(C) The kinetic energy $K$ gained by the particle when accelerated through a potential difference $V$ is given by $K = qV$.Since $K = \frac{1}{2}mv^2$

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$\frac{1}{B} \sqrt{\frac{2mV}{q}}$

(C) The kinetic energy $K$ gained by the particle when accelerated through a potential difference $V$ is given by $K = qV$.Since $K = \frac{1}{2}mv^2$,we have $\frac{1}{2}mv^2 = qV$,which implies $v = \sqrt{\frac{2qV}{m}}$.When a charged particle enters a magnetic field $B$ perpendicularly,it moves in a circular path of radius $R = \frac{mv}{qB}$.Substituting the value of $v$ into the radius formula:$R = \frac{m}{qB} \sqrt{\frac{2qV}{m}} = \frac{1}{B} \sqrt{\frac{m^2 \cdot 2qV}{q^2 \cdot m}} = \frac{1}{B} \sqrt{\frac{2mV}{q}}$.Thus,the correct option is $C$.

Class 12 Physics Moving Charges and Magnetism Motion of Charged Particle In Magnetic Field

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$A$ charged particle of charge $q$ and mass $m$ is accelerated through a potential difference of $V$ volts. It enters a region of orthogonal magnetic field $B$. The radius of its circular path will be:

A
$\sqrt{\frac{2mV}{qB^2}}$
B
$\frac{2mV}{qB^2}$
C
$\frac{1}{B} \sqrt{\frac{2mV}{q}}$
D
$\frac{1}{B} \sqrt{\frac{mV}{q}}$

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Moving Charges and Magnetism

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Motion of Charged Particle In Magnetic Field

$A$ monoenergetic $(18 \;keV)$ electron beam initially in the horizontal direction is subjected to a horizontal magnetic field of $0.04 \;G$ normal to the initial direction. Estimate the up or down deflection of the beam over a distance of $30 \;cm$ $(m_{e} = 9.11 \times 10^{-31} \;kg)$.

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An electron with energy $0.1 \, keV$ moves at a right angle to the Earth's magnetic field of $1 \times 10^{-4} \, Wb/m^2$. The frequency of revolution of the electron will be. (Take mass of electron $= 9.0 \times 10^{-31} \, kg$)

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In a parabola spectrograph,the velocities of four positive ions $P, Q, R,$ and $S$ are $v_1, v_2, v_3,$ and $v_4$ respectively. Based on the provided parabola spectrograph image,determine the relationship between their velocities.

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$A$ point charge is placed in a moving train. $A$ passenger $A$ sitting in the train and a person $B$ standing on the ground observe the fields due to this charge. Then:

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$A$ particle having charge of $1 \, C$,mass $1 \, kg$ and speed $1 \, m/s$ enters a uniform magnetic field,having magnetic induction of $1 \, T$,at an angle $\theta = 30^\circ$ between the velocity vector and the magnetic induction. The pitch of its helical path is (in meters):

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$A$ charged particle of charge $q$ and mass $m$ is accelerated through a potential difference of $V$ volts. It enters a region of orthogonal magnetic field $B$. The radius of its circular path will be:

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$A$ monoenergetic $(18 \;keV)$ electron beam initially in the horizontal direction is subjected to a horizontal magnetic field of $0.04 \;G$ normal to the initial direction. Estimate the up or down deflection of the beam over a distance of $30 \;cm$ $(m_{e} = 9.11 \times 10^{-31} \;kg)$.

An electron with energy $0.1 \, keV$ moves at a right angle to the Earth's magnetic field of $1 \times 10^{-4} \, Wb/m^2$. The frequency of revolution of the electron will be. (Take mass of electron $= 9.0 \times 10^{-31} \, kg$)

In a parabola spectrograph,the velocities of four positive ions $P, Q, R,$ and $S$ are $v_1, v_2, v_3,$ and $v_4$ respectively. Based on the provided parabola spectrograph image,determine the relationship between their velocities.

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