Question

Two particles X and Y having equal charge, after being
accelerated through the same potential difference enter a
region of uniform magnetic field and describe circular paths
of radii R₁ and R₂ respectively. The ratio of the mass of X to
that of Y is
(a) √R₁/R₂ (b) (R₂/R₁)²
(c) (R₁/R₂)² (d) √R₂/R₁

Answer 1

Given :

▪ Radius for particle X = R_1

▪ Radius for particle Y = R_2

▪ Both particles are accelerated through same pd and both have same charge.

To Find :

▪ Ratio of the mass of X to that mass of Y.

Formula :

↗ In a perpendicular magnetic field, the charge follows circular path

$\bigstar\:\underline{\boxed{\bf{\red{R=\dfrac{\sqrt{2mV_o}}{\sqrt{q}\times B}}}}}$

• R denotes radius of circular path
• m denotes mass of particle
• q denotes charge
• Vo denotes pd
• B denotes magnetic field

Calculation :

$\circ\sf\:q_X=q_Y=q\\ \\ \circ\sf\:(V_o)_X=(V_o)_Y=V_o\\ \\ \circ\sf\:B_X=B_Y=B\\ \\ \dashrightarrow\sf\:\red{R\propto \sqrt{m}}\\ \\ \dashrightarrow\sf\:\dfrac{R_X}{R_Y}=\sqrt{\dfrac{m_X}{m_Y}}\\ \\ \dashrightarrow\sf\:\sqrt{\dfrac{m_X}{m_Y}}=\dfrac{R_1}{R_2}\\ \\ \dashrightarrow\underline{\boxed{\bf{\purple{\dfrac{m_X}{m_Y}=\dfrac{(R_1)^2}{(R_2)^2}}}}}\:\orange{\bigstar}$

Answer 2

$\bold{\bf{\blue{R=\frac{\sqrt{2mV_0} }{\sqrt{q}*B } }}}$

where ,

• R denotes radius of circular path
• m denotes mass of particle
• V₀ denotes pd
• q denotes charge
• B denotes the magnetic field

From question ,

$\star \;\; \bold{q_{x}=q_{y}=q}\\\\\star \;\; \bold{(V_0)_x=(V_0)_y=V_0}\\\\\star \;\; \bold{B_x=B_y=B}$

$\bold{Since\;,\;R\;\alpha \sqrt{m}\;\;[\;From\;Formula\;]}\\\\\implies \bold{\frac{R_x}{R_y}= \sqrt{\frac{m_x}{m_y} } }\\\\\implies \bold{\frac{m_x}{m_y}=(\frac{R_x}{R_y} )^2}\\\\\implies \bold{\bf{\red{\frac{m_x}{m_y}=\frac{(R_1)^2}{(R_2)^2} }}}$

So option (c) is correct .