|q_e|=1.6\times10^{-19}\,$$ $$C,$$ $$\mu_0=4\pi\times10^{-7}\,N$$ $$A^{-2},\,1\,ns=10^{-9}\,s]$$
Correct Answer: 250
$$T = \frac{2\pi m_e}{|q_e| \mu_0 n I}$$
$$75 \times 10^{-9} = \frac{(2\pi)(9 \times 10^{-31})}{1.6 \times 10^{-19} \times 4\pi \times 10^{-7} \times n \times 1.5}$$
$$75 \times 10^{-9} = \frac{9 \times 10^{-31}}{1.6 \times 10^{-19} \times 2 \times 10^{-7} \times n \times 1.5}$$
$$75 \times 10^{-9} = \frac{9 \times 10^{-31}}{4.8 \times 10^{-26} \times n} \implies n = 250\text{ m}^{-1}$$
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