A charged particle is moving downward with an initial speed of 5 × 106m/s when it encounters a uniform magnetic field that points east to west.
(a) If the initial force on the particle is to the north, is the particle positive or negative?
(b) If B = .25 T and the particle has a charge- to-mass ratio of 4 × 107C/kg, what is the radius of the particle's path?
(c) What is the particle's speed after it has been in the field for 2 × 10-4 seconds?
To answer these questions, we will use the formula for the magnetic force on a charged particle:
F = q * v * B * sin(theta)
where:
F is the magnetic force
q is the charge of the particle
v is the velocity of the particle
B is the magnitude of the magnetic field
theta is the angle between the velocity and the magnetic field
Now let's go through each question step by step:
(a) To determine whether the particle is positive or negative, we need to look at the direction of the initial force. The force is directed north, which means it is perpendicular to the magnetic field. In the right-hand rule, the force on a positive charged particle moving in a magnetic field points perpendicular to both the velocity and the magnetic field. Since the force is directed upwards (north), the particle must be negative.
(b) To find the radius of the particle's path, we can use the centripetal force equation:
F = (m * v^2) / r
where:
F is the magnetic force
m is the mass of the particle
v is the velocity of the particle
r is the radius of the path
In this case, the magnetic force is equal to the centripetal force:
q * v * B * sin(theta) = (m * v^2) / r
Since we are given the charge-to-mass ratio (q/m) and the magnitude of the magnetic field (B), we can rearrange the equation to solve for the radius (r):
r = (m * v) / (q * B * sin(theta))
Given that q/m = 4 × 10^7 C/kg, B = 0.25 T, and the velocity (v) is given as 5 × 10^6 m/s, we can now calculate the radius.
(c) To find the particle's speed after a certain time, we need to use the equation for centripetal acceleration:
a = v^2 / r
Since we know the radius (r) and the initial speed (v), we can calculate the initial acceleration. However, to find the speed after a certain time, we need to consider the change in velocity due to the force acting on the particle.
Using Newton's second law, we know that the force on the particle will cause it to accelerate. Since we have the acceleration (a), we can multiply it by the time (t) to find the change in velocity (delta v):
delta v = a * t
Finally, we can find the final speed by adding the change in velocity to the initial speed:
v_f = v_i + delta v
Given the time (t) as 2 × 10^-4 s, we can now calculate the final speed of the particle.
By following these steps and plugging in the given values, you should be able to calculate the answers to the given questions.