# An electron of kinetic energy 41 keV moves in a circular orbit perpendicular to a magnetic field of 0.330 T.

(a) Compute the radius of the orbit.

The answer to this part is 2.07mm.

(b) Find the frequency of the motion.

I'm not sure how to find the answer to this part.

Compute the velocity V from the kinetic energy of the electron. The time required to complete 1 orbit is

T = 2 pi R/V

Use the R from part (a)

The frequency of the motion is f = 1/T

## Ha! It's time for a physics joke! Why did the electron go on a diet? Because it wanted to reduce its orbit!

Now, let's get serious and calculate the frequency of the electron's motion.

First, we need to find the velocity of the electron. We can do that by equating the kinetic energy to the half of the mass times the squared velocity:

(1/2)mv^2 = 41 keV

But we need to convert the kinetic energy given in keV into joules:

41 keV = 41 x 1.6 x 10^-19 J

= 6.56 x 10^-18 J

Now, we can rewrite the equation as:

(1/2)mv^2 = 6.56 x 10^-18 J

Using the mass of an electron (9.11 x 10^-31 kg), we can solve for v:

(1/2)(9.11 x 10^-31 kg)v^2 = 6.56 x 10^-18 J

v^2 = (2 x 6.56 x 10^-18 J) / (9.11 x 10^-31 kg)

v^2 = 1.44 x 10^13 m^2/s^2

Taking the square root of both sides, we find:

v ≈ 3.799 x 10^6 m/s

Now that we have the velocity, we can calculate the time taken for one complete orbit using the formula:

T = 2πR / v

Substituting R = 2.07 mm = 2.07 x 10^-3 m and v ≈ 3.799 x 10^6 m/s:

T = (2π)(2.07 x 10^-3 m) / (3.799 x 10^6 m/s)

T ≈ 3.438 x 10^-7 s

Finally, we can find the frequency using the formula:

f = 1 / T

f = 1 / (3.438 x 10^-7 s)

f ≈ 2.91 x 10^6 Hz

So, the frequency of the electron's motion is approximately 2.91 MHz. Keep those electrons spinning!

## To find the frequency of the motion, we can start by calculating the velocity of the electron using its kinetic energy.

Given:

Kinetic energy of the electron, KE = 41 keV

Magnetic field, B = 0.330 T

We know that the kinetic energy of an electron is given by the equation:

KE = (1/2)mv^2

Where m is the mass of the electron and v is its velocity.

Since the electron is moving in a circular orbit, we can equate the magnetic force (qvB) to the centripetal force (mv^2/R), where q is the charge of the electron and R is the radius of the orbit.

Therefore, qvB = mv^2/R

Since the charge of the electron is known, q = -1.6 x 10^-19 C, we can rearrange the equation to solve for the velocity v:

v = BqR/m

The mass of an electron, m = 9.11 x 10^-31 kg.

Substituting the given values into the equation, we have:

v = (0.330 T)(-1.6 x 10^-19 C)(2.07 x 10^-3 m) / (9.11 x 10^-31 kg)

Calculating this equation will give the velocity of the electron. Once we have the velocity, we can compute the time required to complete one orbit using the expression:

T = 2πR/v

Where R is the radius of the orbit. We can use the value obtained in part (a), which is 2.07 mm or 0.00207 m.

Finally, the frequency of the motion can be found by taking the reciprocal of the time:

f = 1/T

Performing these calculations will give the answer to part (b), the frequency of the motion.

## To find the frequency of the motion, you'll need to calculate the orbital velocity of the electron first, using its kinetic energy.

1. Convert the kinetic energy from keV to joules:

41 keV = 41 * 1.6 × 10^-19 J (1 keV = 1.6 × 10^-19 J)

2. Use the formula for kinetic energy: KE = (1/2)mv^2

(1/2)m(v^2) = 41 * 1.6 × 10^-19 J, where m is the mass of the electron.

3. Rearrange the equation to solve for v:

v^2 = (2 * 41 * 1.6 × 10^-19 J) / m

4. The mass of an electron is approximately 9.11 × 10^-31 kg. Substitute this value in and solve for v.

Now that you have the orbital velocity, you can proceed to calculate the frequency.

5. Recall that time required to complete one orbit is given by: T = 2πR/v

6. Use the radius value obtained from part (a): R = 2.07 mm = 2.07 × 10^-3 m

7. Substitute the values of R and v into the equation and solve for T.

8. Finally, calculate the frequency using the formula: f = 1/T

By following these steps, you will be able to find the frequency of the motion of the electron.