A beam of cathode rays (electrons) traveling at a speed of 4.00 \times 10^{6}\text{ m s}^{… — JAMB UTME

A beam of cathode rays (electrons) traveling at a speed of $4.00 \times 10^{6}\text{ m s}^{-1}$ enters a region with a uniform magnetic field of $5.00 \times 10^{-4}\text{ T}$ directed perpendicular to the beam. Taking the specific charge of an electron ( $\frac{e}{m}$ ) to be $1.60 \times 10^{11}\text{ C kg}^{-1}$ , what is the radius of the circular path traced by the cathode rays, in centimeters?

Answer: 5 cm

Answer

The radius of the circular path followed by the cathode rays is 5.0 cm.

When cathode rays enter a uniform magnetic field at right angles, the magnetic force acts as a centripetal force causing the electron beam to trace a circular arc of radius

r = \frac{v}{(e/m)B}

. Substituting

v = 4.00 \times 10^6\text{ m s}^{-1}

e/m = 1.60 \times 10^{11}\text{ C kg}^{-1}

, and

B = 5.00 \times 10^{-4}\text{ T}

yields

r = 0.05\text{ m}

, which equals

5.0\text{ cm}

Step-by-Step Solution

Set the magnetic Lorentz force equal to the required centripetal force for circular motion.

e v B = \frac{m v^2}{r}

Cathode rays consist of moving electrons experiences a magnetic force perpendicular to both their velocity and the magnetic field.

Express the radius

r

in terms of speed

v

, magnetic field

B

, and specific charge

\frac{e}{m}

r = \frac{v}{\left(\frac{e}{m}\right) B}

Simplifying the force balance equation isolates the radius on one side.

Substitute the given numerical values into the expression for

r

r = \frac{4.00 \times 10^{6}\text{ m s}^{-1}}{\left(1.60 \times 10^{11}\text{ C kg}^{-1}\right) \times \left(5.00 \times 10^{-4}\text{ T}\right)} = 0.05\text{ m}

Calculating the denominator gives

8.00 \times 10^{7}\text{ C T kg}^{-1}

, leading to

0.05\text{ m}

Convert the radius from meters to centimeters as requested by the question.

r = 0.05\text{ m} \times 100\text{ cm m}^{-1} = 5.0\text{ cm}

1 meter equals 100 centimeters.

Key Concept

Deflection of Cathode Rays in a Magnetic Field

Estimated Time:1m 30s

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