CBSE 2025, Set 2

Question

A proton and an $\alpha$-particle enter with the same velocity $\vec{v}$ a uniform magnetic field $\vec{B}$ such that $\vec{v} \perp \vec{B}$. The ratio of the radii of their paths is : A). $2$ B). $\frac{1}{2}$ C). $\frac{1}{4}$ D). $4$

Edvaya · Accepted Answer

The correct answer is $\frac{1}{2}$. When a charged particle enters a uniform magnetic field perpendicularly, it moves in a circular path. The radius $r$ of the path is given by:
$$r = \frac{mv}{qB}$$

Since the velocity $v$ and magnetic field $B$ are the same for both particles, $r \propto \frac{m}{q}$.

For a proton:
* Mass $= m_p$
* Charge $= e$
* $r_p \propto \frac{m_p}{e}$

For an $\alpha$-particle (helium nucleus):
* Mass $= 4m_p$
* Charge $= 2e$
* $r_\alpha \propto \frac{4m_p}{2e} = 2 \left(\frac{m_p}{e}ight)$

The ratio of their radii is:
$$\frac{r_p}{r_\alpha} = \frac{(m_p/e)}{2(m_p/e)} = \boxed{\frac{1}{2}}$$

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Physics > Magnetism Induction And Alternating Current > Moving Charges And Magnetism

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