An electron follows a helical path in a uniform mag-netic field of magnitude 0.300 T. The pitch of the path is 6.00 um, and the magnitude of the magnetic force on the elec-tron is 2.00x10^-15 N. What is the electron's speed?
To find the electron's speed, we can use the formula for the magnitude of the magnetic force on a charged particle moving through a magnetic field:
F = qvB
Where:
F is the magnitude of the magnetic force
q is the charge of the particle
v is the velocity of the particle
B is the magnetic field strength
In this case, we know the magnitude of the magnetic force on the electron (F = 2.00x10^-15 N), the magnetic field strength (B = 0.300 T), and the pitch of the path (p = 6.00 um).
The pitch of the helical path can be defined as the distance the electron travels parallel to the magnetic field in one complete revolution. Therefore, the distance traveled parallel to the magnetic field in one complete revolution is equal to the pitch (p).
The distance traveled parallel to the magnetic field is given by the formula:
Distance = Velocity x Time
The time taken for one complete revolution is inversely proportional to the velocity, and it is equal to the period (T) of the helical motion.
The period of the helical motion is given by:
T = 1 / Frequency
Where:
Frequency = n / T
n is the number of complete revolutions per unit of time
The number of complete revolutions per unit of time is equal to the velocity of the electron (v) divided by the pitch (p).
Substituting the values in the formulas, we can solve for the electron's velocity.
Let's calculate it step by step:
Step 1: Determine the frequency (n)
Since the pitch is given in micrometers (um), we need to convert it to meters (m):
p = 6.00 um = 6.00 x 10^-6 m
The frequency is the inverse of the time taken for one complete revolution:
Frequency = 1 / T
Step 2: Determine the period (T)
To calculate the period, we need to find the number of complete revolutions per unit of time:
n = v / p
Substituting the values:
n = v / (6.00 x 10^-6 m)
We are given that n = Frequency x T, so:
T = 1 / (Frequency x (v / p))
T = p / (v x Frequency)
Step 3: Calculate the velocity (v)
Using the magnitude of the magnetic force formula:
F = qvB
Rearranging the formula to solve for the velocity:
v = F / (qB)
Substituting the given values:
v = (2.00 x 10^-15 N) / (q x 0.300 T)
Step 4: Substitute back into the period equation
Plugging the calculated value of v into the period equation:
T = p / (v x Frequency)
Step 5: Calculate the frequency (n)
To find the frequency (n), we substitute the calculated value of v into the equation we derived earlier for n:
n = v / p
Step 6: Substitute into the period equation
Using the calculated values of p, v, and n, we substitute them into the period equation:
T = p / (v x Frequency)
Step 7: Determine the velocity (v)
Finally, we can substitute the calculated values of p, T, and n into the equation to calculate v:
v = (p / (T x n))
By solving these equations step-by-step, we can find the value of the electron's speed.
To find the electron's speed, we can use the equation for the magnitude of the magnetic force on a moving charged particle:
F = qvB
Where:
F = magnitude of the magnetic force
q = charge of the electron (1.6 x 10^-19 C)
v = speed of the electron
B = magnitude of the magnetic field
We are given:
F = 2.00 x 10^-15 N
B = 0.300 T
Rearranging the equation, we can solve for v:
v = F / (qB)
Substituting the given values:
v = (2.00 x 10^-15 N) / [(1.6 x 10^-19 C) (0.300 T)]
Now, let's calculate:
v = (2.00 x 10^-15 N) / (1.6 x 10^-19 C) / (0.300 T)
v ≈ 8.33 x 10^4 m/s
Therefore, the electron's speed is approximately 8.33 x 10^4 m/s.