Circular Motion of Charges in Magnetic Fields Video Lessons

Concept

# Problem: Ions having equal charges but masses of M and 2M are accelerated through the same potential difference and then enter a uniform magnetic field perpendicular to their path. If the heavier ions follow a circular arc of radius R, what is the radius of the arc followed by the lighter?A) 2/⎷2B) R/2C) 3RD) ⎷2RE) 4R

###### FREE Expert Solution

Work done by the electric force in a moving charge:

$\overline{){\mathbf{W}}{\mathbf{=}}{\mathbf{-}}{\mathbf{q}}{\mathbf{∆}}{\mathbf{V}}}$

Kinetic energy:

$\overline{){\mathbf{K}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{mv}}}^{{\mathbf{2}}}}$

Centripetal force:

$\overline{){{\mathbf{F}}}_{{\mathbf{c}}}{\mathbf{=}}\frac{{\mathbf{mv}}^{\mathbf{2}}}{\mathbf{r}}}$

Magnetic force on moving charge:

$\overline{){\mathbf{F}}{\mathbf{=}}{\mathbf{q}}{\mathbf{v}}{\mathbf{B}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{\theta }}}$

The kinetic energy of the ions is equal to the work done by the electric force.

|W| = |K|

qΔV = (1/2)mv2

v = (2qΔV/m)(1/2)

The net force on the ions is due to the magnetic force.

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###### Problem Details

Ions having equal charges but masses of M and 2M are accelerated through the same potential difference and then enter a uniform magnetic field perpendicular to their path. If the heavier ions follow a circular arc of radius R, what is the radius of the arc followed by the lighter?
A) 2/⎷2
B) R/2
C) 3R
D) ⎷2R
E) 4R