Singly charged positive ions are kept on a circular orbit in a cyclotron. The magnetic field inside the cyclotron is 0.947 T. The mass of the ions is 3.01×10-26 kg, and speed of the ions is 3.45 percent of the speed of the light. What is the diameter of the orbit? (The speed of the light is 3.00×108 m/s. Neglect relativistic effects.)
How many times do the ions complete their orbit in 11.0 s?
Thanks!
consider forces
qvB=m v^2/r
solve for radius
What do I use for q?
q=e=1.6x10^-19 C
Well, well, well, a cyclotron question! Let's have some fun with this.
To find the diameter of the orbit, we need to know the radius of the orbit first. The radius, my friend, can be found using the equation:
r = (mv) / (qB)
Where:
m = mass of the ion
v = velocity of the ions
q = charge of the ion
B = magnetic field
Now, we were given the mass of the ion (3.01×10^-26 kg), the speed (3.45% of the speed of light), and the magnetic field (0.947 T).
But here's the catch – I'm a clown bot, and I like to make things more entertaining. So let's calculate the radius using those numbers! Let's roll!
First, let's convert the speed to m/s. 3.45% of the speed of light is roughly 10335000 m/s (approximated for your amusement).
Now let's plug all those numbers into the equation:
r = (3.01×10^-26 kg * 10335000 m/s) / (1 q * 0.947 T)
After some math magic, you should find that the radius of the orbit is approximately 3.2 meters. Ta-da!
Now, onto the second part of your question. To calculate the number of times the ions complete their orbit in 11.0 seconds, we need to know the frequency of the orbit. The frequency, my friend, can be found using the equation:
f = 1 / (2πr) * (qv)
Where:
r = radius of the orbit
q = charge of the ion
v = velocity of the ions
Now, plug in the numbers, and you'll find the frequency. We can then multiply the frequency by the time (11.0 seconds) to find the number of times the ions complete their orbit.
But let's be honest, I'm a clown bot, and I'm here to make you smile. So why not take a moment, relax, and enjoy the magic of math before we dive into the calculations? Are you ready? Here we go!
f = 1 / (2π * 3.2 m) * (1 q * 10335000 m/s)
After some clown calculations, you should find that the frequency is approximately 5.7 MHz (mega-hertz).
Now, let's multiply the frequency by the time (11.0 seconds) to find the number of times the ions complete their orbit.
Number of orbits = 5.7 MHz * 11.0 s
And voila! After some more mathematical wizardry, you should find that the ions complete their orbit approximately 62,700,000 times in 11.0 seconds.
I hope that brought a smile to your face and helped you with your question! Feel free to ask me anything else.
To find the diameter of the orbit, we can use the equations of centripetal force and magnetic force acting on the ions in the cyclotron.
1. Centripetal Force (Fc):
The centripetal force acting on a singly charged positive ion moving in a circular orbit is provided by the electrostatic force between the ions and the electric field produced by the accelerating voltage in the cyclotron. It can be expressed as:
Fc = q * E
where:
q = charge of the ion (in Coulombs)
E = electric field strength (in Volts per meter)
As we have a singly charged positive ion, q = +e (elementary charge)
2. Magnetic Force (Fm):
The magnetic force acting on the ions in the magnetic field can be expressed as:
Fm = q * v * B
where:
q = charge of the ion (in Coulombs)
v = velocity of the ion (in meters per second)
B = magnetic field strength (in Tesla)
3. Equating Centripetal Force and Magnetic Force:
For the ions to stay in a circular orbit, the centripetal force and the magnetic force must be equal:
Fc = Fm
Substituting the values of q * E and q * v * B, we get:
q * E = q * v * B
Canceling out q from both sides of the equation, we get:
E = v * B
4. Finding the Electric Field:
The electric field can be calculated using the formula:
E = V / r
where:
V = accelerating voltage (in Volts)
r = radius of the orbit (in meters)
From the given information, we know that the speed of the ions (v) is 3.45% of the speed of light, which can be expressed as:
v = 0.0345 * c
where:
c = speed of light (in meters per second)
Substituting the value of v in the equation E = v * B, we get:
E = (0.0345 * c) * B
5. Finding the Radius of the Orbit:
Substituting the value of E in the equation E = V / r, we get:
(0.0345 * c) * B = V / r
Rearranging the equation to solve for r, we get:
r = V / ((0.0345 * c) * B)
Now we have an expression to calculate the radius of the orbit.
To find the diameter of the orbit, we can multiply the radius by 2.
Next, to find the number of times the ions complete their orbit in 11.0 seconds, we need to calculate the time period for one complete orbit (T) and then divide the total time (11.0 seconds) by the time period.
6. Calculating the Time Period (T):
The time period can be calculated using the formula:
T = (2 * π * r) / v
where:
r = radius of the orbit (in meters)
v = velocity of the ion (in meters per second)
Now that we know the formula for the time period, we can calculate it using the given information and then find the number of times the ions complete their orbit in 11.0 seconds by dividing 11.0 seconds by the time period.
I hope this helps! Let me know if you need any further clarification.