A particle of mass 6.80 10-8 kg and charge 4.4-µC is traveling due east. It enters perpendicularly a magnetic field whose magnitude is 29.0 T. After entering the field, the particle completes one-half of a circle and exits the field traveling due west. How much time does the particle spend traveling in the magnetic field?
Answer in seconds
To find the time the particle spends traveling in the magnetic field, we can use the equation for the magnetic force experienced by a charged particle in a magnetic field:
F = qvB
Where:
F is the magnetic force,
q is the charge of the particle,
v is the velocity of the particle, and
B is the magnitude of the magnetic field.
In this case, the magnetic force is centripetal, causing the particle to move in a circle.
Since the particle completes only half of the circle, we can equate the magnetic force to the centripetal force:
mv²/r = qvB
Where:
m is the mass of the particle, and
r is the radius of the circle.
We can rearrange this equation to solve for the radius:
r = mv / (qB)
Now, we can calculate the radius:
r = (6.80 * 10^-8 kg * v) / (4.4 * 10^-6 C * 29.0 T)
Next, we need to calculate the velocity of the particle. Since it enters perpendicularly and exits the field due west, we can assume the magnetic force acts as the sole force changing the particle's direction and keeping it in a circular path. Therefore, we can equate the magnetic force to the centripetal force:
qvB = mv²/r
We can rearrange this equation to solve for velocity:
v = qBr / m
Now, we can calculate the velocity:
v = (4.4 * 10^-6 C * 29.0 T * r) / (6.80 * 10^-8 kg)
The time the particle spends traveling in the field can be found using the equation for the period of motion in a circle:
T = 2πr / v
Now, we can substitute the values into the equation and calculate the time:
T = (2 * π * r) / v
Finally, we can determine the time the particle spends traveling in the magnetic field by calculating T.
To solve this problem, we can use the formula for the magnetic force acting on a charged particle in a magnetic field:
F = qvB
Where:
F = magnetic force
q = charge of the particle
v = velocity of the particle
B = magnetic field strength
Since the particle completes half a circle, we know that the magnetic force is providing the centripetal force:
F = mv^2/r
Where:
m = mass of the particle
r = radius of the circular path
Setting these two equations equal to each other, we have:
qvB = mv^2/r
Simplifying the equation, we get:
v = qBr/m
We want to solve for time, so we need to find the velocity of the particle in terms of time. The velocity of the particle can be written as:
v = 2πr / T
Where:
r = radius of the circular path
T = time period
Substituting this into the previous equation, we have:
2πr / T = qBr/m
Now, let's rearrange the equation to solve for T:
T = 2πmr / qB
Given:
m = 6.80 x 10^-8 kg
r = radius of the circle (not given)
q = 4.4 x 10^-6 C
B = 29.0 T
We can substitute these values into the equation to calculate the time spent in the magnetic field. However, we still need to determine the radius of the circular path to get the accurate value.
Please provide the radius of the circular path for the particle.