A beam of electrons moving in the x-direction at 8.3Mm/s enters a region where a uniform 240G magnetic field points in the y-direction. The boundary of the field region is perpendicular to the beam. How far into the field region does the beam penetrate?
distance=mv/qB
To find out how far into the field region the beam penetrates, you need to use the Lorentz force equation.
The Lorentz force equation states that the force experienced by a charged particle moving in a magnetic field is given by the equation:
F = q(v x B)
where F is the force, q is the charge of the particle, v is the velocity vector of the particle, and B is the magnetic field vector.
In this case, the force experienced by the electrons is perpendicular to both the velocity vector and the magnetic field vector. Therefore, the electrons will be deflected in a circular path due to the Lorentz force.
The centripetal force that keeps the electrons in a circular path is provided by the magnetic force, which can be calculated using the equation:
F = mv^2/r
where m is the mass of the electrons, v is the velocity of the electrons, and r is the radius of the circular path.
Since the electrons are moving in the x-direction and the magnetic field is in the y-direction, the force experienced by the electrons will be in the negative z-direction (assuming a right-hand coordinate system).
By equating the centripetal force with the magnetic force, we can solve for the radius of the circular path:
mv^2/r = qvB
Simplifying, we get:
r = mv/(qB)
Now, we can substitute the given values into the equation to find the radius:
m = mass of electron = 9.11 x 10^-31 kg
v = velocity of the beam = 8.3 Mm/s = 8.3 x 10^6 m/s
q = charge of an electron = -1.6 x 10^-19 C
B = magnetic field strength = 240 G = 240 x 10^-4 T (since 1 G = 10^-4 T)
r = (9.11 x 10^-31 kg)(8.3 x 10^6 m/s) / ( -1.6 x 10^-19 C)(240 x 10^-4 T)
Calculating the value of r gives us:
r = 0.3812 meters
Therefore, the radius of the circular path followed by the electrons is 0.3812 meters. Since the electrons are moving in a straight line before they enter the magnetic field region, they will continue to move in a straight line until they reach this radius. So, the distance into the field region that the beam penetrates is equal to the radius of the circular path, which is approximately 0.3812 meters.
To solve this problem, we need to use the formulas for the force experienced by a charged particle moving through a magnetic field and the equation of motion for motion in a straight line.
1. Find the magnetic force experienced by the electrons:
The magnetic force experienced by a charged particle moving through a magnetic field is given by the equation:
F = q * v * B
Where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field.
In this case, the charge of an electron is 1.6 * 10^-19 C, the velocity is 8.3Mm/s (which can be converted to m/s by multiplying by 10^6), and B is 240G (which can be converted to T by dividing by 10^4).
So, the magnetic force experienced by the electrons is:
F = (1.6 * 10^-19 C) * (8.3 * 10^6 m/s) * (240 * 10^-4 T)
2. Find the acceleration of the electrons:
The magnetic force experienced by a charged particle moving through a magnetic field can be written as:
F = q * a
Where a is the acceleration of the particle.
Therefore, we can rearrange the equation to solve for acceleration:
a = F / q
Now, we can substitute the known values to calculate the acceleration.
3. Find the time the electrons spend in the magnetic field region:
The time spent by the electrons in the magnetic field region can be calculated using the equation of motion:
d = v0 * t + (1/2) * a * t^2
Where d is the distance traveled, v0 is the initial velocity, t is the time, and a is the acceleration.
In this case, the initial velocity is 8.3Mm/s (which can be converted to m/s), the distance traveled is the distance into the field region, and the acceleration is calculated in step 2.
Now, rearrange the equation to solve for time:
t = (-v0 ± √(v0^2 - 4 * (1/2) * a * (-d))) / (2 * (1/2) * a)
Since the electrons move in the x-direction and the magnetic field points in the y-direction, the electrons do not experience any force in the x-direction due to the magnetic field.
Therefore, the distance traveled in the x-direction is not affected by the magnetic field, so the distance into the field region is:
d = v0 * t
Now, substitute the known values to calculate the distance.