Question

Two particles, A and B having equal charges, after being accelerated through the same potential difference enter into a region of uniform magnetic field and the particles describe circular paths of radii $R_1$ and $R_2$ respectively. The ratio of the masses of A and B is:

• A. $\sqrt{R_1/R_2}$
• B. $R_1/R_2$
• C. $(R_1/R_2{)}^2$
• D. $(R_2/R_1{)}^2$

Answer 1

Answer: (C)

$(R_1/R_2{)}^2$

Since charge on both particles is same and they are accelerated through the same voltage, work done by electric field will be same for both the charges so the increase in kinetic energy will be same. Assuming that charges started from rest, we have:

$\frac{1}{2}{m}_1{v}_1^2=qV=\frac{1}{2}{m}_2{v}_2^2\Rightarrow \frac{v_1^2}{v_2^2}=\frac{m_2}{m_1}$

where $V$ is the potential difference through which the charges are accelerated.

The magnetic force only provides the necessary centripetal force for circular motion:

$\frac{m_1{v}_1^2}{R_1}=Bq{v}_1\frac{m_2{v}_2^2}{R_2}=Bq{v}_2\Rightarrow \frac{R_1}{R_2}=\frac{m_1{v}_1}{m_2{v}_2}=\frac{m_1^{0.5}}{m_2^{0.5}}\Rightarrow \frac{m_1}{m_2}=\frac{R_1^2}{R_2^2}$