An \alpha-particle and a \beta^--particle emitted from a radioactive source enter a unifor… — JAMB UTME

An $\alpha$ -particle and a $\beta^-$ -particle emitted from a radioactive source enter a uniform magnetic field perpendicularly with equal linear momenta. What is the ratio of the radius of curvature of the trajectory of the $\alpha$ -particle to that of the $\beta^-$ -particle in the magnetic field?

$1 : 2$ Cevap
B
$2 : 1$
C
$4 : 1$
D
$1 : 4$

Cevap

The ratio of the radius of curvature of the path of the

\alpha

-particle to that of the

\beta^-

-particle is

1 : 2

In a uniform magnetic field, the radius of curvature

r

of a moving charged particle is given by

r = \frac{p}{qB}

, where

p

is momentum,

q

is charge magnitude, and

B

is magnetic field flux density. Since both particles have equal momentum in the same field,

r \propto \frac{1}{q}

. The

\alpha

-particle carries charge magnitude

2e

while the

\beta^-

-particle carries charge magnitude

e

. Consequently, the ratio of their radii is

r_\alpha : r_\beta = \frac{1}{2} : 1 = 1 : 2

Adım Adım Çözüm

Relate radius of trajectory to linear momentum and magnetic field

The magnetic force supplies centripetal force:

qvB = \frac{mv^2}{r} \implies r = \frac{mv}{qB} = \frac{p}{qB}

, where

p

is linear momentum.

Expressing radius in terms of momentum directly utilizes the given condition that

p

is equal for both particles.

Identify the magnitude of charge for each emission

For the

\alpha

-particle (

^4_2\text{He}^{2+}

q_\alpha = 2e

. For the

\beta^-

-particle (

^0_{-1}\text{e}

q_\beta = e

Radius of curvature depends on the magnitude of charge carried by each radiation type.

Calculate the ratio of radii

r_\alpha / r_\beta

\frac{r_\alpha}{r_\beta} = \frac{\frac{p}{2eB}}{\frac{p}{eB}} = \frac{e}{2e} = \frac{1}{2}.

Since momentum

p

and field strength

B

are identical, the ratio simplifies directly to the inverse ratio of their charge magnitudes.

Anahtar Kavram

Deflection of radioactive emissions in magnetic fields and radius of curvature

Soru

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