A monoenergetic electron beam with electron speed of 5.20 × 106 m s–1 is subject to a magnetic field of 1.30 × 10–4 T normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals 1.76 × 10^11C kg–1.

To determine the radius of the circular path traced by a monoenergetic electron beam in a magnetic field, we can use the relationship between the magnetic force acting on the electrons and the centripetal force required to keep them moving in a circle. The formula we will use is derived from the Lorentz force equation and the concept of centripetal motion.

Understanding the Forces at Play

When charged particles, like electrons, move through a magnetic field, they experience a magnetic force that acts perpendicular to both their velocity and the magnetic field direction. This force can be expressed as:

F_magnetic = qvB

Where:

• F_magnetic is the magnetic force.
• q is the charge of the electron (approximately -1.6 × 10^-19 C).
• v is the speed of the electron (5.20 × 10⁶ m/s).
• B is the magnetic field strength (1.30 × 10^-4 T).

Relating Magnetic Force to Centripetal Force

The magnetic force provides the necessary centripetal force to keep the electrons moving in a circular path. The centripetal force can be expressed as:

F_centripetal = mv²/r

Where:

• m is the mass of the electron (approximately 9.11 × 10^-31 kg).
• r is the radius of the circular path.

Setting the Forces Equal

Since the magnetic force equals the centripetal force, we can set the two equations equal to each other:

qvB = mv²/r

Solving for the Radius

Rearranging this equation to solve for the radius r gives us:

r = mv/(qB)

Plugging in the Values

Now, we can substitute the known values into this equation:

• m = 9.11 × 10^-31 kg
• v = 5.20 × 10⁶ m/s
• q = -1.6 × 10^-19 C (we can use the absolute value since we are interested in the magnitude)
• B = 1.30 × 10^-4 T

Substituting these values into the radius formula:

r = (9.11 × 10^-31 kg) * (5.20 × 10⁶ m/s) / (1.6 × 10^-19 C * 1.30 × 10^-4 T)

Calculating the Result

Now, performing the calculations step-by-step:

• Calculate the numerator: 9.11 × 10^-31 kg * 5.20 × 10⁶ m/s = 4.7352 × 10^-24 kg·m/s
• Calculate the denominator: 1.6 × 10^-19 C * 1.30 × 10^-4 T = 2.08 × 10^-23 C·T
• Now divide the two results: r = 4.7352 × 10^-24 kg·m/s / 2.08 × 10^-23 C·T ≈ 0.227 m

Final Result

The radius of the circular path traced by the electron beam in the magnetic field is approximately 0.227 meters, or about 22.7 centimeters.