To find the period of revolution of the electron, we can use the equation that relates the centripetal force, magnetic field, and the velocity of a charged particle moving in a circle:
F = (q * B * v) / r
Where:
F is the centripetal force,
q is the charge of the particle,
B is the magnetic field strength,
v is the velocity of the particle, and
r is the radius of the circle.
We know that the proton and the electron experience the same centripetal force, so we can set up the equation for both particles and equate them:
For the electron:
Fe = (qe * Be * ve) / re
For the proton:
Fp = (qp * Bp * vp) / rp
Since the centripetal force is the same for both particles, we have:
Fe = Fp
(qe * Be * ve) / re = (qp * Bp * vp) / rp
We are given the following values:
Mass of electron (me) = 9.11 x 10^-31 kg
Mass of proton (mp) = 1.67 x 10^-27 kg
Velocity of both particles (v) = 2.00 x 10^6 m/s
Radius of the circle (r) = 2.85 cm = 2.85 x 10^-2 m
The charge of an electron (qe) is equal to -1.6 x 10^-19 C, and the charge of a proton (qp) is equal to +1.6 x 10^-19 C. However, we don't need to use these values directly in this problem.
Now let's solve for the period of revolution of the electron using the GUESS method:
1. Gather the given values:
Mass of electron (me) = 9.11 x 10^-31 kg
Velocity of the electron (ve) = 2.00 x 10^6 m/s
Radius of the circle (re) = 2.85 x 10^-2 m
2. Identify the equation to use:
We need to use the equation for centripetal force: F = (q * B * v) / r
3. Substitute the known values into the equation:
Fe = (qe * Be * ve) / re
4. Rearrange the equation to solve for the magnetic field (Be):
Be = (Fe * re) / (qe * ve)
5. Substitute the values into the rearranged equation:
Be = (me * v^2) / (qe * r)
6. Calculate the magnetic field (Be):
Be = (9.11 x 10^-31 kg * (2.00 x 10^6 m/s)^2) / (-1.6 x 10^-19 C * 2.85 x 10^-2 m)
Be ≈ -2.67 x 10^-3 T
Note: The negative sign indicates the direction of the magnetic field.
7. Use the magnetic field (Bp) for the proton to find its period of revolution (Tp):
Fp = (qp * Bp * vp) / rp
Rearranging the equation gives:
Bp = (Fp * rp) / (qp * vp)
Substituting values:
Bp = (mp * v^2) / (qp * r)
Bp = (1.67 x 10^-27 kg * (2.00 x 10^6 m/s)^2) / (1.6 x 10^-19 C * 2.85 x 10^-2 m)
Bp ≈ 1.17 x 10^-2 T
8. Find the period of revolution for the electron (Te):
Te = 2πmeve / (|qe|Be)
Substituting values:
Te = (2π * 9.11 x 10^-31 kg * 2.00 x 10^6 m/s) / (1.6 x 10^-19 C * 2.67 x 10^-3 T)
Te ≈ 1.80 x 10^-7 s
Therefore, the period of revolution of the electron is approximately 1.80 x 10^-7 seconds.