In a cyclotron (one type of particle accelerator), a deuteron (of atomic mass 2.00 u) reaches a final speed of 12.0% of the speed of light while moving in a circular path of radius 0.565 m. The deuteron is maintained in the circular path by a magnetic force. What magnitude of force is required?
To find the magnitude of force required to maintain a deuteron in a circular path in a cyclotron, we can use the equation for centripetal force:
F = (m * v^2) / r
where F is the force required, m is the mass of the deuteron, v is the final speed of the deuteron, and r is the radius of the circular path.
Given:
Mass of deuteron (m) = 2.00 u (atomic mass units)
Final speed of deuteron (v) = 0.12 * c (12.0% of the speed of light)
Radius of circular path (r) = 0.565 m
First, we need to convert the mass of the deuteron from atomic mass units (u) to kilograms (kg). 1 atomic mass unit (u) is equal to 1.67 × 10^-27 kg.
So, m = 2.00 u * (1.67 × 10^-27 kg/u)
= 3.34 × 10^-27 kg
Now, we need to calculate the final speed of the deuteron in meters per second (m/s). The speed of light (c) is approximately 3.00 × 10^8 m/s.
v = 0.12 * c
= 0.12 * 3.00 × 10^8 m/s
= 3.60 × 10^7 m/s
Finally, we can substitute these values into the formula for centripetal force to find the magnitude of force required:
F = (m * v^2) / r
= (3.34 × 10^-27 kg * (3.60 × 10^7 m/s)^2) / 0.565 m
≈ 3.24 × 10^-13 N
Therefore, the magnitude of force required to maintain the deuteron in the circular path is approximately 3.24 × 10^-13 N.
To find the magnitude of the force required to maintain the deuteron in its circular path, we can use the equation for the centripetal force:
F = m * (v^2 / r)
Where:
F is the force.
m is the mass of the deuteron.
v is the velocity of the deuteron.
r is the radius of the circular path.
First, we need to convert the percentage of the speed of light to a decimal:
12.0% = 0.12
Next, we can determine the velocity of the deuteron:
v = 0.12 * c
Where c is the speed of light. The speed of light is approximately 3.00 x 10^8 m/s.
v = 0.12 * 3.00 x 10^8
v ≈ 3.60 x 10^7 m/s
Now we can calculate the force using the given values:
F = (2.00 u) * ((3.60 x 10^7)^2 / 0.565)
To calculate the force accurately, we need to convert the atomic mass unit (u) to kilograms (kg). One atomic mass unit is approximately equal to 1.66 x 10^-27 kg.
m = (2.00 u) * (1.66 x 10^-27 kg/u)
m ≈ 3.32 x 10^-27 kg
F = (3.32 x 10^-27 kg) * ((3.60 x 10^7 m/s)^2 / 0.565)
F ≈ 4.03 x 10^-14 N
Therefore, the magnitude of the force required to maintain the deuteron in the circular path is approximately 4.03 x 10^-14 Newtons.