Question

A hydrogen ion of mass 'm' and charge 'q' travels with a speed 'v' in a circle of radius 'r' in a magnetic field of intensity 'B'. Write the equation in terms of these quantities only relating the force on the ion to the required centripetal force. Hence derive an expression for its time.

Answer 1

Answer:

$r=\frac{mv}{qB}$, $T=\frac{2\pi m}{qB}$

Explanation:

The force experienced by the ion due to the magnetic field is given by:

$F=qvB$

where

q is the charge

v is the speed

B is the intensity of the magnetic field

The cetripetal force is

$F=\frac{mv^2}{r}$

where

m is the mass

r is the radius of the circle

Since the magnetic force acts as centripetal force, we can equate the two expressions:

$qvB=m\frac{v^2}{r}$

Re-arranging it, we find the radius:

$r=\frac{mv}{qB}$

Now, if we want to find the time it takes for the ion to make one complete circle (=the period), we just need  to divide the length of one circumference by the speed:

$T=\frac{2\pi r}{v}$

And susbstituting the expression we found before for r, we find

$T=\frac{2\pi mv}{qBv}=\frac{2\pi m}{qB}$