Question

Two particles X and Y having equal charges, after being accelerated through the same potential difference, enter a region of uniform magnetic field and describe circular paths of radii `R_1 and R_2,` respectively. The ratio of masses of X and Y is
A. `R_1/R_2`
B. `R_2/R_1`
C. `(R_1/R_2)^(1//2)`
D. `(R_1/R_2)^2`

Answer 1

Let `m_1`, `m_2` be the masses of particles X and Y respectively and q be the charge on each particle. Let V be the potential difference under which each particle be accelerated and `v_1`, `v_2` be the velocities acquired by them.
Then, `qV=1/2mv_1v_1^2=1/2m_2v_2^2`
i.e., `m_1v_1^2=m_2v_2^2` or `m_1//m_2=v_2^2//v_1^2`....(1)
Let B be the strength of uniform magnetic field applied
`qv_1B=m_1v_1^2//R_1` ...(2)
and `qv_2B=m_1v_2^2//R_2` ....(3)
`:. v_1/v_2=R_2/R_1` ...(4)
Hence from (1) and (4), `m_1/m_2=(R_1/R_2)^2`