Question

Two particles of masses m₁ and m₂ having charges q₁ and q₂ respectively are projected with the same velocity in a region of uniform magnetic field `vec B` pointing vertically upward. If they describe circular paths as shown in the figure, one may conclude that:

पर्याय

• `m_1/m_2 > q_1/q_2`

• `m_1/m_2 > q_2/q_1`

• `m_1/m_2 < q_1/q_2`

• `m_1/m_2 < q_2/q_1`

Answer 1

`bb(m_1/m_2 > q_1/q_2)`

Explanation:

When a charged particle enters a uniform magnetic field perpendicularly, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path.

The magnetic force provides the centripetal force:

qvB = `(m v^2)/r`

Solving for radius r:

r = `(m v)/(q B)`

Given that both particles have the same velocity v and are in the same magnetic field B, we have:

r ∝ m/q`

From the diagram (as described in the question), we can observe the relative sizes of the circular paths. Let’s denote the radii as r₁ for m₁ and r₂ for m₂.

Based on the figure:

If r₁ > r₂, then,

`m_1/q_1 > m_2/q_2`

If r₁ < r₂, then,

`m_1/q_1 < m_2/q_2`

From `m_1/q_1 > m_2/q_2`, we cross-multiply (all quantities positive):

m₁q₂ > m₂q₁

`m_1/m_2 > q_1/q_2`

∴ The ratio of masses is greater than the ratio of charges when the radius of the path for m₁ is larger than that for m₂.